grade 12 mathematical literacy spring book 2015 · notes: scales mind map; number scales and bar...

Gr.12 ML Spring Book 2015 Karelien Kriel, June 2015©

Grade 12

Mathematical

Literacy

Spring Book

2015

A “crash course” notes (only notes of some topics) and exercise book for Gr.12 learners to study for the September and November exams.

This book consists of a complete CAPS study list, exam type of questions and a complete memorandum.

*Throughout the book: Unless stated otherwise you must round to any applicable amount of decimal places.


This book contains the following: 1. Exam Study list with all CAPS topics 2. Math Lit Formula Sheets (which formulas are given and which you must learn) 3. Conversions Sheet 4. Part A: Basic Skills and Conversions

Notes: Number formats, rounding off very large numbers and rounding off of people, etc.

Notes: Direct proportion method and indirect proportion method

Notes: Percentages – 7 types (the complete notes and mind map)

Notes: Ratio Mind Map

Notes: Converting time and correct time notation

Exam questions and memo: Basic skills topics 5. Part B: Tariff Systems

Notes: Water tariffs with sliding scales

Exam questions and memo: water tariffs and electricity tariffs 6. Part C: Break-even Analysis and Small Business

Notes: Patterns and Relationships – 4 different types (formula, table and graph) – the complete notes

Exam questions and memo: Break-even analysis with 3 lines (formulas, tables, graphs and interpretation) – THREE questions with different contexts

Exam questions and memo: Small business (Budget; Income-and-expenditure statement; cost price, selling price and profit)

7. Part D: Data Handling

Notes: Types of data (numerical, categorical, discreet, continuous)

Notes: Mean, median and mode and which one is best

Notes: Equation type of questions involving average (mean)

Notes: How to estimate the amount if there aren’t clear lines on a graph

Exam questions and memo: Stacked bar graph; data tables; equation: average; quartiles (interpreting box-and-whisker plot)

8. Part E: Finance – Tax

Notes: VAT (complete notes: 4 calculations)

Notes: Personal Income Tax Mind Map + FAQ

Notes: Example of personal Income Tax question answered step-by-step

Exam questions and memo: VAT and personal Income Tax (3 questions with different “set-ups”)


9. Part F: Scales, maps, tables and models (optimal packing)

Notes: Mind Map and example of vehicle operating costs

Notes: Scales Mind map; Number scales and bar scales

Notes: Building

Notes: Optimal packing (fitting in the largest amount of objects possible in real life)

Exam questions and memo: Vehicle operating costs; floor plan with scale questions; optimal packing (2 different types)

10. Part G: Finance – Loans and Investments

Notes: Loans, annuity and Stokvel

Notes: Simple interest and compound interest (investment and depreciation)

Exam questions and memo: Interest questions and investments with spreadsheet type of questions

11. Part H: 2- and 3-dimensional shapes

Notes: All formulae for perimeter, area, TSA and volume

Notes: Example of TSA - more difficult question

Exam questions and memo: More difficult TSA (2 different types); more difficult area and perimeter

12. Part I: Finance – Currency and Exchange Rates

Notes and examples for each situation: Foreign currency – buying and selling, including commission fees

Mind map: Foreign currency – buying and selling

Exam questions and memo: Buying and selling foreign currency 13. Part J: Probability

Notes: Probability – Lottery

Notes: Formula

Notes: Probabilities of multiple events, including tree diagram and two-way table

Exam questions and memo: Lottery; general probability (prediction); tree diagram; two-way table


Grade 12 Mathematical Literacy Study List for September and November Exam

Use this list to make sure you have studied everything. Tick off in the appropriate boxes after you have studied for a certain paper. Remember that you must study everything for each exam paper.

Topic: BASIC SKILLS

Sep Nov

P1 P2 P1 P2 * Basic calculations (order op operations; show all your steps;

includes calculations with calculator; number formats)

* Write common fractions as decimal fractions and decimal fractions as common fractions

* Add, subtract, multiply and divide fractions without a calculator.

* Rounding off * Decimal numbers and money * Rounding off to whole numbers (people, building materials, space issues, animals, etc. * Rounding to the nearest 10, 100, 1 000, etc.

* Estimation (estimate answer of problem by rounding off numbers and then doing calculation – only do this when asked)

* Formulas (must be able to substitute numbers into ANY given formula and calculate the answer)

* Proportion * Direct * Indirect

* Percentages * % of an amount * Write fraction as % * Increase/decrease an amount by a % * Calculate % increase/decrease by using formula * Original amount * VAT calculations * S % of a % of an amount

* Ratio * Write quantities as a ratio and simplify * Parts sum * Calculating a part

* Rate * Speed-distance-time sums (remember to convert time!) * Other rates, e.g. pay rate, fuel consumption, exchange rate, etc.


Topic: MEASUREMENT AND CONVERSIONS

Sep Nov P1 P2 P1 P2

* Distance units: mm, cm, m and km * Mass units: g, kg and t

* Temperature Units: C and F (formulae given)

* Area: * cm2 to ml to litres and m2 to litres * Convert between mm2, cm2 and m2 (conversions given)

* Volume units: * ml and litres * mm3, cm3 and m3 to ml and litres * Convert between mm3, cm3 and m3 (conversions given) * Spoons, cups and ml

* Time units: * Sec, min, hours and days * Days, weeks and months

Topic: PATTERNS AND RELATIONSHIPS (TABLES AND GRAPHS);

BREAK-EVEN ANALYSIS AND SMALL BUSINESS Sep Nov

P1 P2 P1 P2

* Read properly and know how to write formulas from the words (number patterns)

* Use formulas to set up your own tables or to complete given tables

* Use values in tables to draw THREE line graphs on the same set of axes. (E.g. compare 3 different photocopiers; compare 3 different cell phone packages, etc.)

* Reading information from given tables and graphs and interpreting it; describing trends and whether trends of different lines correlate (follow the same pattern in general)

* Income: fixed, variable and occasional

* Expenses: fixed, variable and occasional

* Profit, loss and break-even * Small Business:

* Budget * Income-and-expenditure statement * Cost price, profit and selling price * Determine break-even point (graph)

* Informal small business


Topic: DATA HANDLING

Sep Nov P1 P2 P1 P2

* Types of data * Numerical (continuous or discreet) * Categorical

* Developing questions

* Populations and samples * Instruments of data collection

* Collecting and organizing data *Data tables # Stem-and-leaf # Frequency * Measures of central tendency # Mean # Median # Mode # Which one is best? Mean, median or mode? (Outliers) * Range * Quartiles # Interpret box-and-whisker and/or values in table # Write sentences # Inter-quartile range * Percentiles # Interpretation: sentences # Growth charts and BMI (percentile graphs)

* Representing Data: Data graphs * Compound broken line graphs * Compound bar/column graphs * Stacked bar graphs * Histograms * Pie charts * Scatter plots * Misleading graphs # How to estimate when reading values off a graph that are not on the line # Interpretation / reasoning (full sentences)

* Opposing arguments


Topic: FINANCES - TAXATION

Sep Nov P1 P2 P1 P2

* VAT * Calculate amount including VAT * Calculate amount excluding VAT * Calculate VAT

* UIF * Income Tax

* Gross annual income * Taxable deductions and non-taxable deductions * Taxable income * Income tax payable * Tax threshold * Tax rebates * Medical tax credits * Net annual income * IRP5 Tax forms

Topic: MAPS, PLANS AND TRAVELLING

Sep Nov

P1 P2 P1 P2 * Travelling costs of a CAR

* Operating costs (fixed costs and running costs)

* Travelling costs of a BUS * Map of South Africa * Bus timetable and fare table

* Compass directions * Know directions * Lines of longitude and latitude * Compass directions in house construction

* Scales * Number scales * Bar scales * Calculate real distance / length * Calculate distance / length on map / plan * Advantages and disadvantages of number and bar scales

* Maps * Measure distances on map * Different types of maps # street maps; # road maps; # layout maps * Giving directions * Give grid references


Topic: FINANCE – INTEREST AND INFLATION

Sep Nov P1 P2 P1 P2

* Simple interest * Compound interest

* Loan * Tables (normal & also loan tables with loan factors) * Graphs * Reducing the cost of a loan * The effect of changes in interest rate

* Investments * Annuity / retirement fund * Stokvel & other

* Inflation * Percentage calculations (compound interest) * Graphs; * Formula * Indices (table)

Topic: 2- and 3-DIMENSIONAL SHAPES

Sep Nov P1 P2 P1 P2

* Pythagoras * Perimeter

* Square, rectangle, triangle, circle and any shape * Working backwards (equation-type questions)

* Area * Square, rectangle, triangle, circles, combined shapes and shaded areas * Working backwards (equation-type questions)

* Total surface area * Cubes, rectangular prism and cylinders * Vicks box question (ML NSC Exam 2010, P2)

* Volume *Cubes, rectangular prisms and cylinders and combined prisms

Topic: FINANCE – CURRENCY & EXCHANGE RATES

Sep Nov P1 P2 P1 P2

* Buying and selling currency * Rates (conversion tables and/or graphs) * Methods of selling and buying # Electronic transfer; # Traveller’s cheques; # Notes * Converting currency # Bank buying and # Bank selling * Commission fees (exchange fees)

* Gross Domestic Product


Topic: PLANS AND MODELS

Sep Nov P1 P2 P1 P2

* Plans: * Different types (floor plan; elevation plans; design plans) * Views

Topic: PROBABILITY

Sep Nov P1 P2 P1 P2

* Prediction: Theoretical and practical probability(formula) Write as common fraction, as decimal fraction and/or as percentage

* Expressions of probability in the Press

* Probabilities of multiple events * Tree diagrams * Two-way tables

* Probability of winning the Lottery

Topic: 3D MODELS

Sep Nov P1 P2 P1 P2

* Models (can’t build model for exam, so it’s more scales and plans again)

* Optimal packing * E.g. fitting as many TV boxes as possible into a truck


Mathematical Literacy Formula Sheet Formulas / Rules that I must know (It won’t be given)

Topic Formula / Rule

Finance / Proportion VAT is 14% Finance / Proportion If an amount includes VAT, it is 114% of the amount

Finance / Proportion If an amount excludes VAT, it is 100% of the amount

Finance / Percentages The original / 1st amount is always 100% Percentage increase / decrease

% incr/decr

= (difference of two amounts) original amount 100 Finance Profit / Loss = Income – Expenses

Finance Selling price = cost price + profit

Finance / graph At the break-even point: cost = income Finance (Interest) Simple Interest = principle amount interest rate no. of years

Total amount = principle amount + interest Finance (Interest) Compound Interest: Do calculation each year (same as simple

interest method) but start each new year with the previous year’s total amount and calculate interest on that larger total amount Interest = Final amount – principle amount

Finance (Loans) Real cost = monthly repayment no. of payments made Finance (Tax) UIF is 1% of the gross salary

Finance (Tax) (Gr.12) Taxable Income = Gross Income – Taxable Deductions

Finance (Tax) (Gr.12) Net Income = Gross Income – ALL Deductions on salary slip Finance / Line graphs How to find the formula, e.g. copier example

C = fixed fee + no. of copies price per copy

However, if you get e.g. 800 free copies, then the formula is:

Copies 800 or less: C = 5 000 (if they pay a fixed fee of R5 000 per month)

More than 800 copies: C = 5 000 + (n – 800) price per copy

Topic Formula / Rule

Rates / Proportion Fuel consumption, e.g. 13L/km, then make a triangle to find your own formula or use direct proportion, e.g. 13L 1km

? 60km 13 ÷1 60 = …

Rates Remember “per” means ÷, so you can make your own formula. E.g. R/kg: R/kg = money (R) ÷ mass (kg)


Topic Formula / Rule Data Handling Mean = sum of all data items ÷ no. of data items

Data Handling Median = no. in the middle if data is organised in ascending order. (Two numbers in middle: add together, =; ÷ 2 =)

Data Handling Mode = data item that occurs the most Data Handling Range = largest value – smallest value

Data Handling Pie Chart:

No. of degree = 360total

askediswhat

Percentage = 100total

askediswhat

Amount asked = total360

)(askediswhat

Data Handling (Gr.12) Inter-quartile range = upper quartile – lower quartile

Topic Formula / Rule 2D3D (Pyth) a2 = b2 + c2

2D3D (Pyth) a2 = b2 – c2

Scales D(map) = D(real) ÷ Scale (remember to convert)

Scales D(real) = D(map) Scale (remember to convert)

Formulas / Rules that will be GIVEN

Topic Formula / Rule Given in P1

Given in P2

Finance (Gr.12)

Income Tax Table; rebates; tax threshold and medical tax credits table

yes yes

Finance Vehicle Running costs = (A petrol price(R)) + B + C yes yes

Finance Vehicle Operating Costs = fixed costs + running costs yes yes

Finance Monthly LOAN repayment = loan amount ÷ 1 000 loan factor yes yes

Finance Inflation = 100indexold

index)oldindex(new

yes yes

Data H Box-and-whisker plot (You won’t have to draw it) yes yes

Rate BMI = 2m) in (height

mass(kg) yes yes

Rate Speed (km/h) = Distance (km) ÷ Time (h) yes yes Conversion Temperature formulae yes yes

2D Circumference of circle yes yes 2D Perimeter of square, rectangle and triangle yes no

2D Area of square, rectangle, triangle and circle yes maybe


Mathematical Literacy

Conversions Sheet

Type of Units Conversions I must learn

and know Conversions that will be given

Distance (Metric)

1km = 1 000m 1m = 100cm 1cm = 10mm 1km = 100 000cm (maps) 1km = 1 000 000mm (maps)

Distance (Metric to Imperial)

1 cm = 0,3937 inches 1 m = 3,28,8 feet 1 m = 1,0936 yards 1 km = 0,6214 miles

Distance (Imperial) 1 mile = 1 760 yards 1 yard = 3 feet 1 feet = 12 inches

Mass (Metric)

1t = 1 000kg 1kg = 1 000g

1g = 1000mg

Mass (Metric to Imperial)

1 g = 0,0352 ounces 1 kg = 2,2046 pounds 1 t = 0,9842 UK ton

Mass (Imperial)

1 UK ton = 2 240 pounds 1pound = 16 ounces

Temperature Temp (F) = 1,8 Temp(C) + 32

Temp (C) = [Temp(F) – 32] ÷ 1,8

Area 1m2 = 10 000cm2 1cm2 = 100mm2


Type of Units Conversions I must learn and know

Conversions that will be given

Volume 1ℓ = 1 000mℓ 1kℓ = 1 000ℓ Volume 1cm3 = 1mℓ

1ℓ = 1 000cm3 1m3 = 1 000 000cm3 1cm3 = 1 000mm3

Volume (cooking)

1 cup = 250mℓ 1 table spoon = 15mℓ 1 teaspoon = 5mℓ

Volume (Metric to Imperial)

1 ml = 0,0352 fluid ounces 1 L = 1,7598 pints 1 L = 0,22 gallons

Volume (Imperial)

1 gallon = 8 pints 1 pint = 20 fluid ounces

Cooking units (Volume)

Butter: 1 cup = 230g Flour: 1 cup = 140g Cheddar Cheese: 1 cup = 100g Cottage Cheese: 1 cup = 250g Corn Flour: 1 cup = 120g Rice: 1 cup = 200g Salt: 1 cup = 280g Sugar: 1 cup = 200g Castor Sugar: 1 cup = 210g

Time 1 day = 24 hours 1h = 60 minutes 1min = 60 seconds

Time 1 week = 7 days 1 month = 28/30/31 days 1 month = 4 week(Feb) 1 year = 52 weeks

1 month = 28/30/31 days 1 month = 4/4,5/5 weeks


Part A: Basic Skills and Conversions

Number formats: very large numbers

When you want to express a numbers and words e.g. 2,5 million Rand into number format, you have to know the following:

A million has 6 zeros: R1 000 000

A billion has 9 zeros: R1 000 000 000

Example: Write the following in number format: 1. 3,75 million US Dollar 2. 7,8993 billion Rand

Answers: 1. Step1: For “million” multiply with 1 000 000

3,75 1 000 000 = 3750000 Step 2: Write final answer (with thousand separators) and with unit = $3 750 000

2. Step1: For “billion” multiply with 1 000 000 000

7,8993 1 000 000 = 7899300000 Step 2: Write final answer (with thousand separators) and with unit = R7 899 300 000 ===========================================================================

Rounding (only “other” rounding rules)

Rounding with PEOPLE / animals: People and animals are whole and therefore have to be rounded to the nearest whole

number. In certain situations like catering you must rather make too much food than too little,

therefore you will round UP, even if it is 56,1 people, then you will make it 57 people. Where area is limited, e.g. only 50,6 seats can fit into the plane, you have to round

DOWN because there is not enough space to fit a whole 51th seat into the plane. In STATISTICAL situations you will use normal rounding rules, e.g. 45,3 of people at the

wedding chose the carrot cake above the wedding cake. 45,3 45 people

Rounding BUILDING MATERIALS You always need MORE because if you need 15,3 tiles and you only buy 15, then there

will still be a little untiled piece of floor. Therefore tiles, bricks, paint, etc. will always be rounded UP.


Rounding TIME If it takes you e.g. 4,2 hours to clean and you must round the number to the nearest

hour then you must round UP because you won’t get ALL the work done in just 4 hours. Time to finish a task: always round up otherwise you won’t get the whole task done ======================================================================================================

Direct Proportion

A situation is direct proportion when

both quantities increase OR

when both quantities decrease. For example:

Direct proportion method: Step 1: Write down the information. Remember that things with the SAME units need to be

written underneath each other. Step 2: Find the side that has two “things” and put a 1 between the two numbers. Step 3: Divide by the “top number” and multiply with the “bottom number”. Step 4: What you do on the one side, you have to do on the other side. Step 5: Do the calculation as ONE calculation on your calculator and round off the final

answer.

Example: During a rain session 45 litres of water flows down the storm water pipe in 7 minutes. How long (to the nearest minute) will it take for 100 litres of rainwater to flow through the storm water pipe?

7 ÷ 45 100 = 15,55555… 16 minutes ==================================================================================================================================

Indirect Proportion

A situation is an indirect proportion when

one quantity increase and the other quantity decrease.

Indirect proportion method: Step 1: THINK! Is a “if the one thing gets less, the other thing gets more” situation? This

means that whatever I do on the one side, I will have to do the opposite on the other side.

Step 2: Find the side that has two “things” and put a 1 between the two numbers. Step 3: Divide by the “top number” and multiply with the “bottom number”. Step 4: What you do on the one side – you have to do THE OPPOSITE on the other side. Step 5: Do the calculation as ONE calculation on your calculator and round off the final

answer.


Example: It takes a farmer and his 34 workers 6 weeks to pick all the grapes in the orchard. How many workers are needed if he wants to finish picking all the grapes in 14 days?

35 6 ÷ 2 = 105 workers ===================================================================================================================================

Percentages

1. Writing a fraction as a percentage A percentage is “a number out of 100”, i.e. a fraction with a denominator of 100. That means that if you want to calculate a percentage, you have to multiply by 100.

Example: What percentage of people at a certain company earn more than R5 000 per month if 14 out of every 23 people earn less than R5 000 per month?

Answer: Method 1 Method 2 People who earn more than R5 000 % of people - earn more than R5 000

= 23 – 14 = 23

14 100

= 9 people = 60,8695…% % of people who earn more than R5 000 % of people - earn more than R5 000

= 23

9 100 100% – 60,8695…%

≈ 39,13% ≈ 39,13%

2. Calculating a percentage of an amount

30% is written as a fraction over a 100 100

30

“of” means “multiply”.

Example: 75% of women prefer chunky peanut butter to smooth peanut butter. In a certain room there are 30 women. How many of them like smooth peanut butter?

Answer: Method 1 Method 2 % of women who like smooth No. of women who like chunky = 100% – 75% = 75% of 30 = 25% = 22,5 women


No. of women who like smooth No. of women who like smooth = 25% of 30 = 30 – 22,5

= 0,25 30 = 7,5 = 7,5 ≈ 7 women OR 8 women ≈ 7 women OR 8 women

3. Writing a percentage as a common fraction Divide by 100 and press = on calculator. The calculator will give you the simplified

answer, e.g. for 25 % you press 25 ÷ 100 = ; the calculator will give you 4

1. For some

calculators you have to use your fraction button.

4. Percentage Increase and Percentage Decrease There are TWO TYPES: TYPE 1: the answer in an amount, e.g. in Rand TYPE 2: the answer in a percentage

TYPE 1: You have: the original amount and the percentage. You want: to calculate the new amount

INCREASE Sometimes an amount is increased by a certain percentage, for example, a landlord

might decide to put up the rent of his house by 10% the next year.

Example: Mr Grear decides to increase the monthly rent of R9 800 for his house with 12% for the following year. Calculate the new monthly rent. Round your answer up to the nearest R10.

Answer: Method 1 Method 2 12% of R9 800 New %

= 0,12 9 800 = 100% + 12% = R1 176 = 112%

New monthly rent (add) New monthly rent (% of) = 9 800 + 1 176 112% of R9 800

= R10 976 = 1,12 9

R10 980 = R10 976

R10 980


DECREASE (discount) Sometimes an amount is decreased by a certain percentage, for example, you can pay a

discounted price for an item at a sale.

Example: Siphokazi wants to buy a jacket at a “20% off” sale. The original price of R599 is still on the jacket but she can’t find the discounted price. What will she pay for the jacket?

Answer: Method 1 Method 2 20% of R599 New %

= 0,2 599 = 100% – 20% = R119,80 = 80% New price (subtract) New price (% of) = 599 – 119,80 80% of R599

= R479,20 = 0,8 599 = R479,20

TYPE 2: You have: the original amount and the new amount. You want: to calculate the percentage In this case we always use the following formula:

Percentage increase/decrease = amountoriginal

difference 100

“difference” means that you must subtract the two amounts. “original amount” means the first amount, the one you started with. DECREASE Example: A pair of jeans that cost R299 was marked off to R129. Calculate the percentage discount. Round off your answer to the nearest percentage.

Step 1: Write down the formula

Percentage decrease = amountoriginal

difference 100

Step 2: Substitute the numbers in the place of the words

Percentage decrease =

299

129299 100 = 56,85… ≈ 57%

Remember to put brackets around the two numbers that you subtract to avoid making mistakes on your calculator.

Do the whole calculation on your calculator once and only round at the end. Remember to write the unit! (%)


INCREASE You want: to calculate the percentage In this case we always use the following formula:

Percentage increase/decrease = amountoriginal

difference 100

Example: The school is renting a copier machine. His monthly rent increased from R6 575 to R7 100 this year. Calculate the percentage increase in monthly rent. Round off your answer to the nearest percentage.

Step 1: Write down the formula Percentage increase = amountoriginal

difference 100

Step 2: Substitute the numbers in the place of the words

Percentage increase =

5756

57561007 100 = 7,98… 8%

5. Percentages – The Original Amount To calculate the original amount we will use the direct proportion method.

DECREASE Example: Jessica’s father bought her a new car for R90 000 after getting a 15% discount since he paid in cash for the car. What was the original price of the car?

Step 1: What percentage was actually paid? 100% (original price) – 15% (discount) = 85% (percentage of price paid)

Step 2: Write down all the information.

Note that I said R90 000 is 85% of the price. You got a 15% discount therefore

you only paid 85% of the price. I wrote 100% at the bottom because the original price (asked in the question) is

100% of the car’s price.

Step 3: Do the direct proportion method

R90 000

R ???

85%

100%

85

100

R90 000

R ???

85%

100%

1%

85

100


Step 4: Write down the whole calculation and answer

R90 000 85 100 ≈ R105 882,35

INCREASE Example: John borrowed an amount of money from his twin brother, Aaron. John’s monthly repayment went up with 5% to an amount of R250 this year. What was his monthly repayment last year?

Step 1: What percentage was actually paid? 100% (original price) + 5% (increase) = 105% (new % of repayment)

Step 2: Write down all the information.

Note that I said R250 is 105% of the price. His original payment was 100% of the

payment and now his brother has increased it by 5%.

Step 3: Do the direct proportion method

Step 4: Write down the whole calculation and answer

R250 105 100 ≈ R238,10

R250

R ???

105%

100%

105

100

R250

R ???

105%

100%

1%

105

100


6. Percentage and VAT

A shop owner wants to sell a couch.

His cost price plus profit is R2 500.

Now he still needs to add 14% VAT before he can sell it.

This means that: R2 500 is 100 % of the price The selling price will be 114 % of the price

1,14 R2 500 = R2 850 Therefore R350 VAT is paid.

Let’s work backwards: Example: If a couch is sold for R2 850, including VAT, what would the amount of VAT paid be?

WRONG way to do it: 14 % of R2 850 = R399 (You calculates 14 % of a LARGER amount and therefore do not get the correct answer.)

RIGHT way to do it: R2 850 is 114 % of the price because it includes the 14 % VAT

R2 850 114% 1 ? 14%

R2 850 ÷ 114 14 = R350

7. A Percentage of a Percentage Example: A certain pen’s length is 91,875 % of a Casio calculator’s length. The calculator’s length is 54,2373 % of the text book’s length. How long is the pen if the text book is 29,5 cm long? Round off your answer to 1 decimal place.

Answer: Pen’s length = 91,875 % of 54,2373 % of the length of the text book

= 0,91875 0,542373 29,7 = 14,7000…

14,7 cm


Ratio Mind Map


Time

Conversion of time units It is important to note that 30 minutes is 0,5 hours. Therefore 2h30min ≠ 2,3h but 2h30min = 2,5h! How did I get that?

Steps for converting from hours and minutes to hours: 1. Divide the minutes by 60. 2. Add the hours. 3. Round off to the number of decimal places asked.

Example: Convert 3h17min to hours. Round off your answer to two decimal places.

Step 1: Convert minutes to hours: 17 60 = 0,283333… Step 2: Add hours to answer: 3 + 0,283333… = 3,283333…. Step 3: Write final (rounded) answer: ≈ 3,28h .

Steps for converting from hours and minutes to minutes: 1. Multiply the hours by 60. 2. Add the minutes. 3. Round off to the number of decimal places asked.

Example: Convert 3h17min to minutes. Round off your answer to the nearest min.

Step 1: Convert hours to minutes: 3 60 = 180 Step 2: Add minutes to answer: 180 + 17 Step 3: Write final answer: = 187 minutes

Time calculations on your calculator You can add, subtract, multiply and divide time by using the “time button” on your calculator. It looks like this: or

Example: A train leaves at 06:15 from the Cape Town station and arrives at the Stellenbosch station at 07:03. How long did it take the train to reach its destination? Answer: Step 1: Write the calculation 07:03 – 06:15

Step 2: Put the calculation on your calculator

07 03 – 06 15 =

Step 3: Write the final answer: 48 minutes

(The calculator says 048’0’’ – that means 0 hours 48 minutes and 0 seconds)

‘ ‘’ ‘ ‘’ ‘ ‘’ ‘ ‘’

DMS ‘ ‘’


Correct Time Notation

A physical time is written with a colon. Examples:

quarter past nine in the morning is written as 09:15 (the 0 must be there) OR 09:15 am

quarter past nine in the evening is written as 21:15 OR 09:15 pm (the 0 must be there)

A duration is written in numbers and words (not colon) Example:

I travelled for 9 hours and 34 minutes OR 09h 34min

CORRECT WAY TO ANSWER A QUESTION

Question 1: The train leaves at 9:20. It travels for 2 hours and 45 minutes. At what time does the train arrive?

Correct answer: 09:20 + 2h 45min = 12:05 [2 out of 2]

Wrong (1): 09:20 + 2:45 (wrong) = 12:05(CA) [1 out of 2] Mistake: a DURATION can’t be written as a digital time

Duration is time that passes and is written as 2h 45 min. Time is the actual time that you can read from a watch and is written as 9:20

Wrong (2): 09:20 + 2h 45min = 12h05(wrong) [1 out of 2] Mistake: a TIME can’t be written as a duration

It MUST be 09:20 and not 9:20 (9:20 is “sloppy”!)

Question 2: I leave the house at 07:37 and arrive at work at 08:04. How long did it take me to travel from my home to work?

Correct answer: 08:04 – 07:37 = 27 minutes [2 out of 2]


Part A: Questions

Question 1 [19 marks] Read the following extract and answer the questions that follow:

Scope of Ebola Outbreak Greater Than Stats Show? Officials say new treatment centers in Liberia overflowing with patients WebMD News from HealthDay By HealthDay staff HealthDay Reporter

FRIDAY, Aug. 15, 2014 (HealthDay News) -- The magnitude of the Ebola outbreak in West Africa may be far greater than the current statistics indicate, officials from the World Health Organization said Friday. Patients are flooding treatment centers that have just been opened, and the recorded case and death tolls may "vastly underestimate the magnitude of the outbreak," said WHO spokesman Gregory Hartl, the Associated Press reported. For example, an 80-bed treatment center that recently opened in Liberia's capital filled up immediately, Hartl noted, and dozens of people lined up the next day to be treated for Ebola. The latest WHO figures peg the death toll at 1,069, with nearly 2,000 confirmed cases. At this point, Ebola cases have been reported in Guinea, Liberia, Sierra Leone and Nigeria. In a bit of good news, no new cases have been detected in Nigeria following the deaths of three people in the past month, according to the latest update from WHO. … And food is being delivered to the more than 1 million people … Source: http://www.webmd.com/news/20140815/scope-of-ebola-outbreak-may-be-greater-than-statistics-show-who; 21 August 2014

1.1 How many dozen people can fit into a 80-bed treatment centre? Round your answer to 1 decimal number. (3)

1.2 Write the number 1,069 in the correct number format. (2)

1.3 “nearly 2 000” means that the number has been rounded off. What must the number of the least amount of confirmed cases be to be rounded up to 2 000? (2)

1.4 Write 0,978537 million people as a number. (2)

1.5 Calculate the percentage of diagnosed people that are still alive if there are 2 000 diagnosed people of which 1 069 are deceased. (3)

1.6 If 5 people in the 80-bed facility died today, what fraction of patients are left in the hospital? (3)

1.7 It takes 4 doctors half an hour to visit 10 patients. How long (in hours and minutes, to the nearest minute) will it take them to visit 75 patients? (4)

Question 2 [43 marks]

2.1 Janine is Mr Brown’s new assistant. He got very upset with her when she rounded R678 345 000 to R678 million instead of R678,345 million when typing a PowerPoint presentation for him. Why did he get so upset? Show calculations to prove your answer. Write a full sentence. (4)

http://www.webmd.com/news/20140815/scope-of-ebola-outbreak-may-be-greater-than-statistics-show-who


2.2 Mr Petersen donates a certain amount of disposable gloves to Doctors Without Borders. After a few hours 75% of the gloves that he donated have been used and only 60 gloves remain. How many gloves did he donate? (3)

2.3 A survey was done to determine the home language of people living in a certain province. The results show that there are 9,8% Afrikaans, 42,1% Xhosa, 5,9% Sesotho and 6,4% Zulu speaking people. The rest of the people are all English speaking and there are 9 648 287 people living in this province.

(a) How many people speak English? (4)

(b) How many people are NOT Xhosa speaking? (3)

(c) How many people either speak Afrikaans or Xhosa? (3)

(d) How many people neither speak Sesotho or Zulu? (4)

(e) What is the probability that a person living in this province is Afrikaans speaking? Write your answer as a simplified normal fraction. (2)

2.4 Peter drove 100 miles. It took him 2 hours and 34 minutes to cover this distance. At what speed, in kilometres per hour, was he driving? Use the formula: speed (km/h) = distance (km) ÷ time (h) Given: 1 km = 0,6214 miles (5)

2.5 Megan used 62,5 litres of Diesel to drive a certain distance at an average speed of 108km/h. Her car has a diesel consumption of 15,2 kilometres per litre. How long (in hours and minutes) does it take her to reach her destination? Use the formula: speed (km/h) = distance (km) ÷ time (h) (6)

2.6 Jessica is taking the train from Strand to Blackheath. The train leaves at 06:50 and makes the following stops:

Strand: 06:50 Van der Stel: 06:55 Somerset West: 06:59 Firgrove: 07:04

Faure: 07:10 Eerste River: 07:15 Meltonrose: 07:29 Blackheath: 07:23

(a) At what time does she arrive at the Blackheath station? (2)

(b) Jessica says that the time duration between all the consecutive stops are the same, i.e. it takes the same time to drive from Strand to Van der Stel as it takes to drive from Van der Stel to Somerset West. Verify whether her statement is correct by showing calculations. (3)

2.7 A certain doctor gets up at 05:15 in the morning. It takes him 1 hour and 42 minutes to get ready for and drive to work. What time does he arrive at work? (2)


Part A: Memo Question 1 [19 marks] 1.1 80 ÷ 12(M÷)(A,12) = 6,666… = 6,7 dozen people(CA and R) (3) 1.2 1 069(A) (2) 1.3 1 500(A) (2) 1.4 978 537 people(A) (2) 1.5 % alive

=

100diagnosedall

deaddiagnosedall

=

1000002

06910002

(A,top)(A,bottom and 100) =46,55%(CA and U) (3) 1.6 Fraction of patients left

= total

deadtotal

=

80

580 (A,top)(A,bottom)

=16

15(CA, simplified) (3)

1.7 30 min 10 patients ? 75 patients Direct proportion: the more patients they have to see, the longer it will take.

30 ÷ 10(M) 75(M) = 225 minutes(CA) = 3 hours and 45 minutes(C) (4)

Question 2 [43 marks] 2.1 R678,345 million = R678 345 000 R678 million = R678 000 000 R678 345 000 – R678 000 000(M) = R345 000(CA) When she rounded off she “threw away” R345 000. This is a large amount of money. (J) (4) 2.2 If 75% of the gloves were used, only

25% remain Direct Proportion 25% 60 gloves 100% ?

60 25(M) 100(M) = 240 gloves(CA) (3) 2.3 [*Only penalise ONCE in the whole of

2.3 for not rounding to a whole number.]

(a) 100 – (9,8 + 42,1 + 5,9 + 6,4)(M+) = 100 – 64,2 = 35,8%(CA)

0,358 9 648 287(M) = 3 454 086,746

3 454 087(CA and R*) OR 3 454 086 people (4) (b) 100 – 42,1 = 57,8%(M)

0,578 9 648 287(M) = 5 576 709,886

5 576 710(CA and R*) OR 5 576 709 people (3)


2.3 (c) 9,8 + 42,1 = 51,9%(M)

0,519 9 648 287(M) = 5 007 460,953

5 007 461(CA and R*) OR 5 007 460 (3) (d) 5,9 + 6,4 = 12,3%(M) 100 – 12,3 = 87,7%(M)

0,877 9 648 287(M) = 8 461 547,699

8 461 548(CA and R*) OR 8 461 547 (4)

(e) P(Afr) = 100

9,8(A)

= 500

49(CA) (2)

2.4 Convert miles to km: (direct proportion) 1km 0,6214 miles ? 100 miles

1 0,6214 100(M)

OR 100 0,6214 = 160,9269392km(CA)

Convert h and min to h:

2 + (34 60)(C) = 2,5666666666…h

Speed = distance time

= 160,9269392km 2,5666…h(SF) = 62,69880747km/h

62,7km/h OR 63 OR 60CA) (5)

2.5 Make triangles: Triangle 1: Distance

= 15,2km/L 62,5L(M) = 950km(CA) Triangle 2:

Time = distance speed(F)

= 950km 108km/h(SF) = 8,796296296h(CA)

8 hours and 48 min(CA) (6)

(0,796296296 60 = 47,77...min) 2.6 (a) 07:23 – 06:50(M and notation) = 33 minutes(CA and notation) (2) (b) Strand to Van der Stel: 06:55 – 06:50 = 5 minutes(M)

Van der Stel to Somerset West = 06:59 – 06:55 = 4 minutes(M)

Statement incorrect(CA) (3) [Learner may use any two calculations with different answers.] 2.7 05:15 + 1h 42min(M and notation) = 06:57(CA and notation) (2)

km

km/L L

D

S T

?


Part B: Tariffs How do water tariffs work?

We don’t pay ONE fixed amount per kilolitre of water that we use. We pay according to a sliding scale.

This table works according to the principal: the more you use, the more you pay.

Each municipality uses a special table with water usage rates.

Example of water tariffs of a certain municipality:

Range Price perkL

1 Up to 6kL R0 2 6kL – 30kL R3,25

3 30kL – 60kL R9,50 4 More than 60kL R16,75

+ a fixed fee of R12,50 for infrastructure if you use more than 6kL *kL = kilo litre = 1 000 litres

Explanation of the table First Row: If you use 6kL or less water for the month you don’t pay for the water that you used. You don’t have to pay the fixed fee. Example: If you used e.g. 5kL of water or the month you have to pay R0.

Second Row: If you used e.g. 28,7kL of water for the month you STILL pay R0 per kL FOR THE 1st 6kL and then R3,25 PER kL for the REMAINING water used. Example: Remaining amount of water = 28,7kL – 6kL = 22,7kL

You pay: (6 R0/kL) + (22,7kL R3,25/kL) + R12,50 R86,28

Third Row: If you used e.g. 47kL of water for the month: Example: Usage 47kL for the month. Remaining = 47kL – 30kL = 17kL

You pay: (6 R0/kL) + (24kL R3,25kL) + (17kL R9,50) + R12,50 = R252,00

Fourth Row: If you used e.g. 71,8kL of water for the month you pay: Example: Usage 71,8kL for the month. Remaining = 71,8 – 60 = 11,8kL You pay:

(6 R0/kL) + (24kL R3,25kL) + (30kL R9,50) + (11,8kL R16,75/kL) + R12,50 = R573,15 How do the municipality know how much water you used?

Outside your home/yard there is a tap with a meter on.

The meter will keep on “running” as you use water.

The municipality reads the number from the meter on the same day of each month.

For example, on 1 May 2015 the reading was 0023476 and on 1 June 2015 the reading was 0023529.

The water usage for the month of May 2015 is therefore: 23 529 – 23 476 = 53kL


Part B: Questions Question 1 [9]

Mrs Saver’s municipal bill for July 2015 is shown below. She has pre-paid electricity therefore the bill only shows water usage.

Mrs Q Saver 14 Kiekeriet Street Hermanus

Acc no: 90000025876555 Invoice no: 8632233 Account for: July 2015

Price/kL Amount used (kL) Monthly Charge (R)

Tax 162.08 Water Infrastructure 16.85

Water 0 – 6kL 0.00 6.00 0.00

Water 6 – 30kL 3.00 24.00 72.00 Water 30 – 60kL 10.50 30.00 315.00

Water 60kL+ ? 12.41 280.47 Refuse 122.81

Sewerage Basic 85.96 Sewerage Infrastructure 9.60

Sewerage Consumption 67.55

Sub total ? 14% VAT ?

TOTAL PAYABLE BY 20/07/2015 ?

1.1 Name two ways of how Mrs Saver can be sure that this is her municipal bill. (2)

1.2 How much does the municipality charge for water usage per kilo litre (in Rand/kilo litre) if more than 60 kilo litres have been used? (3)

1.3 Calculate the total amount payable for July 2015. (5)


Question 2 [14] In the Cape Town area there are two types of electricity. The one is called Lifeline and the other one Domestic. You are eligible for Lifeline if you are an existing pre-paid customer who receives up to 450kWh of electricity per month, including any free electricity. It is also applicable to prospective customers who receives an average of up to 450kWh of electricity per month, has a pre-paid electricity meter and have a municipal property valuation of less than R300 000. The rest of the Cape Town residents have to pay Domestic tariffs. ***From 1 June 2013 consumers receiving less than 250kWh per month on average will receive 60kWh free electricity supply per month. Consumers receiving more than 250kWh but less than 450kWh per month on average will receive 25kWh free electricity supply per month.

Here are the tariffs for these two options:

Lifeline

Block 1 Block 2 Free basic energy Cost: no charge to the customer ***See notes on previous page

No free electricity

Usage balance up to 350kWh per calendar month Cost = 91,06c/kWh, VAT excluded or 103,81c/kWh, VAT included

Usage above 350kWh per calendar month Cost = 252,12c/kWh, VAT excluded or 287,42c/kWh, VAT included

Domestic

Block 1 Block 2 0 – 600kWh per calendar month Cost = 154,30c/kWh, VAT excluded or 175,90c/kWh, VAT included

Usage above 600kWh per calendar month Cost = 187,63c/kWh, VAT excluded

2.1 Mr Molefe lives in a township in the Cape Town area. The municipal value of his home is R280 000. He used 367kWh of electricity in March 2015. How much must he pay (in Rand) for electricity? This amount must include VAT. (4)

2.2 Miss Ngwenya lives in Michells Plain. By using the amount of electricity that she used each month during 2014 she determined that she uses about 703kWh per month. For which electricity option will she be charged? (2)

2.3 In July 2015 she used 765kWh of electricity. Determine her total cost, incl. VAT. (6)

2.4 Name one possible reason why she would use an amount of electricity that is way above her average monthly usage. (2)


Question 3 [15] The water tariffs for a certain town are given below: Range kL used Price perkL

1 Up to 6kL 6 R0

2 6kL – 26kL 20 R4,18 3 26kL – 46kL 20 R9,60

4 46kL – 66kL 20 R20,05 5 More than 66kL R53,00

+ a fixed fee of R17,77 for infrastructure if you use more than 6kL *kL = kilo litre = 1 000 litres *All prices exclude 14% VAT 3.1 Lerato used 48,9kL of water during July. How much did she pay in total for her

water usage during July? (7)

3.2 During the December holiday the whole (extended) Geldenhuys family stayed at their beach house. They used 67,2kL of water during December. In January Jack Geldenhuys gets the shock of his life when he receives his municipal bill. How much did he pay in total for his water usage during December? (8)


Part B: Memo Question 1 [9]

1.1

She must check that it is the correct address.

Her name must be on top of the bill.

She can check if the account number is correct. (2)

1.2 R280,47 R12,41(RT,A)(M)

R22,60(CA) (3) 1.3 162.08 + 16.85 + 72 + 315 + 280.47 + 122.47 + 122.81 + 85.96 + 9.60 + 67.55 = R1 132,32(A)

0,14 1 132,32(M) = 158,5248 1 132,32 + 158,5248 (M)

R1 290,84(CA) (4)

Question 2 [14]

2.1 Lifeline First 25kWh = R0 Balance = 367 – 25 = 342kWh(A)

He has to pay for 342kWh of electricity.

It is less than 350kWh, so it all falls under Block 1.

342kWh 103,81c/kWh(M) = 35 503,02c(CA)

R355,03(C) (4) 2.2 Domestic(RT,A) (2)

2.3 “Balance” = 765 – 600 = 165kWh Block 2: (incl. VAT)

1,14 187,63c(M) = 213,8982c(CA) Cost

= (600 175,90)(M)

+ (165 213,8982)(M, A for 165) = 140 833,203c(CA)

R1 408,33(C) (6) 2.4 It is winter (July). She used her heater a

lot or tumble dried her clothes. (2)

Question 3 [15]

3.1 (6 0)

+ (20 4,18)(A)

+ (20 9,60) (A)

+ (2,9 20,05) (A) + 17,77(A) = R351,515(CA)

1,14 351,515(M)

R400,73(CA) (7)

3.2 (6 0)

+ (20 4,18) (A)

+ (20 9,60) (A)

+ (20 20,05) (A)

+ (1,2 53) (A) + 17,77(A) = R25 796,97(CA)

1,14 25 796,97(M)

R29 408,55(CA) (8)


Part C: Graphs, Break-even Analysis and Small Business

Tables and graphs

Patterns and Relationships: Tables and Formulas

Patterns & relationships can be represented by using FORMULAS, TABLES & GRAPHS.

If you want to draw a graph you need values from a table. In order to make sense of tables we first need to understand how number patterns work.

There are four different patterns and graphs we will discuss:

1. Constant / Fixed Relationship: This means that the one quantity will stay the same no matter what the other quantity is. It will give us a straight horizontal line.

2. Constant Difference: This is a direct proportion where the straight line will start at the origin. The line will have a slope.

3. Constant Difference with Fixed Amount: This is a direct proportion where the straight line will start at a number above the origin. The line will have a slope.

4. Indirect proportion: This means that the one quantity will increase as the other quantity decrease. It will give us a curve.


Number Patterns Type 1: Constant / Fixed Relationship

In this relationship there is NO difference. That means the “answer” will stay the same amount all the time.

Example: The school hires a 60-seater bus for the day and it costs them R10 000. The company says they can travel a maximum of 2 000 km. It doesn’t matter how far they travel (as long as it is less than 2 000 km) or how many learners there are on the bus the price stays R10 000 for the day.

We can show this information in a table:

Number of children (n) 0 10 20 30 60

Price for the day in Rand (P) 10 000 10 000 10 000 A 10 000

Finding the formula to work out the travel cost for the day:

P = 10 000 (distance < 2 000 km; children less or equal to 60)

Find the value of A: A = R10 000

The graph of this situation will look like this:

The line stops at 60 because it is impossible to fit more than 60 children into the bus.

Learners can only be whole numbers. That means the values are discreet and therefore the line is a dotted line.

The first dot is an open dot because there are NO learners. Someone won’t hire a bus if there are NO learners to go onto the bus.

15 000

12 500

10 000

7 500

5 000

2 500

0 10 20 30 40 50 60 70

Number of children

Cost per day for hiring a 60-seater bus

Co

st (

Ran

d)


Number Patterns Type 2: Constant Difference (Direct Proportion)

“difference” means to subtract “constant difference” means that I get the same answer each time I subtract consecutive

answers. For example, it costs R12 for 1 liter of petrol. How much will it cost to fill a car with a 45

liter tank? We can show this information in a table:

Amount of petrol (p) 0 1 2 3 45 Cost in Rand (C) 0 12 24 36 B

Note that if I buy NO petrol I will pay NO money. (Makes sense, right?) Finding the formula to work out the total cost of buying a certain amount of petrol:

C = 12 p Note that: 12 – 0 = 12; 24 – 12 = 12; 26 – 24 = 12; so I can just add 12 every time until I

find, e.g. the answer of B. However, this will take a long time, so it’s easier to use the formula.

Find the value of B: B = 12 45 = R540 (remember the unit!) The graph of this situation will look like this:

Note that the line stops at 45 ℓ. The petrol tank can only take a maximum of 45 ℓ of

petrol.

The first dot is an open dot because at that point you haven’t purchased any petrol.

The line is a solid line (i.e. not dotted) because we are dealing here with continuous values. For example, it is possible to purchase 34,7 litres of petrol.

Cost of filling a car (petrol)

0 5 10 15 20 25 30 35 40 45

Amount of petrol (litres)

Time (weeks)

600

500

400

300

200

100

Co

st (

Ran

d)


Number Patterns Type 3: Constant Difference with Fixed Amount (Dir. Prop.)

For this type of pattern you will always be given all the information in WORDS and then you have to write the formula and complete the table by using that information.

For example, the school hires a photocopier for R1 500 per month. It costs them R0,50 to make one copy. Find the formula for determining the cost (C) after making n amount of copies.

In words: Cost (in Rand) = monthly rent + cost per copy

In symbols (the formula): C = 1 500 + 0,50 n Use the formula to find the missing values in the table:

Number of copies made (n) 0 100 200 500 1 000 Cost in Rand (C) 1 500 1 550 1 600 D 2 000

Note that the school has to pay R1 500 for the machine for the month even if they made no copies. (The machine is at the school and they still have to pay rent for it, even if it is e.g. December holidays.)

Find the value of D: D = 1 500 + 0,50 500 = R1 750 The graph of this situation will look like this:

Note that:

The line starts at R1 500 when you made 0 copies.

The first dot is an open dot because you haven’t made any copies yet.

The line is dotted because the number of copies is discreet values. You can’t make e.g. half a copy.

The line has an arrow at the end showing that you can make more than a 1 000 copies.

Cost of renting copier for the month

0 100 200 300 400 500 600 700 800 900 1 000

Amount of copies made

Time (weeks)

3 000

2 500

2 000

1 500

1 000

500

Co

st (

Ran

d)


Number Patterns Type 4: Indirect Proportion (Inverse Proportion)

Remember that for indirect proportion the one quantity will increase while the other one decreases.

For example, a taxi company says it will cost R2 200 to take teachers to and from work daily. How much will it cost per teacher to travel to and from work by this taxi per month? The taxi can take a maximum of 12 people.

We can show this information in a table:

Number of teachers in taxi (n) 1 2 3 4 12

Cost in Rand per person (C) 2 200 1 100 R734 E R184

Finding the formula to work out the cost per person:

C = 2 200 ÷ n

Find the value of E: E = 2 200 ÷ 4 = R550 per person

The graph of this situation will look like this:

The number of people is discreet.

Normally if values are discreet we will draw a dotted line.

However, in this case a line is not needed as we make a dot for each person on the graph (and there can’t be anything in between – we don’t get a fraction of a person).

Cost per person to travel to and from work per month

Time (weeks)

2 500

2 000

1 500

1 000

500

Co

st (

Ran

d)

0 2 4 6 8 10 12 Number of people


Number Patterns: Working Backwards to Find Missing Values

When we have the formula we normally follow these steps to find the answer:

Note that we follow the “order of operations”, i.e. or ÷ first and then + or –.

If we have the answer and we want to know what number we started with we have to work backwards.

Note that we follow the OPPOSITE of the “order of operations”, i.e. + or – first and then

or ÷.

Number Patterns Type 1: Constant / Fixed Relationship

Example: the school hires a 60-seater bus for the day and it costs them R10 000. The company says they can travel a maximum of 2 000 km. It doesn’t matter how far they travel (as long as it is less than 2 000 km) or how many learners there are on the bus the price stays R10 000 for the day.

Number of children (n) 0 10 20 A 60

Price for the day in Rand (P) 10 000 10 000 10 000 10 000 10 000

Find the value of A: A = any number from 0 to 60

Number Patterns Type 2: Constant Difference (Direct Proportion)

Example: It costs R12 for 1 litre of petrol.

Amount of petrol (p) 0 1 2 3 B

Cost in Rand (C) 0 12 24 36 432

Formula: C = 12 p

Find the value of B: The opposite of is ÷

B = 432 ÷ 12 = 36 ℓ

n or ÷ + or – answer

(opposite) (opposite)

n or ÷ + or – answer


Number Patterns Type 3: Constant Difference with Fixed Amount (Dir. Prop.)

Example: the school hires a photocopier for R1 500 per month. It costs them R0,50 to make one copy.

Number of copies made (n) 0 100 200 D 1 000

Cost in Rand (C) 1 500 1 550 1 600 1 680 2 000

Formula: C = 1 500 + 0,50 n

Find the value of D:

To find D we must do the opposite:

1 680 – 1 500 = 180 OR (1 680 – 1 500) 0,50

180 ÷ 0,50 = 360 = 360

D = 360 copies

Number Patterns Type 4: Indirect Proportion (Inverse Proportion)

Example: a taxi company says it will cost R2 200 to take teachers to and from work daily. How much will it cost per teacher to travel to and from work by this taxi per month? The taxi can take a maximum of 12 people.

Number of teachers in taxi (n) 1 2 3 E 12

Cost in Rand per person (C) 2 200 1 100 R734 R367 R184

Formula: C = 2 200 ÷ n

Find the value of E: For type 4 we don’t use the opposite, we still use ÷

E = 2 200 ÷ 367 = 5,99… 6 people

n 0,50 + 1 500 answer

n ÷ 0,50 – 1 500 answer


Part C: Questions

Question 1: Break-even Analysis – Printers [19 marks] Faatimah has her own design company. She must rent a colour laser printer. Her assistant gets 3 quotations. Quotation 1: There is no fixed monthly fee and she has to pay of R8 per page. Quotation 2: Her personal assistant lost the quotation. Quotation 3: There is a fixed monthly fee of R6 000 and a fee of R1,67 per page. The first 600 pages are free. Before her assistant lost the quotation she used the information for all three quotations to draw a graph. (See graph on next page.) 1.1 Write a formula to determine the total cost for printing a certain number of pages

for Quotation 1. (2)

1.2 Write a formula to determine the total cost for printing a certain number of pages for Quotation 2. (5)

1.3 Write TWO formulas to determine the total cost for printing a certain number of pages for Quotation 3. (3)

1.4 The break-even points divide the graph up into four regions. Describe those regions in terms of the number of pages printed. (4)

1.5 Faatimah and her colleagues do a maximum of 4 design projects per month. They print more or less 40 coloured pages per design project. Which quotation will suit her best? Motivate your answer. (5)


1

2

3


Question 2: Break-even Analysis – Venues [37 marks] On Farheinhof, a wine farm, there is a restaurant with a part that specifically caters for kids’ parties. There is an indoor and outdoor playground. There are three options available to parents: (All the options include food and drinks for the kids but not for the parents.)

*Option 1: Fixed Fee A fixed fee of R2 400 is charged. A maximum of 30 children are allowed.

*Option 2: Partial Fixed Fee A fixed fee of R800 is charged for the first 10 children. Thereafter an additional R60 is charged per child. A maximum of 40 children are allowed.

*Option 3: Charge Per Person No fixed fee is charged. The cost is R100 per child. A maximum of 40 children are allowed.

The fees for all three options are shown in the table below:

No. of children 0 5 10 D 25 30 40 Cost Option 1 2 400 A 2 400 2 400 2 400 2 400 N/A

Cost Option 2 800 800 800 1 340 B 2 600 3 200

Cost Option 3 0 500 1 000 1 900 6 250 C 4 000 *Let C = Cost in Rand and n = number of children 2.1 Write an equation to determine the total cost for Option 1. (2)

2.2 Write an equation to determine the total cost for Option 2. (3)

2.3 Write an equation to determine the total cost for Option 3. (2)

2.4 Use the equations above or any other method to determine the values of A, B, C and D in the table. Show ALL your calculations. (7)

2.5 Use the graph paper on the next page to draw 3 line graphs on the same set of axes. Plot ONLY points that correspond with the underlined values in the table. (13)

2.6 Write the number of children as well as the total cost for each break-even point. (6)

2.7 Which option will be the most cost effective for the client if there are 20 or less kids at the party? (2)

2.8 Which option will be the most cost effective for the client if there are between 30 and 40 kids at the party? (2)


Cost for Kids’ Party

Number of children

5 10 15 20 25 30 35 40

Tota

l Co

st f

or

par

ty (

Ran

d)

4 000

3 750

3 500

3 250

3 000

2 750

2 500

2 250

2 000

1 750

1 500

1 250

1 000

750

500

250


Question 3: Break-even Analysis – Laundry [31 marks] It is time for Darren to upgrade his cell phone contract. His service provider gives him the following 3 options: (All the packages have “per second” billing.) Package 1: They charge a flat rate of R1 000 per month. Formula given: C = 1 000, where C = total monthly cost in Rand Package 2: They charge a fixed monthly fee of R200 plus an additional amount of R0,03 per second. The first 100 minutes (6 000 seconds) are free. Package 3: They charge a fixed monthly fee of R100 plus an additional amount of R0,04 per second. The first 50 minutes (3 000 seconds) are free.

3.1 Write formulas to determine the total monthly cost for Package 2. Use n for the amount of seconds called. (4)

3.2 Write formulas to determine the total monthly cost for Package 2. (3)

3.3 The tables below shows the total monthly cost after a certain amount of seconds has been called: Package 1: Time in seconds (n) 0 600 1 200 30 000

Total cost in Rand (C) 1 000 1 000 A 1 000

Package 2: Time in seconds (n) 0 6 000 6 960 30 000

Total cost in Rand (C) 200 200 B 920

Package 3:

Time in seconds (n) 0 3 000 C 30 000 Total cost in Rand (C) 50 50 56 1 180

Use the formulas above (3.1 and 3.2) or any other means to determine the correct values of A, B and C in the tables above. (7)

3.4 The shop assistant shows Darren a graph containing the information of the 3 different packages. On the graph the lines are solid, meaning that the data is continuous. Why do you think they see it as continuous data? (2)

3.5 Use the graph paper on the next page to draw line graphs for Package 2 and Package 3 on the same set of axes. The line for Package 1 is already drawn. Do NOT plot the three points that contains A, B and C in the table. (11)

3.6 Darren decides to choose Package 3. For how many minutes per month does he talk on his cell phone, i.e. for which region is Package 3 the cheapest? (4)


Comparing three cell phone packages

Time (seconds) 6 000 12 000 18 000 24 000 30 000

Tota

l mo

nth

ly c

ost

(R

and

) 1500

1400

1300

1200

1100

1000

900

800

700

600

500

400

300

200

100


Question 4: Small Business – Logistics Company [26 marks] Leroy’s Logistics is a logistical company (they transport goods). Below is their budget for June 2015.

LEROY’S LOGISTICS: BUDGET FOR JUNE 2015 Income (Rand)

Expenditure (Rand)

National deliveries 875 000 Salaries 200 000 International deliveries 500 000 Overtime 100 000

Fuel 220 000

Repairs and services A Office supplies 500

Cleaning of trucks 4 500

Rent: printers 3 400

Pre-paid cell vouchers 9 500

TOTAL: 1 375 000 TOTAL: 687 900

*International deliveries are made to Namibia, Botswana and Mozambique

4.1 Show with calculations whether the company makes a profit or a loss for June 2015. (3)

4.2 Why are all the numbers rounded off to the nearest 100, 1 000 or 10 000? (2)

4.3 Calculate the value of A. (3)

Below is an example of an invoice that they gave to a client:

Leroy’s Logistics 16 Long Street Tel: 021 952 4578 Cape Town Fax: 021 952 4545 8 000 E-mail: [email protected]

VAT number: 97521632133

Sender: Mr M Smith 34 Church Street Cape Town 8 000; Western Cape

Recipient: Mr J Baadjies 7 van Riebeeck Street Springbok 8240; Northern Cape

Date: 15 June 2015

Package details: Contents: fragile Height: 30cm Length: 50cm Width: 26cm Weight: 8,72kg

Cost: 8,72kg @ R9,99/kg

+ 14% VAT

TOTAL

A

B

C

mailto:[email protected]


4.4 Calculate how much the client must pay (i.e. the value of C). (5)

4.5 Leroy adds a profit of 35% to the cost price of all his quotations. If the “selling price” for fragile cargo is R9,99/kg, what is the cost price of that cargo? (3)

Below is the income-and-expenditure statement for June 2015.

LEROY’S LOGISTICS: INCOME-AND-EXPENDITURE STATEMENT FOR JUNE 2015 Income (Rand)

Expenditure (Rand)

National deliveries 874 872,56 Salaries 231 845,00 International deliveries 602 834,87 Overtime 114 638,95

Fuel 226 310,03

Repairs and services 182 222,57 Office supplies 478,75

Cleaning of trucks 3 985,00 Rent: printers 3 400,00

Pre-paid cell vouchers 8 756,25

TOTAL: A TOTAL: B

4.6 What is the difference between a budget and an income-and-expenditure

statement? Write a sentence. (2)

4.7 What percentage is the total expenditure of the total income? (5)

4.8 Which item’s amount stayed exactly the same in the budget and the income-and-expenditure statement? Why do you think this is the case? (3)


Part C: Memo Question 1 [19 marks] *If no =, it’s not a formula 0 marks *Wrong formula 0 marks

1.1 C = 8 n(A) Where C = total cost and n = number of

printed pages(A) (2)

1.2 Use any point: 1 000 pages cost R8 000

C = 3 000 + ? n

8 000 = 3 000 + ? 1 000 (M, using a point)

(8 000 – 3 000) 1 000(M)(M) = R5/page (CA)

C = 3 000 + 5 n(CA) (5)

1.3 For n = or < 600: C = 6 000 (A) For n > 600: C = 6 000 + 1,67(n – 600) (A) (A) *Penalise once for no limitations given (3)

1.4 0 – 600 pages(A) 601 – 780 pages(A) 780 – 1 000 pages(A) More than 1 000 pages(A) (4) *780 not exact amount: accept 760 - 790

1.5 4 40(M) = 160 pages(CA) Quotation 1(CA) For less than 600 pages printed per month Quotation 1’s line lies the lowest meaning that it is the cheapest option.(J) (5)

Question 2 [37 marks] 2.1 C = 2 400 (max 30 children) (2)

2.2 For n =<10: C = 800(A) For n > 10 but n=<40: C = 800 + 60(n – 10)(A) (3)

2.3 C = 100 n(A) (2) 2.4 *remember that it’s marked according

to the mistakes made in 2.1 – 2.3 if learners wrote wrong formulas.

A = R2 400(A)

B = 800 + 60(25 – 10)(M) = R 1700(CA)

C = 100 30(M) = R3 000(CA)

D Use Option 2:

(1 340 – 800) 60 + 10(M) = 19(CA) OR

Use Option 3: 1 900 100(M) = 19(CA) (7)

2.5 [graph: see next page] (13) 2.6 *CA according to learner’s graph (8 children; R800)(RG) (24 children; R2400) (RG) (30 children; R2400) (RG) (6) *must write both no. of children and cost to

get mark for that B-E point. 2.7 Option 2(RG, CA acc to graph) (2) 2.8 Option 2(RG, CA acc to graph) (2)


Cost for Kids’ Party

Number of children

5 10 15 20 25 30 35 40

Tota

l Co

st f

or

par

ty (

Ran

d)

4 000

3 750

3 500

3 250

3 000

2 750

2 500

2 250

2 000

1 750

1 500

1 250

1 000

750

500

250

Option 1

Option 2

Option 3

names

all lines dotted (kids discreet)

last dot

last dot

last dot

all lines STOP (no arrow points)

1st dot

1st dot

1st dot

Tip: Each block is R250;

therefore each small block is

250 5 = R50

flat line

flat line

straight line (Op2)

straight line (Op3)


Question 3 [31 marks] *Penalise once in 3.1 and 3.2 for no

limitations given.

3.1 Let n = amount of seconds(A) For n<=6 000 seconds: C = 100(A) For n > 6 000 seconds: C = 100 + 0,03(n – 6 000) (A) (A)

(4)

3.2 For n<= 3 000 seconds: C = 200(A) For n > 3 000 seconds: C = 200 + 0,04(n – 3 000) (A) (3)

3.3 *mark according to learner’s formulas in 3.1 and 3.2

A = R1 000(A)

B = 100 + 0,03(6 960 – 6 000)(SF) = R128,80(CA) [no 0 by 80, no CA mark]

C: C = 50 + 0,05(n – 3 000) 56 – 50(M) = 6

6 0,05(M) = 120 120 + 3 000(M) = 3 120 seconds(CA) (7)

3.4 If they billed for whole minutes or part thereof we would be working with whole numbers (discreet). But now the billing is per second and in real life we don’t work with anything smaller than a second, therefore the time as well as the cost is continuous. (2)

OR Any appropriate reasoning

3.5 [graph: see next page] (11)

3.6 Lowest possible value = 5 900RG Accept [5 800 – 5 999, but not 6 000 as

the line lies before the 6 000 dot] Highest possible value = 32 400RG Accept 32 000 – 33 000] (Each small block = 1 200) Convert:

5 900 60(C) = 98 OR 99 minutes

32 400 60 = 540 minutes He can call between 98 and 540

minutes.(CA) (4) *Answers may vary a lot according to RG of

learner. *RG = value read from graph


Comparing three cell phone packages

Time (seconds) 6 000 12 000 18 000 24 000 30 000

Tota

l mo

nth

ly c

ost

(R

and

)

1500

1400

1300

1200

1100

1000

900

800

700

600

500

400

300

200

100

Package 3

Package 1

Package 2 correct point

correct point

1st point

1st point (100; 3000)

(200; 6000)

flat line

flat line

straight line: P3

straight line: P2

names

Annexure B: Question 3.5


Question 4 [26 marks] 4.1 Profit/Loss = Income – Expenses = 1 375 000 – 687 900(M) = R687 100(CA) profit(CA) (3) 4.2 These are estimated amounts.(J) (2) 4.3 200 000 + 100 000 + 220 000 + 500 + 4 500 + 3 400 + 9 500(M) = R537 900 687 900 – 537 900(M) = R150 000(CA) (3)

4.4 8,72 9,99(M) = 87,1128(CA) VAT

0,14 87,1128(M) = 12,195792(CA) Total = 87,1128 + 12,195792 = 99,308592

R99,30(CA and R) (5) (not R99,31 because we don’t have a 1c to pay with.) 4.5 SELLING R9,99/kg 135% COST ? 100%

9,99 135(M) 100(M) = R7,40(CA) (3)

4.6 You set up a budget BEFORE the month begins and you ESTIMATE the amounts.

You set up an income-and-expenditure statement AFTER the month is over and you use the ACTUAL amounts. (2)

4.7 Total income = R1 477 707,43(A) Total expenditure = R771 636,55(A)

100incometotal

eexpenditurtotal

= 100707,434771

636,55771 (M) (M)

= 52,21849294

52% OR 52,2% OR 52,22%(CA) (5) 4.8 Rent: printer(A) They probably have a contract that tells

them the exact monthly payment. (J) (3)


Part D: Data Handling

Types of data

We need to know this because…

If the data is continuous, then we will draw a solid line

If the data is discreet, we will draw a dotted line

DATA

numerical categorical

continuous discreet

Categorical data means that the data will fall in a certain category. For example when answering the question “male/female?”, the person will either be in the male category or the female category.

Numerical data means the data consists of numbers. (Numerical data can be sorted in two different ways: single amounts and groups.)

Continuous numerical data may contain decimal numbers, e.g. weights, rainfall measurements, temperatures, etc.

Discrete numerical data are number that can only be whole numbers, e.g. number of people, animals, cars, etc.


Mean, median or mode?

Which one is the best – mean, median or mode? Outliers: An outlier is an observation that lies an abnormal distance from other values in a random sample from a population, i.e. this number differs from the other numbers in the set with a large amount. Data set It has outliers It doesn’t have outliers

The median is best The mean is best

MEDIAN example: Ten learners were asked how much cash they had on them at that moment. The results were: R15, R6, R10, R4, R0, R25, R17, R0, R9 and R2000 (The last guy is buying a mp3-player cash after school.) Which measure of central tendency describes this situation best – mean or median?

Mean = 10

0862= R208,60

Taking the mean we might conclude that the average learner carries R207,60 in his/her pocket all the time. However, that doesn’t make sense! Nine out of the ten students had less than R30! Therefore the mean is not the best indicator of how much money a learner carries around. R0, R0, R4, R6, R9, R10, R15, R17, R25, R2000

Median =

2

109 = R9,50

It makes more sense that any learner will rather carry around R9,50 than R208,60. Answer: The median(answer) describes this situation best because there is an outlier(reason: outliers). If you calculate the mean it gives you an unrealistic high amount that is carried around by any learner. (reason: real life)

MEAN example: Question: A dog breeder had the following amount of puppies in his litters the past 6 years: 3, 4, 3, 2, 5, 3, 3, 2, 4, 4, 2 Which measure of central tendency describes this situation best – mean or median? Answer: The mean(answer) describes this situation best because there are no outliers(reason: outliers). The number of puppies in the litter does not fluctuate between extreme limits. (reason: real ife) The mode is only the best indicator in situations where you want to know the most or the least. For example, if you are a shop owner and you want to know which product sold the best this week you will look at the mode.

Mode


Equation type of questions involving average

(Working backwards)

Example 1: Question: In August 2014 a 2ℓ bottle of Fanta Orange Zero cost the following at each of the four shops:

Checkers Hyper in the Blue Route Mall: R12,99

Checkers Super Market in Kenilworth Centre: R15,49

Pick-‘n-Pay Hyper in Ottery: R???

Super Spar in Rosemead: R13,99 Source: Price Check; http://retailpricewatch.co.za/redgekko/product/carbonated-soft-drinks/fanta/orange-bottle/45115#.VOWIE2AcRMt; 19 Feb 2015

The average price of a 2ℓ Fanta Orange Zero in August 2014 was R14,115. How much did a 2ℓ Fanta Orange Zero cost at Pick-‘n-Pay Hyper in Ottery? (3)

Use the formula: 4

R13,99PnPR15,49R12,99R14,115

Answer:

12,99 + 15,49 + 13,99 = R42,47

4

42,4714,115

PnP [1st: “Get rid of” the ÷ 4 by doing 4 on both sides]

56,46 = 42,47 + PnP [2nd: “Get rid of” + 42,47 by doing – 42,47 on both sides]

R13,99 = PnP

Example 2: Question: It is one of the Rhenish girls’ favourite days again – Valentine’s Day! The Paul

Roos boys deliver the following amount of roses to girls in the Gr.12 class. Jess receives 12 roses, Faatimah receives 7 roses, Pharryn receives ??? roses, Thandi receives 9 roses and Courtney receives 16 roses. The average amount of roses received per girl, is 10,8. How many roses did Pharryn receive?

Use the formula: 5

169P71210,8

Answer: 12 + 7 + 9 + 16 = 44

5

P4410,8

[1st: “Get rid of” the ÷ 5 by doing 5 on both sides]

54 = 44 + P [2nd: “Get rid of” + 44 by doing – 44 on both sides]

10 roses = Pharryn

http://retailpricewatch.co.za/redgekko/product/carbonated-soft-drinks/fanta/orange-bottle/45115#.VOWIE2AcRMt


How to estimate when reading amounts from a graph Example:

Property sales: days on the market before sold

0

10

20

30

40

50

60

70

Jan-1

1

Feb-1

1

Mar-

11

Apr-

11

May-1

1

Jun-1

1

Jul-

11

Aug-1

1

Sep-1

1

Oct-

11

Nov-1

1

Dec-1

1

Month and year

Nu

mb

er

of

days

Bachelor's flat

1 bedroom flat

2 bedroom flat

Vertical axis:

Between 0 and 10 there are 5 spaces. Therefore 1 space = 10 5 = 2 days

The “Bachelor’s flat” for January 2011: the dot is in the middle of the 3rd space

That means that it lies at 2 + 2 + (half of 2) = 5 days

STEPS OF HOW TO ESTIMATE NUMBERS If there are lines: Step 1: Find the difference between the numbers of two consecutive “large” lines. (E.g. in

the graph above we can also use between 20 and 30. 30 – 20 = 10) Step 2: Divide that number (from step 1) by the number of spaces. Now you will know how

much to add for each line. (E.g. if a dot lies on the 4th line above 50, then it is

50 + 4 2 = 58)

If there are no “smaller” lines in between: Step 1: Measure with your ruler and draw your own pencil line in the middle of the two

larger lines. This will help you to see whether the dot lies on, below or above the middle.

Step 2: If you still struggle to read off the value you can draw a further line, e.g. in the middle between the bottom and the middle line.


Part D : Questions

Question 1 [14 Marks] The table below shows the number of goals scored by each team from 2010 to 2014.

1.1 How many goals did Team B score in 2010? (3)

1.2 How many goals did Team B score in 2012? (3)

1.3 What is the relationship between your answer in 1.1 and your answer in 1.2? (2)

1.4 Jessica makes the statement that Team A’s scores increased by 100% from 2010 to 2014. Verify whether this statement is true. Show all your calculations. (4)

1.5 What is the advantage of showing the number of goals in a stacked bar graph instead of a multiple bar graph with the bars next to each other? (2)

Number of soccer goals scored for the season: 2010 - 2014

Year 2010 2011 2012 2013 2014 N

um

ber

of

goal

s sc

ore

d

70

60

50

40

30

20

10

Team A

Team B

Team C


Question 2 [26 Marks] Study the table below and then answer the questions that follow.

Source: http://www.scielo.org.za/img/revistas/sajems/v13n4/a03tab04.jpg; 21 August 2014

2.1 What is this table about? (2)

2.2 How much was spent on Correctional Services in 1980? (2)

2.3 Is this data discreet or continuous? Motivate your answer. (3)

A

http://www.scielo.org.za/img/revistas/sajems/v13n4/a03tab04.jpg


2.4 What was the mean amount of money spent on Justice from 1980 to 2006? (3)

2.5 What was the median amount of money spent on Justice from 1980 to 2006? (2)

2.6 Which measure of spread – the mean or the median – describes this situation best? Motivate your answer. (3)

2.7 Round the government expenditure on Police in 2000 to the nearest 100 000. (2)

2.8 Find the value of A. (3)

2.9 What percentage of the total expenditure was spent on Police during 1980 – 2006? (3)

2.10 What was the percentage increase of Government expenditure on Justice from 1980 to 2 000? (3)


3.1 Jordan did a survey to see how many cars were parked in a certain parking lot at 12:00 of each day. She did this survey for a week. On Monday there were ??? cars, on Tuesday 37 on Wednesday 65, on Thursday 58, on Friday 89, on Saturday 92 and on Sunday 68 cars. The average amount of cars parked in that parking lot at 12:00 was 67,5714. How many cars were parked there on Monday?

Use the formula: 7

689289586537M67,5714

(4)

3.2 Girls just love their shoes! Schae has 24 pairs of shoes, Shelley has 31, Tammy has 27, Kelly has 17, Shinaed has 22, Rebekah has ???, Sasha has 45 and Taylor has 29 pairs of shoes. The average amount of pairs of shoes owned by a girl in this group is 28,875. How many pairs of shoes does Rebekah own?

Use the formula: 8

2945R221727312428,875

(4)

3.3 The Grade 12’s are taking part in a recycling competition. The goal is to see which class can recycle the most cool drink cans and plastic bottles in 4 weeks. Last week the Gr.12A’s had 345 items (“items” are cool drink cans and/or plastic bottles), the Gr.12B’s had 256, the Gr.12C’s had 718, the Gr.12D’s had 592 and the Gr.12E’s had ??? items. The average amount of items recycled by a Grade 12 class last week was 530. How many items did the Gr.12E’s recycle last week? (4)


3.4 Five women were asked how many handbags they have owned during their lives. Their answers were: 45, 24, 67, ??? and 92. The average amount of handbags owned by a woman in this group is 54. How many handbags has the 4th woman owned? (4)

3.5 A group of 10 boys, who each owns an i-pod, were selected. Each boy was asked how many songs he has on his i-pod. Gareth has 945, Liam has 475, Jonathan has 758, Nicholas has 1 264, Matthew has 888, John has 2 564, Kyle has 758, Henry has 670, Paul has 1 754 and Michael has ??? songs on his i-pod. The average amount of songs per i-pod is 1 205,1. How many songs does Michael have on his i-pod? (4)

Question 4 [26 marks] The table below shows the religious beliefs of South Africans at that point in time.

Religion GAU% NW% FS% NC% MPU% LIM% KZN% EC% WC%

African 22.2 16.2 19.3 12.6 18.9 18.7 18 17 8.8 Atheism 0.1 0 0.04 0.07 0 0 0.04 0.2 0.02

Bahaism 0 0 0.04 0 0.1 0 0.04 0 0 Buddhism 0.03 0 0 0 0 0 0.04 0 0.02

Christianity 72.9 81.1 78.2 85.4 79.1 79.5 74.3 80.3 88.4 Hinduism 0.8 0.4 0.3 0.13 0.4 0.3 4.7 0.2 0.3

Islam 0.3 0.2 0.08 0.3 0.2 0.3 0.7 0.3 0.8

Judaism 0.3 0 0 0 0 0 0.08 0.2 0 Scientology 0.3 0.09 0.11 0.07 0 0 0 0 0.02

Other 3.1 2 2 1.4 1.3 1.3 2.3 2 1.6 Source: http://www.scielo.org.za/img/revistas/hts/v69n1/01t05.jpg; 21 August 2014

4.1 What was the mean percentage of people belonging to a religion in Gauteng? (3)

4.2 Re-arrange the percentages for Gauteng in ascending order. (2)

4.3 What is the mode of this data set in 4.2? (2)

4.4 What was the median percentage of people belonging to a religion in Gauteng? (2)

4.5 Which measure of spread – the mean or the median – describes this situation best? Motivate your answer. (3)

4.6 What is the mean percentage per province of people whose religion is Christianity? (3)

http://www.scielo.org.za/img/revistas/hts/v69n1/01t05.jpg


4.7 Below is a box-and-whisker plot that shows the information of the percentage people per province that follows African Religion.

Smallest % 8,8

Lower quartile 14,4 Median 18

Upper quartile 19,1

Highest % 22,2

(2) Comment on the middle 50% of people practicing African Religion.

4.8 Calculate the range of this data set. (2)

4.9 Calculate the inter-quartile range of this data set. (2)

4.10 Why can it be useful to determine the inter-quartile range? (2)

4.11 Mary’s finishing time in a cycling race is 5 hours and 3 minutes. Her time lies at the 4th percentile. Is this a good or a bad time? Motivate your answer. (3)


Part D: Memo Question 1 [14 marks]

1.1 30 – 10(RT)(M-) = 20 goals(CA) (3)

1.2 33 – 13(RT)(M-) = 20 goals(CA) (3) 1.3 They are the same.(CA) (2)

1.4 % increase = 100original

difference

=

10010

0120

(A,top)(A,bottom and 100) = 100%(CA) The statement is correct.(CA) (4)

1.5 You can easily see how many goals was scored in total by all the teams together during that year.(J) (2)

Question 2 [26 marks] 2.1 It is about the amount of money that

the government spent on the criminal justice system from 1980 to 2006. (A) (2)

2.2 R99 793 000(A, amount)(U) [no 000 at end, no mark!] (2)

2.3 Continuous. (A) Money can be written as a decimal number. (J)

(3)

2.4 Mean = yearsof no.

amounttotal

= 27

00059629344

(A,top)(A,bottom)

R1 640 503 556 per year(CA) (3)

2.5 R862 697 000(A) (2)

2.6 The mean. (A) There are no outliers. (J) (3) [Although the numbers looks “far apart” they are very large numbers and therefore they are relatively “close together”.]

2.7 R14 572 459 000

R14 572 500 000(A) (2)

2.8 A = 77 217 849 – sum of amounts of 1980-1997 and 1999-2006 = 77 217 749 – 73 255 549(M-)(A,+)

= R3 962 300 R3 962 300 000(CA) (3)

2.9 % = 100eexpenditurTotal

Total Police

= 100493171354

596660232

(A,top)(A,bottom and 100)

= 65,691… 65,69%(CA) (3)

2.10 % increase = 100original

difference

=

1001980exp

1980exp0002exp

=

10087448

874483856542

(A,top)(A,bottom and 100) = 5 331,077…

5 331,08% increase!!(CA) (3)

Question 3 [16 marks] 3.1 37 + 65 + 58 + 89 + 92 + 68(M) = 409

67,5714 7(M) = 472,9998 472,9998 – 409(M)

= 63,9998 64 cars (OR 63) (CA and R) (4)


3.2 24 + 31 + 27 + 17 + 22 + 45 + 29(M) = 195

28,875 8(M) = 231 231 – 195(M) = 36(CA) (4)

3.3

5

E592718256345530

345 + 256 + 718 + 592(M) = 1 911

530 5(M) = 2 650 2 650 – 1 911(M) = 739 (CA) (4)

3.4

5

92H67244554

45 + 24 + 67 + 92(M) = 228

54 5(M) = 270 270 – 228(M) = 42(CA) (4)

3.5 945 + 475 + 758 + 1 264 + 888 + 2 564 + 758 + 670 + 1 754(M) = 10 076

1 205,1 10(M) = 12 051 12 051 – 10 076(M) = 1 975(CA) (4)

Question 4 [26 marks] 4.1 Mean

= religions10

Gauteng in religions allof%allofsum

= 10

1.33.03.03.08.09.7203.001.02.22

(M,+all)(A,10)

= 10

100,03

= 10,003 10%(CA) (3)

4.2 0 0.03 0.1 0.3 0.3 0.3 0.8 3.1 22.2 72.9 (O, -1 per mistake until 0) (2) 4.3 Mode = 0,3%(A) (2)

4.4 Median =

2

0,3 0,3 = 0,3%(A) (2)

4.5 The median(A) because there are outliers(J) [22,2 and 72,9] (3)

4.6 Mean

= provinces9

provinces allintyChristinaiof%allofsum

= 9

2,719(M,+all)(A,9)

= 79,9111… 79,91%(CA) (3)

(A,top)(A,bottom)

R1 640 503 556 per year(CA) (3)

4.7 Between 14,4% and 19,2% people practice African religion per province and that makes our 50% of people practicing African religion(A)

[There is more than one correct way to say this sentence, so each answer must be judged individually.] (2)

4.8 Range = 22,2 – 8,8 = 13,4%(A) (2)

4.9 Inter-quartile Range = 19,2 – 14,4 = 4,8%(A) (2)

4.10 It gives us more information about the middle of the data than the range. (A) (2)

4.11 It is a good time(A) because it means that only 4% of the people in the race finished in the same time as her or before her. (J) OR It is a good time(A) because it means that 96% of the people in the race finished after her. (J) (3)


Part E: Finance - Tax

Value Added Tax (VAT)

Percentages

The original price is 100%. This price excludes VAT.

VAT is 14% of the original price.

A VAT inclusive price is 114% of the original price.

Calculating VAT inclusive prices Example: Mr White gets a quotation from Mr Handyman for R27 500 to fix a few things in and around his house. The quotation does not include VAT. How much is the quotation in total, including VAT?

Answer: 114% of R27 500

= 100

114 27 500

= R31 350

Calculating the original price (VAT exclusive prices) Example: Miss Green pays R1 245 for her groceries at Pick ‘n Pay. None of the products she bought is VAT exempt products. What was the original price of her shopping, excluding VAT?

Answer: R1 245 114% 1 ? 100%

1 245 ÷ 114 100 R1 092,11

Calculating VAT Example 1: Mr White gets a quotation from Mr Handyman for R27 500 to fix a few things in and around his house. The quotation does not include VAT. How much VAT must Mr White pay?

Answer: 14% of R27 500

= 100

14 27 500

= R3 850

Example 2: Miss Green pays R1 245 for her groceries at Pick ‘n Pay. None of the products she bought is VAT exempt products. How much VAT did she pay?

Answer: R1 245 114% 1 ? 14%

1 245 ÷ 114 14 R152,89

100 100

÷ 114 ÷ 114

14 14

÷ 114 ÷ 114


Personal Income Tax

Mind Map: steps to follow when calculating Income Tax

Persons under the age of 65 Persons 65 and older

Annual Gross

Salary

Minus pension (max 7,5% of

gross)

= Taxable Income

Tax Table (use taxable income value)

Minus primary rebate = Annual Income Tax

Minus Medical Tax Credits = “Net” Income Tax

(the amount you have to pay to SARS)

Annual Gross Salary

Minus pension (max 7,5% of gross)

Minus all medical expenses

= Taxable Income

Tax Table (use taxable income value)

Minus all appropriate rebates

= Annual Income Tax (the amount you have

to pay to SARS)


Frequently Asked Questions….

Q: What is the difference between “Medical Tax Credits” and “Medical Expenses”? A: Medical Tax Credits is like a discount that the government gives you. If you belong to a medical aid you can get this discount, which is an after-tax deduction. That is why we only subtract it right at the end. Medical expenses are the physical medical expenses you have, e.g. doctor’s bills, hospital bills, prescribed medication (invoices), your medical aid contribution, etc.

Q: Why must I first deduct the rebate and then the Medical Tax Credits (in that order?) A: The rebate is part of the Income Tax calculation whereas the Medical Tax Credits is an after-tax deduction. (This question was also answered in the previous question.)

Q: Which value must I use in the formula in the Income Tax table? A: You must use Taxable Income.

Q: Why must I subtract a maximum of 7,5% for pension when calculating Taxable Income? A: This is like a 7,5% discount. The government says you don’t have to pay Income Tax on your full pension amount, only on the amount that is more than 7,5% of your gross salary.

Q: Why must I subtract the full pension amount when calculating Net Income? A: When you calculate net income you must subtract all the deductions (from the gross salary) as they are on the salary slip. And on the salary slip e.g. 10% of your gross salary was deducted each month for pension. Therefore you have to subtract the full 10%.

Q: What must I subtract and where if my pension percentage is less than 7,5%, e.g. 5%? A: Then you subtract 5% of gross salary to work out the Taxable Income and you also subtract 5% of gross salary when calculating net income.

Q: Must I calculate and deduct 1% of gross salary for UIF for calculating Net Income even if they haven’t asked me to do so? A: Yes

Q: I don’t agree with your mind map. Our teacher told us something else OR my parent is a Chartered Accountant and tells me that this is not how things work exactly in real life. Please explain. A: For a person under the age of 65: In real life when you calculate your tax deductible expenses, which is only pension in the mind map cloud, you can also deduct medical expenses. If a person had medical expenses of more than 7,5% of his/her annual gross income, he/she can deduct the amount that is more than the 7,5% also. In other words, if your gross income is R100 000 and you have medical expenses of more than 7,5% (R7 500), e.g. R8 300, then you can subtract the difference (8 300 – 7 500 = R800) as a tax deductible expense. However, this is too complicated for Grade 12 Mathematical Literacy learners, so we leave it out. I also haven’t seen this calculation done in any CAPS text book.


INCOME TAX FOR INDIVIDUALS

Income tax: Tax rates for individuals – 2015/2016

Taxable income Rates of Tax

0 – 181 900 18 % of each Rand (taxable income)

181 901 – 284 100 R32 742 + 26% of the amount above R181 900 284 101 – 393 200 R59 314 + 31% of the amount above R284 100


701 301 and above R208 587 + 41% of the amount above R701 300

Tax rebates (Amounts deductible from the tax payable) Primary rebate (persons under 65): R13 257 Secondary rebate (persons 65 – 74): R7 407 Tertiary rebate (persons 75 and older): R2 466

Tax Thresholds (If you earn less/equal to that amount per year you will not pay tax.) Persons under 65: R73 650 Persons 65 – 74: R114 800 Persons 75 and older: R128 500

Tax Rebates

All people who pay Income Tax get the Tax Rebate

For persons under 65 you deduct R13 257 after tax has been calculated on the taxable income. (Primary rebate)

For persons 65 – 74 years old you deduct R13 257 + R7 407 after tax has been calculated on the taxable income. (R20 664) (Primary + secondary rebate)

For persons 75 years and older years old you deduct R13 257 + R7 407 + R2 466 after tax has been calculated on the taxable income. (R23 130) (Primary + sec. + ter.rebate)

Sometimes Tax Rebates are given as follows:

Tax Rebates for 2015/2016 Tax Year Below age 65 R13 257

Age 65 – 74 R20 664 Age 75 and over R23 130

This means that they have already added the different rebates together. Then you just subtract the ONE amount.

Medical Tax Credits for 2015/2016 Tax Year

Per month Per annum

Taxpayer younger than 65 years R270 R3 240 Taxpayer’s 1st dependent R270 R3 240

Taxpayer’s additional dependents R181 each R2 172 each


Example: Calculating Personal Income Tax

Question Mary is 46 years old and earns R34 000 monthly. She is divorced and has two children. They all belong to a medical aid. Her pension fund is 8% of her annual salary. The only other deduction on her pay slip is for UIF. Calculate:

(a) her gross annual income

(b) her total annual tax deductible expenses due to pension scheme

(c) her taxable income

(d) her total annual tax credits for medical aid contributions

(e) her annual income tax payable after medical tax credits have been deducted (net Income Tax)

(f) her net annual income (income AFTER all expenses that are deducted on the pay slip, are deducted)

(g) her net monthly income Answer (a) Gross annual salary

= R34 000 12 = R408 000

(b) Pension scheme

8% is above the maximum of 7,5%, therefore she can only deduct 7,5% of her gross annual salary

Tax deductible expenses

= 0,075 R408 000 = R30 600

(c) Taxable Income = Gross annual salary – tax deductible expenses = R408 000 – R30 600 = R377 400


(d) Medical aid Mary (taxpayer): Tax Credit = R3 240 1st child (1st dependent): Tax Credit = R3 240 2nd child (additional dependent): Tax Credit = R2 172

Total Tax Credits = R3 240 + R3 240 + R2 172 = R8 652

(e) Annual Income Tax payable

Step 1: In which tax bracket does the person’s annual salary after deductions fall? It falls in the 3rd bracket.

Taxable income Rates of Tax 284 101 – 393 200 R59 314 + 31% of the amount above R284 100

Step 2: Follow the steps as in that tax bracket. (Write the % as a decimal fraction.)

R59 314 + 0,31 (377 400 – 284 100) = R88 237

Step 3: Subtract the rebate (younger than 65 = primary rebate) R88 237 – R13 257 = R74 980 (Income Tax)

Step 3: Subtract the medical tax credits R74 980 – R8 652 = R66 328 (Net Income Tax)

Annual Income Tax payable = R66 328

(f) UIF = 1% of gross income = 0,01 408 000 = R4 080

Pension = 8% of gross income = 0,08 408 000 = R32 640

Net annual income = Gross annual income – Annual Income Tax – UIF – pension = R408 000 – R66 328 – R4 080 – R32 640 = R304 952

(g) Net monthly income = Net annual income ÷ 12 = R304 952 ÷ 12

R25 412,67


Part E: Questions Question 1 [14 marks]

1.1 Jacques bought a new car from a dealership. He paid a cash amount of R123 575 this week after paying a deposit of R50 000 last week. He also had to pay an additional amount of R1 200 for administration purposes. Calculate how much VAT was paid if the total price of the car (including administration) already includes VAT. (5)

1.2 Harold’s brother bought him a new camera for R5 475. Harold has to pay him back. However, because it is family, his brother decides to only let him repay the original amount, excluding VAT. How much must Harold repay his brother? Round your answer to the nearest R100. (4)

1.3 Susan earns R12 768,54 per month. Calculate the annual contribution made to UIF by her and her employer. (5)


Income tax: Tax rates for individuals – 2015/2016

Taxable income Rates of Tax

0 – 181 900 18 % of each Rand (taxable income) 181 901 – 284 100 R32 742 + 26% of the amount above R181 900

284 101 – 393 200 R59 314 + 31% of the amount above R284 100


701 301 and above R208 587 + 41% of the amount above R701 300

Tax rebates (Amounts deductible from the tax payable) Primary rebate (persons under 65): R13 257 Secondary rebate (persons 65 – 74): R7 407 Tertiary rebate (persons 75 and older): R2 466

Medical Tax Credits for 2015/2016 Tax Year Per month Per annum

Taxpayer younger than 65 years R270 R3 240 Taxpayer’s 1st dependent R270 R3 240

Taxpayer’s additional dependents R181 each R2 172 each


For each of the questions in Question 2 you must answer the following:

Calculate:

(a) His/her annual gross income

(b) His/her total annual tax deductible expenses (pension and medical expenses where appropriate)

(c) His/her taxable income

(d) His/her total annual medical tax credits

(e) His/her annual income tax payable (after medical tax credits have been deducted)

(f) His/her net annual income 2.1 Valerie is a 22 year old single mother of one child. She earns R10 000 per

month. She belongs to a medical aid and her child is the first dependant on her aid. She only contributes 2% of her salary towards a pension fund. (a = 2 marks) (b = 2) (c = 2) (d = 2) (e = 5) (f = 5) (18)

2.2 Jacob is a 34 year old accountant who earns R47 650 per month. He has a wife

and three children, all for whom he pays medical aid fees (R69 000 for the year, a deduction on his pay slip). 10% of his gross salary is deducted from his pay slip for a pension fund. (a = 2 marks) (b = 2) (c = 2) (d = 2) (e = 5) (f = 7) (20)

2.3 Simon is 75 years old. He still earns a salary of R43 200 per month. He had

medical expenses of R45 783 for the year and he contributes 5% of his salary towards a pension fund. Leave out (d). (a = 2 marks) (b = 4) (c = 2) (d = 0) (e = 6) (f = 5) (19)


Part E: Memo


1.1 123 575 + 50 000 + 1 200(M) = R174 775(CA)

Direct proportion R174 775 114% ? 14%

174 775 114(M) 14(M) = 21 463,59649

R21 463,60(CA) (5) 1.2 R5 475 114% ? 100%

5 475 ÷ 114(M,A) 100(M,A) = R4 802,63…(CA)

R4 800(R) (4) 1.3 1% + 1% = 2%(A)

R12 768,54 12(M) = R153 222,48(CA)

0,02 R153 222,48(M) = R3 064,4496

R3 064,45(CA) (5)


2.1

(a) 10 000 12(M) = R120 000(CA) (2)

(b) 0,02 120 000(M) = R2 400(CA) (2)

(c) 120 000 – 2 400(M) = R117 600(CA) (2)

(d) 3 240 + 3 240(M) = R6 480(CA) (2)

(e) 1st row

0,18 117 600(M) = R21 168(CA)

21 168 – 13 257(M) = R7 911

7 911 – 6 480(M) = R1 431(CA) (5) (f) Pension = R2 400

UIF = 0,01 120 000 = R1 200(M) Income tax = 1 431

Net income = Gross – pension – UIF – Income Tax = 120 000 – 2400(M) – 1 200(M) – 1 431(M) = R114 969(CA) (5) 2.2

(a) 47 650 12(M) = R571 800(CA) (2)

(b) 0,075 571 800(M) = R42 885(CA) (2)

(c) 571 800 – 42 885(M) = R528 915(CA) (2)

(d) 3 240 + 3 240 + 3 2 172(M) = R12 996(CA) (2)

(e) 4th row 93 135 + 0,36(528 915 – 393 200)(M) = R141 992,40(CA)

141 992,40 – 13 257(M) = R128 725,40

128 725,40 – 12 996(M) = R115 739,40(CA) (5)


2.2

(f) Pension = 0,1 571 800 = R57 180(M)

UIF = 0,01 571 800 = R5 718(M) Income tax = R115 739,40 School fund = R69 000

Net income = Gross – pension – UIF – Income Tax – School fund = 571 800 – 57 180(M) – 5 718(M) – 115 739,40(M) – 69 000(M) = R324 162,60(CA) (7) 2.3 (a) Gross annual income

= R43 200 12(M) = R518 400(CA) (2)

(b) Deductible expenses Pension = 5% of gross

= 0,05 518 400(M) = R25 920(CA)

Medical expenses = R45 783

Total = R25 920 + R45 783(M) = R71 703(CA) (4)

(c) Taxable income = gross – deductible expenses = 518 400 – 71 703(M) = R446 697(CA) (2)

(d) Leave out

(e) Annual income tax payable Tax bracket: 4th 93 135 + 0,36(446 697 – R393 200)(SF) = R112 393,92(CA)

Deduct rebates [primary + secondary + tertiary] R112 393,92 – R13 257(M) – R7 407(M) – R2 466(M) = R89 263,92(CA) (6)

(f) UIF = 0,01 518 400 = R5 184(M) Net annual income = gross – pension – UIF – income tax = 518 400 – 25 920(M) – 5 184(M) – 89 263,92(M) = R398 032,08(CA) (5)


Part F: Scales, Maps and Tables

Vehicles: operating costs

Table 1: Fixed costs Fixed Costs Table

Average fixed cost (c/km) [All costs inclusive of VAT]

Annual distance travelled (km)

Purchase Price(R) (including VAT)

< 10 000

10 001 - 15 000

15 001 - 20 000

20 001 - 25 000

25 001 - 30 000

30 0001 - 35 000

35 001 - 40 000

> 40 000

R50 001 - R75 000 237 158 119 96 81 71 63 57

R75 001 - R100 00 318 213 160 129 108 95 84 76

R100 001 - R125 000 344 230 173 140 118 103 92 83

R125 001 - R150 000 415 277 209 169 142 124 110 100

R150 000 - R175 000 487 325 245 198 166 146 130 117

R175 001 - R200 000 560 374 281 227 191 168 149 136

R200 001 - R250 000 704 470 354 286 240 211 187 169

R250 001 - R300 000 788 526 396 320 269 237 210 191

R300 001 - R350 000 927 619 466 377 317 279 247 224

R350 001 - R400 000 1067 713 536 434 365 321 285 258

more than R400 001 1183 790 594 481 404 356 316 286 Source: AA petrol vehicle fixed costs

Operating costs

Fixed costs Running costs

Fuel (Petrol/ Diesel)

Service &

Repair

Tyres

Depreciation of vehicle

and

Licensing Formula: Running costs (c/km)

= (petrol factor petrol price) + S&R + Tyres


Table 2: Running Costs Table - Petrol vehicles

Average running cost (c/km) [All costs inclusive of VAT]

FUEL MAINTENANCE

Engine Capacity (cc)

Petrol factor Service and repair

costs (c) Tyre costs (c)

A B C

< 1 300 6,62 17,18 8,98

1 301 - 1 500 7,39 19,76 13,1

1 501 - 1 800 8,03 22,73 16,7

1 801 - 2 000 9,24 24,29 22

2 001 - 2 500 10,6 29,17 25,2

2 500 - 3 000 10,96 35,97 31,7

3 001 - 4 000 12,02 38,02 32,5

> 4001 14,4 52,34 41

Running Costs calculation (c/km) = (A multiplied with petrol price (R/litre) + B + C

Example: Tyrone owns a 1 600cc Toyota Corolla. The purchase price of his car was R195 000. He drove 23 560km this year. The current petrol price is R13,25 per litre. Calculate his vehicle’s operating cost (Rand) for the whole distance he has driven this year. Formulas given: Operating cost (c/km) = fixed cost + running costs

Running costs (c/km) = (petrol factor petrol price) + S&R + Tyres

Answer: Step 1: Find the fixed costs for 1km from table 1 Distance travelled = 20 001 to 25 000km; Purchase price = R175 001 – R200 000 Fixed cost = 227c/km = R2,27/km

Step 2: Find the running costs for 1km from table 2 Engen size: 1 600: Petrol factor = 8,03

Running costs = (petrol factor petrol price) + S&R + Tyres

= 8,03 13,25 + 22,73 + 16,70 = 145,8275c/km = R1,458275/km

Step 3: Find the operating costs for 1km Operating cost = fixed cost + running costs = R2,27/km + R1,458275/km = R3,728275/km

Step 4: Find the operating costs for the annual distance, i.e. 23 560km Annual operating cost

= (cost per km) (km)

= R3,728275/km 23 560km = R87 838,159 R87 838,16


Scales and Building

A scale is written as a ratio in the form 1 : …, e.g. 1 : 250 000 (called a number scale)

Since the ratio is written without units, you can use any units you like.

Meaning: 1 : 250 000 can mean that 1 cm on the map represents 250 000 cm in real life.

It can also mean than 1 mm on the map represents 250 000 mm in real life. Calculating the distance

Remember: * If your answer is in cm you must ÷ 100 000 to get answer in km

* If your answer is in mm you must ÷ 1000 000 to get answer in km

Scales on maps

Want: REAL distance (distance in real life)

large

have to

Want: MAP distance (or distance on scale drawing)

small

have to ÷

Formula:

Dreal = Dmap scale Formula: Dmap = Dreal ÷ scale

Remember: * both distances must have the same unit * For example, if the scale is 1:2 500 000 and the real distance is 30 km, then:

* both units must be in km: 25km

30km

* or both units must be in cm

000cm5002

000cm0003


Finding the SCALE of the map Step 1: Write amounts as a ratio, always writing the measurement of the map (the smaller

amount) first. For example, 30 cm : 96 km Step 2: Convert the second amount to the same unit as the first amount For example, 30 cm : 9 600 000 cm Step 3: Simplify the ratio For example, 1 : 320 000 What if it is an “ugly number”, e.g. 1 : 266 666,66666…? Then you round it off to the nearest whole number, e.g. 1 : 266 667 or to number of decimal places asked for in the question. Bar Scales A bar scale is a picture of a bar that shows you what the actual distance will be for that length of the bar on the map. For a bar scale you first have to measure how long the whole bar is. For example, the bar below is 4,2 cm long.

That means that 4,2 cm represents 126 km. Now follow the same steps as above to get the scale: 4,2 cm : 126 km = 4,2 cm : 12 600 000 cm = 1 : 3 000 000 Advantages and disadvantages of different scales

Number scale Bar scale

Advantage Disadvantage Advantage Disadvantage You can do calculations with numbers without worrying that you measured wrong.

If the size of the map is changed, the scale is wrong

If the size of the map is changed it won’t matter because the size of the bar will change too. The scale will still be correct.

You have to measure the bar and can make errors. It also requires more work than the number scale.

0 km 63 km 126 km


Building notes: Building can be complicated for people who are not builders, like us!

You need concrete: concrete is made from cement, sand and stones

Dig foundation and lay foundation for exterior and interior walls

Need: floor plan + elevation plans of each side of the house, with correct measurements

Building a house

Build the walls

Make concrete and concrete is laid in foundation via metal/wooden trenches

To calculate the number of bricks needed for 1 wall: (ignore doors & windows)

Length(wall) ÷ Length(brick + mortar)

Height(wall) ÷ Height(brick + mortar)

Multiply the two answers

Round your answer up to the nearest brick

Lay black plastic over the length of the foundation (to prevent rising damp)

Exterior walls have DOUBLE brick walls

Interior walls have SINGLE brick walls

Mortar is the cement mix put between bricks

Put on the roof Install the roof trusses (wooden beams) and put roof sheeting on wooden trusses. Each sheet overlaps with another.

Window and door openings will be built into walls


Bricks and Mortar Below is a picture of a brick:

Source: http://s3.amazonaws.com/rapgenius/single-brick.jpg; 23 August 2014

Mortar is normally 12mm thick.

That means the length of the brick plus mortar = 312mm

The width of the brick plus mortar = 142mm

The height of the brick plus mortar = 132mm

When we do calculations, we do it with the mortar included since the mortar does take up space in real life.

Length 300mm

Breadth / Width 130mm

Height 120mm

http://s3.amazonaws.com/rapgenius/single-brick.jpg


Models: fitting in objects

Optimal Packing

When you e.g. pack products into a box or boxes into a truck you want to pack as many as possible.

This makes sense because if you can load more products in the truck, you can make less trips and it will cost the company less money (less fuel; fewer working hours; etc.)

In terms of packing products into boxes it also makes sense because you want to use the least packaging material as possible – again, to make it as cheap as possible for the company.

It is important to remember that you can only pack WHOLE products into boxes / trucks. For example, if 14,8 juice boxes fit into the box, you can only pack in 14 boxes. If you cut another box to fit 0,8 of the box in, the juice will spill out!

Methods for optimal packing

When you want to fit e.g. boxes into a truck, you can’t use volume:

Wrong method: Volume of truck ÷ volume of 1 box = number of boxes that can fit in

This method is wrong because when you physically pack the boxes into the truck, there will be some open spaces, e.g. 2½ boxes can fit into the width. Since you can’t cut e.g. a box with a TV in, in half, you can only fit 2 boxes into the width, which means there will be a lot of open space.

Volume takes the maximum boxes into account but the following (correct) method takes reality into account.

Correct method to fit objects into a larger space: Step 1: Length of big container divided by side of small container. Round down to the

nearest whole number (NOT up). Step 2: Breadth of big container divided by side of small container Round down to the

nearest whole number (NOT up). Step 3: Height of big container divided by height of small container. Round down to the

nearest whole number (NOT up). Step 4: Multiply the three answers.


Example: Logo Logistics transport televisions. They have a truck with the following dimensions:

Side view View from back

One television box has the following dimensions:

The boxes have to be put into the truck in an upright position.

(a) How many boxes can fit into the truck if you pack the boxes into the truck so that the face of the box faces the door? (I.e. The 1 m side of the box is put into the width of the truck.)

(b) How many boxes can fit into the truck if you pack the boxes into the truck so that the face of the box faces the side? (I.e. The 1 m side of box is put into length of truck.)

(c) Hence, determine which way will be the best to fit the maximum amount of boxes into the truck.

Answer: (a) In the breadth of the truck: B(truck) ÷ front of box = 2,6m ÷ 1m = 2,6

2 boxes

1m 40cm

0,9 m Flat screen TV

W = 2,6 m

H =

3 m

9 m


In the length of the truck L(truck) ÷ side of box = 9m ÷ 0,4m [we converted 40cm!] = 22,5

22 boxes In the height of the truck H(truck) ÷ H of box = 3m ÷ 0,9m = 3,333…

3 boxes Total amount of boxes that can fit in

= 2 22 3 = 132 boxes (b) In the breadth of the truck: B(truck) ÷ side of box = 2,6m ÷ 0,4m = 6,5

6 boxes In the length of the truck L(truck) ÷ front of box = 9m ÷ 1m = 9 boxes In the height of the truck H(truck) ÷ H of box = 3m ÷ 0,9m = 3,333…

3 boxes Total amount of boxes that can fit in

= 6 9 3 = 162 boxes (c) Option (b) is the best way to fit the TV boxes in the truck because you can load in

more boxes than with option (a).


Part F: Questions Question 1 [16 marks] Table 1: Fixed costs

Fixed Costs Table

Average fixed cost (c/km) [All costs inclusive of VAT]

Annual distance travelled (km)

Purchase Price(R) (including VAT)

< 10 000

10 001 - 15 000

15 001 - 20 000

20 001 - 25 000

25 001 - 30 000

30 0001 - 35 000

35 001 - 40 000

> 40 000

R50 001 - R75 000 237 158 119 96 81 71 63 57

R75 001 - R100 00 318 213 160 129 108 95 84 76

R100 001 - R125 000 344 230 173 140 118 103 92 83

R125 001 - R150 000 415 277 209 169 142 124 110 100

R150 000 - R175 000 487 325 245 198 166 146 130 117

R175 001 - R200 000 560 374 281 227 191 168 149 136

R200 001 - R250 000 704 470 354 286 240 211 187 169

R250 001 - R300 000 788 526 396 320 269 237 210 191

R300 001 - R350 000 927 619 466 377 317 279 247 224

R350 001 - R400 000 1067 713 536 434 365 321 285 258

more than R400 001 1183 790 594 481 404 356 316 286 Source: AA petrol vehicle fixed costs

Table 2: Running Costs

Running Costs Table - Petrol vehicles

Average running cost (c/km) [All costs inclusive of VAT]

FUEL MAINTENANCE

Engine Capacity (cc)

Petrol factor Service and repair

costs (c) Tyre costs (c)

A B C

< 1 300 6,62 17,18 8,98

1 301 - 1 500 7,39 19,76 13,1

1 501 - 1 800 8,03 22,73 16,7

1 801 - 2 000 9,24 24,29 22

2 001 - 2 500 10,6 29,17 25,2

2 500 - 3 000 10,96 35,97 31,7

3 001 - 4 000 12,02 38,02 32,5

> 4001 14,4 52,34 41

Running Costs calculation (c/km) = (A multiplied with petrol price (R/litre) + B + C

1.1 Justin Mostert owns a 2 000cc 3-series BMW. He bought the car for R450 000. He drove 57 830 km for the year. The current petrol price is R13,09 per litre. Use the tables above to calculate his vehicle’s operating cost (Rand) for the whole distance he has driven for the year. Formulas given: Operating cost 9c/km) = fixed cost + running costs

Running costs (c/km) = (petrol factor petrol price) + S&R + Tyres (8)


1.2 Meryl owns a 1 300cc Toyota Yaris. The purchase price of the car was R175 000. She drove a distance of 84 315km during this year. The current petrol price is R12,15 per litre. Use the tables on the previous page to calculate her vehicle’s operating cost (Rand) for the whole distance she has driven this year. Formulas given: Operating cost (c/km) = fixed cost + running costs

Running costs (c/km) = (petrol factor petrol price) + S&R + Tyres (8)

Question 2 [19 marks] Below is a scale drawing of a house.

2.1 Determine the number scale. Write your answer as 1 : … (3)

2.2 Measure the wall (in cm) of the East elevation. (2)

2.3 How long (in metres) is the wall on the East elevation in reality? (3)

2.4 The wall on the left-hand-side of the house has a rectangular shape and is 9m long and 3,2m high. The dimensions of 1 brick are: length = 300mm, breadth = 130mm and height = 120mm. The bricklayer puts 12mm of mortar on each side of the brick when he builds the wall. How many bricks are needed to build this wall if it is a double brick wall? (8)

Wall Balustrade Window Door

0 1 2m W

S

E

N


2.5 An architect named Sue drew a scale drawing of the parking area using the scale 1 : 250. If the actual length of the parking area is 12m, determine the length (in cm) of the parking on Sue’s scale drawing. (3)


Fruit juice boxes are packed into a larger box so that it can be transported to shops without getting damaged. The dimensions of 1 large box are: length = 50cm, breadth = 30cm and height = 35cm The dimensions of 1 juice box are: length = 4,5cm, breadth = 3,5cm and height = 6cm The juice boxes can be packed into the larger boxes in two ways: Option 1 Option 2

In option 1 the length of the juice box is packed into the length of the large box. In option 2 the breadth of the juice box is packed into the length of the large box.

3.1 How many juice boxes can fit into the larger box for option 1? (8)

3.2 How many juice boxes can fit into the larger box for option 2? (8)

3.3 Hence, determine which option is best. Motivate your answer. (3)

B

L

H

L B

H

B B

H

B

L

H

L B

H



4.1 Katherine’s brother is a carpenter. He builds shelves for her to pack her books into. Here is the sketch of one such a shelve:

The whole shelve (wall) is 4,2m high in total. There are 3 shelves of 2cm thick each. The books are packed on their sides with the spine lying horizontally. The maximum width of each book’s spine is 0,05m.

How many books can fit into the height of the wall? (6)

4.2 When Katherine receives a large order from a bookstore, she packs the books into boxes.

The dimensions of one box are: breadth = 32cm, length = 76cm and height = 32cm.

The dimensions of one book are: breadth = 10cm, length = 25cm and height = 5cm.

The books are packed lying flat into the box, with the length of the book lying in the length of the box, as shown in the sketch below:

How many books can be packed into this box? (8)

0,05m

BOOK

8m

4,2m 2cm

2cm

2cm

76cm

25cm

32cm

32cm 5cm


Part F: Memo

Question 1 [16 marks] 1.1 Fixed cost = 286c/km(A, RT) = R2,86/km

Running costs


= 9,24 13,09 + 24,29 + 22(RT and SF) = 167,2416c/km(CA) = R1,672416/km(C)

Operating cost = fixed cost + running costs = R2,86/km + R1,672416/km(M+) = R4,532416/km(CA)

Annual operating cost


= R4,532416/km 57 830km(M) = R262 109,6173

R262 109,62(CA) (8) 1.2 Fixed cost = 117c/km(A, RT) = R1,17/km

Running costs


= 7,39 12,15 + 19,76 + 13,1(RT and SF) = 122,6485/km(CA) = R1,226485/km(C)

Operating cost = fixed cost + running costs = R1,17/km + R1,226485/km(M+) = R2,396485/km(CA)

Annual operating cost


= R2,396485m 84 315km(M)

= R202 059,6328 R202 059,63(CA) (8)

Question 2 [19 marks] 2.1 1cm = 1m(M) 1cm = 100m(C) 1 : 100(CA) (3) 2.2 5,2cm(A) [accept 5,1cm also] (2) 2.3 D(real)

= D(plan) scale ÷ 100 (convert to m)

= 5,2 100 ÷ 100 (M) = 5,2m(CA) or (A) (3) 2.4 Length of wall: Length of wall ÷ length of brick and mortar = 9m ÷ (300mm + 12mm) (M) = 9m ÷ 0,312m(C) = 28,84…

29 bricks(CA and R) Height of wall: Height of wall ÷ Height of brick and mortar = 3,2m ÷ (120mm + 12mm) (M) = 3,2m ÷ 0,132m = 24,24…

25 bricks(CA and R) Total amount of brick (double brick wall)

= (29 25) (M) 2(M)

= 725 2 = 1 450 bricks(CA) (8)

2.5 12 100 = 1 200cm(C)

Distance on map

= Real distance scale

= 1 200 250(M) = 4,8cm(CA) (3)


Question 3 [19 marks] 3.1 In the length of the box Length of box ÷ length of juice box = 50cm ÷ 4,5cm(M) = 11,111…

11 boxes(CA and R)

In the breadth of the box Breadth of box ÷ breadth of juice box = 30cm ÷ 3,5cm(M) = 8,571…

8 boxes(CA and R)

In the height of the box Height of box ÷ height of juice box = 35cm ÷ 6cm(M) = 5,833…

5 boxes(CA and R)

Total amount of boxes that can fit in

= 11 8 5(M) = 440 juice boxes(CA) (8)

3.2 In the length of the box Length of box ÷ breadth of juice box = 50cm ÷ 3,5cm(M) = 14,285…

14 boxes(CA and R)

In the breadth of the box Breadth of box ÷ length of juice box = 30cm ÷ 4,5cm(M) = 6,666…

6 boxes(CA and R)

In the height of the box Height of box ÷ height of juice box = 35cm ÷ 6cm(M) = 5,833…

5 boxes(CA and R)

Total amount of boxes that can fit in

= 14 6 5(M) = 420 juice boxes(CA) (8) 3.3 Option 1(CA ) is best because you

can fit more juice boxes into the large box than for option 2. (J)

(3)

Question 4 [14 marks] 4.1 Height of wall – shelves

= 4,2m – 3(0,02m) (M-)(M) = 4,14m(A) 4,14m ÷ 0,05m/book(M) [OR 414 ÷ 5] = 82,8(CA)

82 books (R DOWN)AF (6) 4.2 In the breadth: B(box) ÷ B(book) = 32cm ÷ 10cm(M) = 3,2

3 books(CA and R) In the length: L(box) ÷ L(book) = 76cm ÷ 25cm(M) = 3,04

3 books(CA and R) In the height: H(box) ÷ H(book) = 32cm ÷ 5cm(M) = 6,4

6 books(CA and R) Total amount of books:

3 3 6(M) = 54 books(CA)AF (8)


Part G: Finance – Loans and Investments

Loans

You borrow an amount of money from a financial institution.

You have to repay a fixed amount per month for a certain number of months.

The payments already include (a lot of) interest.

Advantage: You get a large amount of money now.

Disadvantage: You pay a large amount of interest.

Investment: Annuity

You have to make a monthly payment until you retire

Compound interest: the more years you invest the more the amount of interest you received.

At retirement age (or age stated by the annuity contract) you will receive a lump sum of money as well as a monthly “allowance” for the rest of your life.

Investment made by an individual.

Investment: Stokvel

More than one person (group of people) put the same amount of money together as regular intervals, e.g. every month.

The money is invested.

Each person receives more interest in the end than what they would have gotten if they invested only their share on their own. (Larger amount of money invested means more interest received.)

An amount of money can be withdrawn from the investment and divided between the members of the Stokvel if all the members agree to it.

Simple Interest

When you invest an amount of money at the bank, the bank pays you a certain amount of interest.

Example: Patience invests R10 000 at ABC-Bank at 12,5 % simple interest per year. How much money would she receive in total after a period of five years? At the beginning of the 1st year she has: R10 000 At the end of the 1st year she has:

R10 000 + 12,5 % interest on R10 000

= 10 000 + (0,125 10 000) = 10 000 + 1 250 = R11 250


The interest per year is R1 250. With simple interest the amount of interest stays the same each year.

At the end of the 2nd year she will have: R11 250 + R1 250 = R12 500 At the end of the 3rd year she will have: R12 500 + R1 250 = R13 750 At the end of the 4th year she will have: R13 750 + R1 250 = R15 000 At the end of the 5th year she will have: R15 000 + R1 250 = R16 250

Isn’t there a shorter way of doing this??? Yes!

R10 000 + 5 R1 250 = R16 250

If you only want to know how much INTEREST she received:

Method 1: 0,125 10 000 5 = R6 250 Method 2: R16 250 – R10 000 = R6 250

Compound Interest: Investment

Example: Patience invests R10 000 at ABC-Bank at 12,5 % compound interest per year. How much money would she receive in total after a period of five years?

At the beginning of the 1st year she has: R10 000 At the end of the 1st year she has:

R10 000 + 12,5 % interest on R10 000

= 10 000 + (0,125 10 000) = 10 000 + 1 250 = R11 250

With compound interest the amount of interest increases each year.

At the end of the 2nd year she will have:

11 250 + (0,125 11 250) = 11 250 + 1 406,25 = R12 656,25

At the end of the 3rd year she will have:

= 12 656,25 + (0,125 12 656,25) = 12 656,25 + 1 582,03 = R14 238,28

At the end of the 4th year she will have:

= 14 238,28 + (0,125 14 238,28) = 14 238,28 + 1 779,79 = R16 018,07


At the end of the 5th year she will have:

= 16 018,07 + (0,125 16 018,07) = 16 018,07 + 2 002,26 = R18 020,33

Isn’t there a shorter way of doing this??? NO But a shorter way of writing it is:

1,125 R10 000 = R11 250

1,125 R11 250 = R12 656,25, etc.

If you only want to know how much INTEREST she received: Method 1: 1 250 + 1 406,25 + 1 582,03 + 1 779,79 + 2 002,26 = R8 020,33 Method 2: R18 020,33 – R10 000 = R8 020,33

Compound Interest: Depreciation

The value of some items assets will depreciate every year, e.g. the value of your car, lawn mower, computer, etc. Example: Susan buys a car for R205 000 in 2014. How much will her car be worth in 2017 if it loses 18% of its value every year? 100% - 18% = 82%

Value of car at the end of the 2014:

= 205 000 – 0,18 205 000 OR 0,82 205 000 = R168 000 = R168 100


= 168 100 – 0,18 168 100 OR 0,82 186 100 = R137 842 = R137 842


= 137 842 – 0,18 137 842 OR 0,82 137 842 = R75 609,52145 = R75 609,52145


= 75 609,52145 – 0,18 75 609,52145 OR 0,82 75 609,52145

R61 999,81 R61 999,81


Part G: Questions


1.1 Geraldine owns a printer. She bought it for R1 289 in 2012. The value of her printer depreciates with 20% each year. What is her printer worth in 2014? (5)

1.2 Zinzi invests R450 000 in a high interest investment account. She receives 0,8%

interest per month, compounded monthly. How much money will she have in total after 4 months? (6)


Thabile borrows R5 000 from a bank. As she only has a small monthly income she chose to repay her loan over the maximum period of 60 months. Interest of 0,8% is charged per month. Below is a spreadsheet that shows some of the information for repaying her loan:

Month Opening balance

Interest Balance + Interest

Payment Closing balance

(Rand) (Rand) (Rand) (Rand) (Rand)

1 14 582.66 116.66 14 699.32 307.00 14 392.32

2 14 392.32 115.14 14 507.46 307.00 14 200.46

3 A B C D E

4 112.06 14 119.12 307.00 13 812.12

… … … … … …

57 F 9.62 1 211.67 307.00 904.67

58 904.67 7.24 911.91 307.00 604.91

59 604.91 4.84 609.75 307.00 302.75

60 302.75 2.42 305.17 305.17 0.00

2.1 How much interest will she pay in total over the 60 month period? (2) 2.2 Determine the values of A, B, C, D and E in the table. Show all you calculations.

You may round off each answer to two decimal places. (8) 2.3 Calculate the value of F. Show all calculations. (3)



Jane, Meryl, Caren, Paula and Helen start a Stokvel. They each invest R500 per month. The money is put into a bank account where they receive 2,15% interest per month, compounded monthly. Below is a spreadsheet that shows some of the information for their investment:

Table 1: Stokvel account Table 2: Individual’s account


Interest Closing balance


Interest Closing balance

(Rand) (Rand) (Rand) (Rand) (Rand) (Rand)

1 2 500.00 53.75 2 553.75 1 500 10.75 510.80

2 2 553.75 54.91 2 608.66 2 510.80 10.98 521.70

3 2 608.66 56.09 2 664.74 3 521.70 11.22 532.90

4 2 664.74 57.29 2 722.03 4 532.90 11.46 544.40

… … … … … ... … …

34 5 044.36 108.45 5 152.82 34 1 009 21.69 1 031

35 5 152.82 110.79 5 263.60 35 1 031 22.16 1 053

36 5 263.60 113.17 5 376.77 36 1 053 22.63 1 075

3.1 Verify whether the interest of R54,91 was correctly calculated for month 2 for

the Stokvel account. (3) 3.2 How much interest did the Stokvel account receive after 36 months? (2) 3.3 How much interest did the individual’s account receive after 36 months? (3) 3.4 Write the interest received by the individual (answer of 3.3) as a ratio of the

interest received by the Stokvel account (answer of 3.2). Write your answer in the form 1: ___ and round your answer to the nearest whole number. (2)

3.5 Use your answer in 3.4 to motivate why it is much better to invest in a Stokvel

account than in an individual’s account. (2) 3.6 Name one disadvantage of a Stokvel. (2) 3.7 Show, with calculations, how much the Stokvel account will have after 38

months. You may round off the answer of each calculation. (6)


Part G: Memo

Question 1 [11 marks] 1.1 Value of car at the end of the 2012:

= 1 289 – 0,2 1 289 OR 0,8 1 289

(M-) (M, 20% of) (M,80%) (M,) = R1 031,20(CA) = R1 031,20(CA)


= 1 031,20 – 0,2 1 031,20 OR 0,8 1 031,20 = R824,96(CA) = R824,96(CA)


= 824,96 – 0,2 824,96 OR 0,8 824,96

R659,97(CA) R659,97(CA) (5)

1.2 At the end of the 1st month she will have:

450 000 + (0,008 450 000) OR 1,008 450 000

(M+) (M) (M, 1,008) (M) = R453 600(CA)

At the end of the 2nd month she will have:

453 600 + (0,008 453 600) OR 1,008 453 600 = R457 228,80(CA)

At the end of the 3rd month she will have:

457 228,80 + (0,008 457 228,80) OR 1,008 457 228,80 = R460 886,6304(CA)

At the end of the 4th month she will have:

460 886,6304 + (0,008 460 886,6304) OR 1,008 460 886,6304

R464 573,72(CA) (6)


Question 2 [13 marks] 2.1 14 582,66 – 5 000(M) = R9 582,66(CA) (2) 2.2 A = R14 200,46(A)

B = 0,8 100 14 200,46(M) = 113,60368

R113,60(CA) C = 14 200,46 + 113,60(M) = R14 314,06(CA) D = R307(A) E = 14 314,06 – 307(M) = R14 007,06(CA) (8) 2.3 R1 211,67 100,8% ? 100%

1 211,67 100,8(M) 100(M) = R1 202,05(CA) (3) OR 0,8% of F = 9,62

F100

0,8 = 9,62 [100(M)]

0,8 F = 962

F = 962 0,8(M) = R1 202,05(CA) (3)


3.1 2,15 100 2 553,75(M) = 54,905625

R54,91(CA) Correct(CA) (3) 3.2 5 376,77 – 2 500(M) = R2 876,77(CA) (2) 3.3 1 075 – 500(M) = R575(CA) (2) 3.4 575 : 2876,77(CA) = 1 : 5(CA and R) (2) 3.5 You can get 5 times more interest with

the Stokvel account than with an individual’s account.(J) (2)

3.6 You must set up a contract so that one

person doesn’t disappear with all the money.(J) (2)

OR Any appropriate reason. 3.7 Month 37

2,15 100 5 376,77(M) = R115,60 5 376,77 + 115,60(M) = R5 492,37(CA) Month 38

2,15 100 5 492,37(M) = R118,09 5 492,37 + 118,09(M) = R5 610,46(CA) (6)


Part H: 2- and 3-dimensional shapes

All formulae

Perimeter

Square: P = 4 side

Rectangle: P = 2(B + L) OR P = 2B + 2L

Triangle: P = a + b + c, where a, b and c are the three sides of the triangle

Circle: C = 2 r, where = 3,142 OR C = d

Any shape: P = add all the exterior sides together

Area

Square: A = side2

Rectangle: A = B L

Triangle: A = ½ b h, where b = base and h = perpendicular height

Circle: A = r2, where = 3,142

Total Surface Area

Cube: TSA = 6 side2

Rectangular prism: TSA = 2 (B L) + 2 (B H) + 2 (L H)

Cylinder: TSA = 2 ( r2) + (2 r) H

Volume

General formula: V = base area height

Cube: V = side3

Rectangular prism: V = B L H

Cylinder: V = r2 H, where = 3,142


Total Surface Area

The difficulty in this topic is: Finding unknown sides when percentage and direct proportion is involved

Knowing which formula is the correct one to use

Knowing which numbers to substitute into the formula Example: Michael bought his teacher a bell. It is a metal bell and it was packaged in a thick clear plastic box. The picture is shown below:

The bottom of the bell is a circle with a diameter of 75mm. The length of the side of the base of the box is 1% larger that the diameter of the bell. The height of the box is 134mm and the height of the bell is 2,5% less than the height of the box. (a) Calculate the length (in mm) of the side of the bottom of the box. (2)

Answer: Side length = 101% of the diameter [OR 0,01 75 = 0,75

= 1,01 75 75 + 0,75 = 75,74mm] = 75,75mm

(b) Calculate the height (in mm) of the bell. (3)

Answer: 134mm 102,5% ? 100% [The original one (smaller) is 100%]

134 ÷ 102,5 100 = 130,7317…

130,73mm

Length of side

Length of side

Height

Height

Squared base

diameter


Below is a picture of the net of the box (opened up):

Formulas given: Area of square = side2

Area of rectangle = breadth length

Area of circle = radius2, where = 3,142 Area of open box = 4A + 2B + 4C + D + E +2F + G + H Other information given: Area of 1D = 130mm2 Area of 1F = 975mm2

Area of 1G = 450mm2

Area of 1H = 2 062,5mm2 1cm2 = 100mm2 1m2 = 1 000 000mm2

(c) How much plastic (in mm2) is needed to make one box? (11)

Answer: Area of A

= side length height

= 75,75 134 = 10 150,5mm2

A A A A

B

B

C C

C C

D

E

F F

G

H

Side length

he

igh

t

Side length Side length Side length

Diameter of E = 2,5cm

3,5cm


Area of B = side2

= 75,752 = 5 738,0625mm2 Area of C

= side length 35mm

= 75,75 35 = 2 651,25mm2 Area of E

= ½ r2

= ½ 3,142 37,52 = 2 209,21875mm2 Total area = 4A + 2B + 4C + D + E +2F + G + H = 4(10 150,5) + 2(5 738,0625) + 4(2 651,25) + 130 + 2 209,21875 +2(975) + 450 + 2 062,5

69 484,84mm2

(d) This particular plastic costs R9,50 per m2. Use the formula below to calculate the cost of making one box.

Formula: Cost of box (Rand) = R24,50 + area R9,50/m2 (3)

Answer: 69 484,84mm2 ÷ 1 000 000 = 0,06948484m2 Cost of box

= R24,50 + area R9,50/m2

= 24,50 + 0,06948484 9,50

R10,37


Part H: Questions


A Purity cereal box has the following dimensions: Length = 14,5cm; breadth = 5cm and height = 193mm

Inside the box there is a foil bag containing the cereal. The length of the bag is 29

28 of the

length of the box. The height of the bag is 7% shorter than the box. The breadth of the bag at the bottom is 4cm.

Below is a picture of the net of the box (opened up):

Formulas given:

Area of rectangle = breadth length Area of open box = 2(A + B) + 4(C + D) + E

Volume of rectangular prism = B L H

CB

D

A A B B

he

igh

t

length breadth

CB

CB

CB

D

D D

E

length

breadth

height


Other information given: Area of 1C = 43,5cm2 1cm2 = 100mm2 Area of 1D = 12,5cm2 1m2 = 1 0 000cm2

Area of 1E = 11,6cm2

1.1 How much cardboard (in cm2) is needed to make one box? (7)

1.2 This particular cardboard is cut from a sheet that is 90cm by 90cm. How many boxes will you be able to cut out of this sheet? Assume C is 3cm wide and E is 1cm wide. (6)

1.3 The sheet of cardboard costs R5/m2. Use the formula below to calculate the cost of making the number of boxes that you calculated in 1.2.

Formula: Cost of box (Rand) = R17,20 + area R5/m2 (4)

1.4 In order for the company to save money, the difference between the volume of the box and the volume of the foil bag must between 350cm3 and 400cm3

Assume that the foil bag is a rectangular prism. Verify, by showing calculations, whether this cardboard box and foil bag satisfies the guidelines. (12)


Cedric buys his son a cylindrical “Retro addition” Smartie box at the Dubai airport. The measurements of the box are as follow:

H = 5mm

d = 48mm Plastic Lid

Sm

ar

tie

s

Re

tr

o A

dd

itio

n

H = 248mm

d = 4,7cm

d = 47mm

F F

F

F

F F

F

F

Bottom of box


The cylindrical box is made from cardboard and has a height of 248mm and a diameter of 4,7cm.

At the bottom of the cylinder is a circle with 8 identical flaps. The diameter of the bottom is 47mm. At the top the cylinder has a blue plastic lid which is also a cylinder.

The lid is open at the bottom. The plastic lid has a diameter of 48mm and a height of 5mm.

Formulae given:

= 3,142

Area of cylinder that is open on both sides = 2 radius Height

Area of a circle = radius2

Area of ONE flap (F) = 8

1 d height, where the height of each flap is 10mm

Area of cylinder that is open on ONE side = ( radius2) + (2 radius Height) Conversions given: 1cm2 = 100mm2 1m2 = 1 000 000mm2 2.1 How much cardboard (in mm2) is used to make one Smartie box? This includes

the cylinder (“body” of the box) as well as the bottom of the box. (11)

2.2 How much plastic (in mm2) is used to make one Smartie box’s lid? (3)

2.3 Calculate the cost (in AED – United Emirates Dirham) for making one box if the following formula is used to work out the cost

Formula: Cost (AED) = AED 2 + (area of cardboard in m2 AED 0,98/m2)

+ (area of plastic in m2 AED 1,25/m2) (4)


Mrs Fourie and her waitresses are helping out at the derby day at the school. They need to work out how many cups of juice or hot chocolate they can take in one go to the players after each game.

3.1 The circular tray has a diameter of 30cm. Calculate the circumference of your

tray. Formula given: C = 2 r, where = 3,142 (3)

4cm


3.2 They pack the styrofoam cups along the edge of the circular tray, as well as some along the imaginary interior circle (circle with dotted line in the sketch below). The sketch below shows just some of the cups that are placed on the tray:

Calculate how many Styrofoam cups can placed on the tray if the inner circle

has 5 cups. Use 94,26cm as the circumference of the tray. Round your answer off to the nearest cup. (5)

3.3 A Styrofoam cup consists of a small circular base and larger circular top.

Formula: Volume = r2 height, where = 3,142

(a) Calculate the volume of the cylinder if the base has a radius of 2cm and the height of the cup is 9,5cm. (2)

(b) Calculate the volume of the cylinder if the top circular opening has a radius of3cm and the height of the cup is 9,5cm. (2)

(c) The volumes are 119,4ml for the small cylinder and 268,64ml for the larger cylinder. Use the formula below to calculate the volume of the cup. Formula: volume small cylinder +[(volume large cylinder – volume small cylinder) x 0,5] (3)

3.4 Calculate how many litres of water you need to buy if your cup can hold roughly 194ml and you can fit 20 cups on a tray. (4)

3.5 You are mixing hot chocolate in your cups. You put 4 teaspoons of cocoa, 10 ml milk and 160 ml hot water in each cup.

a) Write a simplified ratio for cocoa : milk : hot water (2) b) What volume for the cup is not being used if a cup can hold 250ml? (3) c) If you were given 375ml of cocoa, how much milk and water will you

need? (3)

6 cm

6 cm

6 cm


Part H: Memo

Question 1 [34 marks] 1.1 Area of A

= height length

= 19,3 14,5(M)(C) = 279,85cm2 (CA) Area of B

= height breadth

= 19,3 5(M) = 96,5cm2 (CA) Total area = 2(A + B) + 4(C + D) + E = 2(279,85 + 96,5) + 4 (43,5 +12,5 ) + 11,6(SF) = 988,3cm2

(CA) (7) 1.2 In the length = 90cm ÷ (1 + 14,5 + 5 + 14,5 + 5)(M÷)(M+) = 90cm ÷ 40 = 2,25

2 boxes(CA and R) In the breadth = 90cm ÷ (3 + 19,3 + 3)(M÷)(M+) = 90cm ÷ 25,3 = 3,557…

3 boxes(CA and R)

In total 2 3(M) = 6(CA) fruit juice boxes can be cut from 1 sheet. (6)

1.3 Cost of box

= R17,20 + 0,09883m2 R5/m2(SF) (C)

= R17,69415

6 R17,69415(M)

R106,16(CA) (3) 1.4 Length of the bag

= 29

28 14,5(M)

= 14cm(CA)

Height of the bag 19,3cm 107% ? 100%

19,3 ÷ 107(M) 100(M) = 18,03738318cm(CA)

Volume of the bag

= B L H

= 4 14 18,03738318(M) = 1 010,093458cm3

(CA)

Volume of the box

= B L H

= 5 14,5 19,3(M) = 1 399,25cm3

(CA)

Difference = 1 399,25cm3 – 1 010,093458cm3

(M)

389,16cm3(CA)

Yes, it does satisfy the guidelines. (CA) (12)


Question 2 [18 marks] 2.1 CARDBOARD d = 4,7cm = 47mm(C) r = 47 ÷ 2 = 23,5mm(A,r)

Area of Body = cylinder open on both sides

= 2 radius Height

= 2 3,142 23,5 248(SF) = 36 623,152mm2 (CA) [36 618,40397]

Area of circle = ( radius2)

= (3,142 23,52) (SF) = 1 735,1695mm2 (CA) [1 734,944543]

Area of 8 flaps = 8 (8

1 d height)

= 8 (8

1 3,142 47 10) (M,8) (SF)

= 1 476,74mm2 (CA) [1 476,548547]

Total area = 36 623,152 + 1 735,1695 + 1 476,74(M)

39 835,06mm2 (CA) [39 829,90] (11)

2.2 PLASTIC Area of open cylinder (one side)

= ( radius2) + (2 radius Height)

= (3,142 242) + (2 3,142 24 5) (A,r)(SF) = 2 563,97mm2 (CA) [2 563,54] (3)

2.3 CONVERT: Cardboard: 39 835,06mm2 ÷ 1 000 000 = 0,03983506m2 (C) [0,03982990]

Plastic: 2 563,97mm2 ÷ 1 000 000 = 0,00256397m2 (C) [0,00256354] Cost = AED 2

+ (A of cardboard in m2 AED 0,98)

+ (A of plastic in m2 AED 1,25)

= 2 + (0,03983506 0,98)

+ (0,00256397 1,25) (SF)

AED 2,04(CA) (4)


3.1 r = 30 2 = 15cm(A)

C = 2 3,142 15(SF) = 94,26cm(CA) (3)

3.2 (94,26 6)(M)(A,6)

= 15,71 15(CA) 15 + 5(M) = 20 cups(CA) (5)

3.3

(a) V = 3,142 22 9,5(SF) = 119,396

119,4cm3(CA and R) (2)

(b) V = 3,142 32 9,5(SF) = 268,64cm3

(CA and U) (2)

(c) V = 119,4 + [(268,64 – 119,4)0,5] (SF) (SF) = 194,02ml (CA) (3)

3.4 20 194(M) = 3 880ml(CA)

3 880 1 000(C) = 3,88L(CA and U) (4)

3.5 (a) C : M : H20 = 20ml : 10ml : 160ml (M) = 2 : 1 : 16 (CA) (2) (b) 5 + 10 + 160(M) = 175

250 – 175 (M) = 75ml(CA) (3)

(c) 1 part = 375 2 = 187,5ml(CA) 187,5ml milk(CA)

187,5 16 = 3 000ml water(CA) (3)


Part I: Finance – Currency and Exchange Rates

Foreign Exchange

Exchange rate tables (excluding exchange fees) (The following information was obtained from ABSA’s website on 23 July 2014)

Table 1: EXCHANGE RATE OF RAND PER UNIT FOREIGN CURRENCY

Bank Buying Bank selling

Transfers Cheques Notes Transfers &

Cheques Notes

US Dollar ($) 10,3212 10,3012 10,2962 10,6482 10,6532

Euro (€) 13,9056 13,8726 13,7826 14,3549 14,3874 British Pound (£) 17,6064 17,5764 17,4864 18,1229 18,2129

* “Unit foreign currency” means 1 of the other currency, not R1, e.g. $1.

Table 2: EXCHANGE RATE OF FOREIGN CURRENCY PER RAND UNIT

Bank Buying Bank selling

Transfers Cheques Notes Transfers

& Cheques

Notes

AED United Arab Emirates Dirham

0,0 0,0 0,3804 0,3406 0,3166

AUD Australian Dollar 0,1044 0,1054 0,1064 0,0974 0,0964

BWP Botswana Pula 0,8832 0,8852 0,8892 0,7934 0,7504

CNY China Yuan Renminbi 0,6358 0,0 0,6558 0,5436 0,5236 MWK Malawian Kwacha 0,0 0,0 0,0 34,9683 34,8933

NGN Nigerian Naira 0,0 0,0 0,0 13,81 0,0 Source: http://www.absa.co.za/Absacoza/Indices/Absa-Exchange-Rates; 23 July 2014

You must always look at a problem from the BANK’s perspective and in terms of the foreign currency:

I have I want The bank does this with

the foreign currency Which column do I use?

Rand Foreign currency,

e.g. Euro The bank SELLS Euro to

me Bank Selling (light grey)

Foreign currency, e.g. Euro

Rand The bank BUYS the foreign

currency back from me Bank Buying (dark grey)

In other words: (a) If I have Rand the bank SELLS foreign currency to me. (b) If I have foreign currency the bank BUYS foreign currency from me

http://www.absa.co.za/Absacoza/Indices/Absa-Exchange-Rates


Example 1: I have R30 000 that I want to exchange for US Dollars via an electronic transfer. How many Dollars will I receive?

THINK I have I want The bank does this Which column do I use?

Rand US Dollar The bank SELLS US Dollar to me

Bank selling (light grey)

DO

Step 1 Find the value of $1 in table 1 in the “bank selling” section and at “transfer”

Step 2 Write down information as direct proportion

Step 3 Do calculation and find final answer $1 R10,6482 $? R30 000

30 000 ÷ 10,6482 [OR 1 ÷ 10,6482 30 000]

$2 817,38 Example 2: I have AED 1 530 that I want to exchange back to Rand in notes. How many Rand will I receive?

THINK I have I want The bank does this Which column do I use?

AED Rand BUYS AED back from me Bank buying (dark grey)

DO

Step 1 Find the value AED for R1 in table 2 in the “bank buying” section and at “notes”

Step 2 Write down information as direct proportion Step 3 Do calculation and find final answer

R1 AED 0,3804 R? AED 1 530

1 530 ÷ 0,3804 [OR 1 ÷ 0,3804 1 530]

R4 022,08


Calculating the Exchange Fee The big question is: Do I subtract the fee before or after converting currency? Answer: The exchange fee is ALWAYS subtracted from the RAND amount.

I have I want Do

Rand Foreign currency, e.g. Euro

Subtract exchange fee 1st from Rand and then convert to Euro

Foreign currency, e.g. Euro

Rand Convert to Rand 1st and then subtract exchange fee

Exchange fee table

Buying Foreign Currency Selling Foreign

Currency

Transaction Type Fee Min. Cost

Additional tele-communication fees

Fee Min. Cost

Foreign Notes 1,95% R70 1,95% R70 Travellers’ Cheques 1,65% R65,50 1,65% R65,50

T/T (Electronic payments)

0,35% R120 R75

Type 1: I have Rand (and I know how much)

THINK I have I want

Rand Foreign

DO

Step 1 Calculate fee (%) of Rand amount

Step 2 Subtract fee from Rand amount

Step 3 Convert money

Example: I want to exchange R16 000 to American Dollar via a transfer. How many American Dollar ($) will I receive after deducting the exchange fee? Think I want to change Rand to Dollar (Transfer). The bank sells me Dollars (Light

grey). Remember to first subtract the fee. Do Exchange fee

1,85% of R16 000

= 1,85 ÷ 100 16 000

Rand left over = R16 000 – R296 = R15 704

Exchange to Dollar $1 R10,6482 $? R15 704

15 704 ÷ 10,6842 $1 474,80


Type 2: I have foreign currency (and I know how much)

THINK I have I want

Foreign Rand

DO

Step 1 Convert money to Rand

Step 2 Calculate fee (%) of Rand amount Step 3 Subtract fee from Rand amount

Example: I have €720 and I want to exchange it to Rand via notes. How many Rand can I get after the exchange fee has been deducted?

Think I want to change Euro to Rand. The bank buys Euro from me Euro. Dark grey Remember to subtract the fee at the end.

Do Exchange to Rand Exchange fee Rand left over €1 R13,7826 1,85% of R9 923,472 9 923,472 – 183,58…

€720 R? = 1,85 ÷ 100 9 923,472 R9 739,89

720 13,7826 = R183,584232 = R9 923,472

Type 3: I have Rand (and I DON’T know how much)

THINK I have I want

Rand (but I don’t know how many Rand I have) Foreign

DO Step 1 Convert money to Rand Step 2 Calculate fee (%) of Rand amount

Step 3 ADD fee from Rand amount Example 3: I want to exchange Rand to $100 via a transfer. How many Rand (excluding the exchange fee) do I need to exchange to get $100?

Think I want to change Rand to Dollar. The bank sells me Dollars. Light grey Remember to ADD the exchange fee.

Do Exchange to Rand Exchange fee Rand left over $1 R10,6482 1,85% of R1 064,82 1 064,82 + 19,69917

$100 R? = 1,85 ÷ 100 1 064,82 R1 084,52

100 10,6482 = R19,69917 = R1 064,82


What does the bank do with the foreign currency?

The bank buys foreign currency from me

The bank sells foreign currency to me

I Have: Foreign I want: Rand

I Have: Rand I want: Foreign

Exchange money

BANK SELLING BANK BUYING

I Have: Foreign I Have: Rand

Exchange fee

STEP 1: Calculate fee

STEP 2: Subtract fee from Rand amount

STEP 3: Exchange to foreign

STEP 1: Exchange to Rand

STEP 2: Calculate fee from Rand amount

STEP 3: Subtract fee


Part I: Questions


Use the exchange rate tables in the notes as well as the table below (showing the exchange fees charged) to answer question 1.1 – 1.6.

Transaction type

Buying Foreign Currency Selling Foreign

Currency

fee Min. Additional

Telecommuni-cation fees

Fee Min.

Foreign notes 1,85% R70,00 1,85% R60,00

Travellers cheque 1,75% R57,50 1,75% R57,50

T/T and electronic payments

0,45% R130,00 R95

1.1 Megan wants to exchange Rand for €15 000 (electronic transfer). Determine how many Rand she needs. No exchange fees are charged. (2)

1.2 Xolani just came back from his holiday in New York. He wants to exchange $45(traveller’s checque) for Rand. How many Rand will he receive? No exchange fees are charged. (2)

1.3 Jannie has R20 000 (cash) that he wants to exchange to €. He is going to visit his sister, who lives in London, next week. How many Euros will he receive after exchange fees have been deducted? (5)

1.4 Lynne wants to exchange AED400 000 notes to Rand. How many Rand does she receive after the exchange fees have been deducted? (5)

1.5 Paul has 50 000 Australian Dollar that he want to exchange to Rand via an electronic transfer. How much money (R) will he have after exchange fees have been deducted? (6)

1.6 Zaheerah has an amount of Rand that she wants to exchange to 20 000 Botswana Pula via a traveller’s cheque. Exchange fees are deducted. How many Rand does she have? (5)


Part I: Memo Question 1 [25 marks] 1.1Has: Rand (and don’t know amount) Want: Foreign Bank selling (Transfer) €1 R14,3549 €15 000 ?

15 000 14,3549(RT and M) = R215 323,50(CA) (2)

1.2 Has: Foreign; Want: Rand Bank buying (Traveller’s cheque) $1 R10,3012 S45 ?

45 10,3012(RT and M)

R463,55(CA) (2)

1.3 Has: Rand (and know the amount) Want: Foreign Bank selling (cash) Fee = 1,85% of R20 000

= 1,85 ÷ 100 20 000(M) = R370(CA)

Rand amount after fee is subtracted R20 000 – R370 = R19 630(CA) €1 R14,3874 ? R19 630 19 630 ÷ 14,3874(RT and M)

[OR 1 ÷ 14,3874 19 630]

€1 364,39(CA) (5)

1.4 Has: Foreign; Want: Rand Bank buying (notes) Exchange R1 AED0,3804 ? AED400 000

400 000 ÷ 0,3804(RT and M)

[OR 1 ÷ 0,3804 400 000] = R1 051 524,711(CA) Fee = 1,85% of R1 051 524,711

= 1,85 ÷ 100 1 051 524,711(M) = R19 453,20715(CA)

Deduct fee 1 051 524,711 - R19 453,20715

R1 032 071,50(CA) (5) 1.5 Has: Foreign; Want: Rand Bank buying (electronic transfer)

Exchange R1 AUD0,1044 ? AUD50 000 50 000 ÷ 0,1044(RT and M)

[OR 1 ÷ 0,1044 50 000] = R478 927,2031(CA)

Fee = 0,45% of R478 927,2031

= 0,45 ÷ 100 478 927,2031(M) = R2 155,172414(CA)

Deduct fee 478 927,2031 – R2 155,172414 – R95(M)

R476 677,03(CA) (6)

1.6 Has: Rand (and she doesn’t know the amount) Want: Foreign Bank selling (travellers cheque)

Exchange R1 BWP0,7934 ? BWP20 000 20 000 ÷ 0,7934(RT and M)

[OR 1 ÷ 0,7934 20 000] = R25 207,96752(CA)

Fee = 1,75% of R25 207,96752

= 1,75 ÷ 100 R25 207,96752(M) = R411,1394001(CA)

ADD fee R25 207,96752 + R411,1394001

R25 649,11(CA) (5)


Part J: Probability

Probability: Lottery

The national lottery in South Africa is called Lotto. Winning Categories & Numbers

1st Category 6 Correct Numbers

2nd Category 5 Correct Numbers + Bonus Ball

3rd Category 5 Correct Numbers

4th Category 4 Correct Numbers + Bonus Ball

5th Category 4 Correct Numbers

6th Category 3 Correct Numbers + Bonus Ball

7th Category 3 Correct Numbers Winning Categories & %Distribution of winnings

1st Category 18.25%

2nd Category 4.00%

3rd Category 9.00%

4th Category 5.00%

5th Category 16.75%

6th Category 11.00%

7th Category 36.00% Source: https://www.nationallottery.co.za/lotto_home/how_to_play.asp

How does the Lotto work?

They spin the large see-through ball. In the large ball there are the numbers 1 to 49.

Before any ball fall out, there are 49 BALLS and you need 6 CORRECT NUMBERS TO WIN.

Therefore the probability to get the first ball correct is 49

6

Reason: There are 6 possible correct numbers that you can get, so you have ”6 chances” out of a total of 49 balls.

When they press the button, the 1st ball falls out. One correct number has been drawn, so now only 48 balls and 5 correct numbers remain.

The probability of getting the second ball right is now 48

5

When they press the button, the 2nd ball falls out. TWO correct numbers have been drawn and only 47 balls and 4 correct numbers remain.

The probability of getting the 3rd ball right is now 47

4

https://www.nationallottery.co.za/lotto_home/how_to_play.asp


When they press the button, the 3rd ball falls out. THREE correct numbers have been drawn and only 46 balls and 3 correct numbers remain.

The probability of getting the 4th ball right is now 46

3

When they press the button, the 4th ball falls out. FOUR correct numbers have been drawn and only 45 balls and 2 correct numbers remain.


2

When they press the button, the 5th ball falls out. FIVE correct numbers have been drawn and only 44 balls and 1 correct numbers remain.


1

Now let’s put everything together: The probability of winning the Lotto

= 44

1

45

2

46

3

47

4

48

5

49

6

= 0,00000007151123842

0,0000072% (Not a very good chance!)

General Formula for Probability

The probability that an event will happen is:

P(event) = eventtheforoutcomespossibletotal

happencaneventanwayspossibleofnumber

Example: I throw a die. What is the probability that it will land with a 3 on top? Answer:

The total possible outcomes are 6 because the die has six sides

The no. of possible ways the event can happen is 1 because there is only ONE 3 on the die.

Therefore P(3) = 6

1


Probabilities of Multiple Events

In probability we use the word OR when we have ONE object.

For example: you throw ONE die. The probability it will either land with 2 on top OR land

with 6 op top, is: P(2 or 6) = 6

1 + 6

1 = 6

2 = ⅓

“OR” means you must ADD. In probability we use the word AND when we have more than one object.

For example: you throw TWO dice. The probability one die will land with 2 on top AND

the other die will land with 6 op top, is: P(2 and 6) = 6

1 6

1 = 36

1

“AND” means you must MULTIPLY.

Important to note: Normally “and” means + but in probability “and” means

Drawing TWO cards (putting it back) I have a pack of cards. I draw one card, REPLACE it in the pack and then draw another card. What is the probability that I will draw a 6 od hearts and then any diamond?

P(6 and ) = 208

1

52

13

52

1

Note that both fractions is out of 52. I have 52 cards; I draw one card and put it back; therefore I have 52 cards again.

There are 13 diamonds because the first card we draw is a heart; we haven’t touched the diamonds; so there are still 13 diamonds left in the pack with 51 cards.

Drawing TWO cards (NOT putting it back) I have a pack of cards. I draw one card, DON’T REPLACE it in the pack and then draw another card. What is the probability that I will draw a 6 od hearts and then any diamond?

P(6 and ) = 204

1

51

13

52

1

Note that the first fraction is out of 52 and the second fraction out of 51. I have 52 cards; I draw one card and DON’T replace it; therefore I have 51 cards left to draw the second card from.


Tree diagrams

A tree diagram is a picture (in the shape of a tree with branches) that shows all the possible outcomes of an event.

Each new “branch” of the tree is a new event happening.

Example: There are 5 red balls, 6 blue balls and 2 green balls in a bag. One ball is drawn and not replaced. Then a second ball is drawn from the bag.

(a) Draw a tree diagram of the situation.

(b) How many outcomes are there? List them all. (R = red, B = blue and G = green) There are 9 outcomes: RR, RB, RG, BR, BB, BG, GR, GB and GG (c) What is the probability that the first ball that you will draw is blue? P(B) = 6/13 (d) What is the probability that I will draw a red and a green ball IN THAT ORDER? Write

your answer as a percentage. P(R and G) = 13

5

12

2 =

78

5;

78

5 100 6,41%

Start

Red

6

12

Blue

Green

Red

Blue

Green

Red

Blue

Green

Red

Blue

Green

13

5

13

2

12

2

12

5

12

5

12

2

12

5

12

1

13

6

12

4

6

12


(e) What is the probability that I will draw a red and a green ball IN ANY ORDER?

P(R and G OR G and R) =

12

5

13

2

12

2

13

5 =

39

5

(Do the whole calculation in one step on your calculator.) (f) What is the probability that I will draw two balls of the SAME COLOUR? Write your

answer as a decimal fraction.

P(RR or BB or GG) =

12

1

13

2

12

5

13

6

12

4

13

5=

3

1 0,33

(g) What is the probability that I will draw two balls of DIFFERENT COLOURS? Write

your answer as a fraction. P(1 – same colour) = 1 – 3

1 =

3

2

Two-way tables

Instead of writing the information on a tree diagram we write it in a table.

This table is called a two-way table because you can read it in two different ways.

For example: I toss two coins and all the possible outcomes are…

Head

Head

Tail

Tail

HH

TH

HT

TT

Coin 1

Coin 2


Part J: Questions


1.1 The school launches their own Lottery game to collect money for revamping the teachers’ parking area. In this game there are 30 numbers (1 to 30). To win the game someone has to guess 4 correct numbers. The 30 numbers are placed in a bag, shaken and then the first number is drawn and not replaced. What is the probability that Sharon will win the game? (Assume she bought a ticket! ) Write your answer as a percentage and round it off to the nearest percentage. (5)

1.2 According to Statistics South Africa 1,7% of household expenditure in

2005/2006 contributed to health. The table below shows the annual household consumption expenditure on private medical services in South Africa.

Expenditure item Black / African

Co-loured

Indian / Asian

White

Pharmaceutical products 33,4% 40,3% 38,2% 34,5%

Other medical products 1,3% 1,5% 1,5% 1,0% Therapeutic appliances and equipment 1,6% 3,8% 4,1% 3,5% Medical services 50,1% 33,6% 22,5% 31,4%

Dental services 2,4% 3,0% 4,4% 8,6%

Paramedic services 2,6% 6,1% 20,9% 8,0% Hospital services 8,6% 11,8% 8,4% 13,1%

Source: Stats SA; Statistical release P0100 (income and expenditure on households; 2005/2006; p.17

What is the probability that a coloured person will spend money on paramedic services? Write your answer as a fraction. (3)

1.3 Lotto results – 12 February 2011

8 12 25 35 46 47 +39 Prize division No. of winners Amount (per person)

Six numbers 0 Five numbers + bonus 2 R161 798

Five numbers 58 R12 553

Four numbers + bonus 158 R2 560 Four numbers 3 424 R395

Three numbers + bonus 4 758 R187 Three numbers 70 937 R41


(a) How many Lotto winners were there in total? (2) (b) Calculate the total amount of money that was won this week. (2) (c) What do you notice about the number of winners and the amount received

per person if you look from the top to the bottom of the table?

(2) (d) If the company realise that they have made a mistake and there were actually

59 winners in the “five numbers” prize division, how much would each winner now get?

(3)


A bag contains four red balls, six blue balls and three green balls. A ball is drawn, and not replaced and then another ball is drawn again. 2.1 What is the total number of balls in the container? (2)

2.2 Find the values of missing probabilities A, B and C. (3)

2.3 Use the tree diagram to determine the probabilities of the following:

(a) Drawing a blue ball in the first draw (2)

(b) Drawing a red ball and then a green ball (2)

(c) Drawing two balls of the same colour (4)

(d) Drawing a black ball (2)

Start

4

13

6

13

3

13

Red

A

6

12

3

12

Blue

4

12

B

3

12

Green

4

12

6

12

C

Red

Blue

Green

RR

RB

RG

BR

BB

BG

GR

GB

GG

Red

Blue

Green

Red

Blue

Green


Question 3 [10 marks] Jessica has four pairs of high heeled shoes: black, grey, red and tan coloured. She has three jackets: black, red and denim. The two-way table below shows the options of wearing a certain jacket and a certain pair of shoes together.

BS GS RS TS BJ (BS,BJ) (GS, BJ) (RS, BJ) (TS, BJ)

RJ (BS, RJ) (GS, RJ) (RS, RJ) (TS, RJ) DJ A (GS, DJ) (RS, DJ) (TS, DJ)

KEY Shoes Jackets

BS = black shoes GS = grey shoes RS = red shoes TS = tan shoes

BJ = black jacket RJ = red jacket DJ = denim jacket

3.1 Determine A. (2)

3.2 What is the probability that she will wear a denim jacket with black shoes? (2)

3.3 Determine the number of possible outfits that she can wear if she wanted to wear a jacket and shoes of the same colour. (2)

3.4 How many combinations does she have if she doesn’t want to wear anything that is red? (2)

3.5 Determine the probability that she will at least wear one black item. (2)


Part J: Memo

Question 1 [17 marks] 1.1 P(win)

= 27

1

28

2

29

3

30

4 (M)

= 0,0000364896166(CA)

0,0000364896166 100(M) = 0,00364896166%(CA)

0% (R) (5) 1.2 6,1%(RT,A) = 6,1/100(M) = 61/1 000(CA) (3) 1.3 (a) 2 + 58 + 158 + 3 424 + 4 758 + 70 937(M) = 79 337 winners(CA) (2)

(b) (R161 798 2) + (R12 553 58) + …(M)

= R6 606 793(CA) (2) (c) When the number of winners increase

the amount won per person decrease. (J)(J) (2)

(Zero or two marks: if they only say increase or only decrease, then no marks. Must say both.)

(d) 58 12 553(M) = R728 074

R728 074 59(M) R12 340,24 per person(CA+R) (3)

Question 2 [15 marks] 2.1 4 + 6 + 3(M) = 13(CA) (2)

2.2 A = 12

3(A) OR ¼

B = 12

5(A)

C = 12

2(A) OR

6

1 (3)

2.3

(a) 13

6(A) (2)

(b) 13

4

12

3(M)

= 13

1(CA) (2)

(c) (13

4

12

3)(M)

+ (13

6

12

5)(M)

+ (13

3

12

2)(M)

= 13

4(CA) (4)

(d) 0 or 0%(A) (2) (0/13 is wrong)

Question 3 [10] 3.1 (BS, DJ)(A) (2)

3.2 12

1(A) (2)

3.3 2(A) (2) 3.4 7(A) (2) 3.5 6/12(top)(bottom) (2) OR ½

grade 12 mathematical literacy spring book 2015 · notes: scales mind map; number scales and bar...

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