42510011 0010 1010 1101 0001 0100 1011

US5235Solving Number Solving Number

ProblemsProblems

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SIGNIFICANT FIGURESSIGNIFICANT FIGURES- Count from the first non-zero number

e.g. State the number of significant figures (s.f.) in the following: a) 7553b) 4.06c) 0.012

4 s.f.3 s.f.2 s.f.

Zero’s at the front are known as place holders and are not counted

- A way of representing numbers

DECIMAL PLACESDECIMAL PLACES- Count from the first number after the decimal point

e.g. State the number of decimal places (d.p.) in the following: a) 70.652 3 d.p.

- Another way of representing numbers

b) 0.021 3 d.p.c) 46 0 d.p.

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ROUNDINGROUNDING1. DECIMAL PLACES (d.p.)i) Count the number of places needed AFTER the decimal pointii) Look at the next digit

- If it’s a 5 or more, add 1 to the previous digit- If it’s less than 5, leave previous digit unchanged

iii) Drop off any extra digits

e.g. Round 6.12538 to:

a) 1 decimal place (1 d.p.) b) 4 d.p.

Next digit = 2= leave unchanged

= 6.1

Next digit = 8= add 1

= 6.1254

The number of places you have to round to should tell you how many digits are left after the decimal point in your answer. i.e. 3 d.p. = 3 digits after the decimal point.

When rounding decimals, you DO NOT move digits

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2. SIGNIFCANT FIGURES (s.f.)

i) Count the number of places needed from the first NON-ZERO digitii) Look at the next digit

- If it’s a 5 or more, add 1 to the previous digit- If it’s less than 5, leave previous digit unchanged

iii) If needed, add zeros as placeholders to keep the number the same size

e.g. Round 0.00564 to:

a) 1 significant figure (1 s.f.) a) 2 s.f.

Next digit = 2= leave unchanged

= 6.1

Next digit = 7= add 1

= 19

e.g. Round 18730 to:

000

Don’t forget to include zeros if your are rounding digits BEFORE the decimal point. Your answer should still be around the same place value

- ALWAYS round sensibly i.e. Money is rounded to 2 d.p.

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FRACTIONSFRACTIONS

1. SIMPLIFYING FRACTIONS

- Fractions must ALWAYS be simplified where possible

e.g. Simplify

a) 5 = 10

b) 6 = 9

c) 45 = 60

12

23

34

Make use of the a b/c button on your calculator

- When fraction has been entered into calculator, simply press equals

2. CALCULATIONS INVOLVING FRACTIONS

- Enter calculation as seen in question using the fraction button

a) 1 + 2 4 5

= 13 20

b) 1 1 2 2 3

2× = 4

e.g. Simplify

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3. MIXED NUMBERS and IMPROPER FRACTIONS- Mixed numbers are combinations of whole numbers and fractions.

e.g. Change into improper fractions:

a) 3 4

b) 1 3

4 = 19 4

6 = 19 3

- To change into an improper fraction use d/c button. (shift key and a b/c button)- Improper fractions have the top number bigger than the bottom.

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PERCENTAGESPERCENTAGES1. ONE AMOUNT AS A PERCENTAGE OF ANOTHER- A number of similar strategies such as setting up a fraction and

multiplying by 100 exist.

e.g. Paul got 28 out of 50. What percentage is this?100 ÷ 50 = 2 (each mark is worth 2%) 28 × 2 = 56%

e.g. Mark got 39 out of 50. What percentage is this?

39 50

× 100 = 78%

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- To calculate percentage increase/decrease we can use:

Percentage increase/decrease = decrease/increase × 100

original amount

e.g. Mikes wages increased from $11 to $13.50 an hour.

a) How much was the increase?

b) Calculate the percentage increase

13.50 - 11 = $2.50

2.50 11

× 100 = 22.7% (1 d.p.)

e.g. A car originally brought for $4500 is resold for $2800. What was the percentage decrease in price?

Decrease = $1700 = 4500 - 2800 Percentage Decrease = 1700

4500 × 100

= 37.8% (1 d.p.)

2. PERCENTAGE INCREASE/DECREASE

To spot these types of questions, look for the word ‘percentage’

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3. WORKING OUT ORIGINAL QUANITIES- Convert the final amount’s percentage into a decimal.- Divide the final amount by the decimal.

e.g. 16 is 20% of an amount. What is this amount

e.g. A price of $85 includes a tax mark-up of 15%. Calculate the pre-tax price.

20% as a decimal = 0.2 Amount = 16 ÷ 0.2 = 80

Final amount as a percentage = 100 + 15 =115

Final amount as a decimal = 1.15

Pre-tax price = 85 ÷ 1.15 = $73.91

To spot these types of questions, look for words such as ‘pre’, ‘before’ or ‘original’

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RATIOSRATIOS- Compare amounts of two quantities of similar units- Written with a colon- Can be simplified just like fractions and should always contain whole numbers

e.g. Simplify 200 mL : 800 mL÷200 ÷200

1 mL : 4 mL

e.g. Simplify 600 m : 2 km

Must have the same units!

600 m : 2000 m÷200 ÷200

3 m : 10 m

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2. SPLITTING IN GIVEN RATIOS- Steps: i) Add parts

ii) Divide total into amount being split iii) Multiply answer by parts in given ratio

e.g. Split $1400 between two people in the ratio 2:5

e.g. What is the smallest ratio when $2500 is split in the ratio 5:3:2

Total parts:2 + 5 = 7Divide into amount: 1400 ÷ 7 = 200Multiply by parts: 200 × 2 = $400 200 × 5 = $1000

Answer: $400 : $1000

Order of a ratio is very important

Total parts:5 + 3 + 2 = 10Divide into amount: 2000 ÷ 10Multiply by parts: 250 × 2 = $500

Answer: $500

= 250

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PROPORTIONSPROPORTIONS- If less for 1, divide then multiply. - If more for 1, multiply then divide.

e.g. If 4 oranges cost $3.60, how much will 9 cost?

It costs less money for 1 orange so divide then multiply

3.60 ÷ 4 = 0.909 × 0.90 = $8.10

e.g. If it takes 6 painters 15 days to paint a school, how long will it take for 10?

It takes more time for 1 painter so multiply then divide

6 × 15 = 9090 ÷ 10 = 9 days

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0011 0010 1010 1101 0001 0100 1011ESTIMATIONESTIMATION- Involves guessing what the real answer may be close to by working with whole numbers

e.g. Estimate

a) 4.986 × 7.003 = b) 413 × 2.96 =5 × 7 400 × 3= 35 = 1200

- Generally we round numbers to 1 significant figure first