fractions, - mr. macmillan mathematics -...

MAT2LGrade 10, Essentials –

Mathematics

Lesson 11

Fractions, Decimals and Percent

MAT2L – Mathematics Unit 3 – Lesson 11

Lesson Eleven Concepts

determine the relationship among fractions, decimals and percentages by constructing diagrams

recall from memory the most commonly used equivalences or approximations between fractions and percentages

read, interpret, and explain, orally and in writing, data displayed in tables and graphs

Percent

Percent means “per one hundred” or “hundredths”.

5% means “5 per one hundred” and is shown visually below:

34% means “34 per one hundred” and is also shown visually below:

Example

What percent do each of the shaded sections of the diagrams below represent?

a. b.

Copyright © 2005, Durham Continuing Education of 34


Answer:

a. Since each column and row represents 10 units then there are 80 units shaded. 80% of the diagram is shaded.

b. Since each column and row represents 10 units then there are 27 units shaded. 27% of the diagram is shaded.

Support Questions

Percent as a Decimal

Example

a. Write 35% as a decimal.

Answer. 35 100 = 0.35

To convert into a decimal, rewrite the percentage without the % then divide that number by 100.

b. Write 47% as a decimal.

Answer:



Support Questions

2. Write each percent as a decimal.

a. 3% b. 9% c. 0.5% d. 23% e. 112%

Writing Fractions as Decimals

To convert a fraction into a decimal you divide the numerator (top) of the fraction by the denominator (bottom) of the fraction.

Example

a. Write as a decimal.

Answer: 3 8 = 0.375

b. Write as a decimal.

Answer:

Support Questions

3. Write each fraction in decimal form.

a. b. c. d. e.


Numerator Denominator

Fraction

Decimal


Example

c. Write as a decimal. Round to the hundredths place value.

Answer:

Example

d. Write as a decimal. Round to the hundredths place value.

Answer:

Support Questions

4. Write each fraction in decimal form rounded to the hundredths place value.

a. b. c. d. e.

Writing Fractions as Percent

First convert the fraction into a decimal.

Then multiply that decimal by 100.


The bar means that the 3 repeats forever. It is a repeating decimal.

This symbol means approximately.


Example

a. Write as a percent.

Answer:

b. Write as a percent. Round to the nearest percent.

Answer:

Support Questions

5. Estimate then calculate each fraction below as a percent. Round to the nearest percent.

a. b. c. d. e.


First convert to a decimal.

Always include the percent sign (%) after converting into percent.

Multiply the decimal by 100 to convert into percent.


Percent of a Number

Example

a. What is 25% of 200?

Answer:

Convert the percent into a decimal THEN multiply that decimal by the value given.

Therefore, 25% of 200 is 50.

b. What is 17% of 130?

Answer:

Therefore 17% of 130 is 22.1

Support Questions

6. Answer the following ‘percent of a number’ questions.



c. What is 0.5% of 1000?

d. What is 10.5% of 75?

e. What is 35% of 40?

f. What is 125% of 100?


In mathematics the word “of” means multiply.


Key Question #11

1. What percent do each of the shaded sections of the diagrams below represent?

2. Write each percent as a decimal.

a. 5% b. 8% c. 7% d. 15% e. 87%

3. Write each fraction in decimal form.

a. b. c. d. e.

4. Write each fraction in decimal form rounded to the hundredths place value.

a. b. c. d. e.

5. Estimate then calculate each fraction below as a percent. Round to the nearest percent.

a. b. c. d. e.

6. Answer the following ‘percent of a number’ questions.



c. What is 67% of 1000?


Discount

Lesson 12


Lesson Twelve Concepts

solve problems involving the most commonly used equivalences between fractions and percentages

round decimal values appropriately in solving problems drawn from everyday situations

solve problems involving fractions and percentages in practical situations verbalize their observations and reflections regarding money sense and ask

questions to clarify their understanding explain their reasoning used in problem solving and in judging reasonableness communicate, orally and in writing, the solutions to money problems and the results

of investigations, using appropriate terminology, symbols, and form

Discount

Discount uses percent of a number.

Example

a. What is the discount amount on an item that is 40% off and has a regular price of $125.00?

Answer: Discount amount = discount % x cost of item

= 40% x $125.00

= 0.40 x $125.00

= $50.00

b. What is the sale price of an item that is ¼ off and has a regular price of $60.00?

Answer: Discount amount = discount % x cost of item

= ¼ x $60.00

= 0.25 x $60.00

= $15.00


Remember 40% = 40 100 = 0.40

1 4 = 0.25


Support Questions

1. Calculate the amount of the discount for each of the following items.

a. b.

c. d.

e. f.



Sale Price

The sale price of the item is the regular price, subtract the discount.

Example

a. What is the sale price of an item that regularly sells for $150.00 and is 30% off?

Answer: Discount = 30% x 150.00= 0.30 x 150.00= $45.00

Sale Price = Regular price – Discount= $150.00 - $45.00= $105.00

Support Questions

2. Calculate the sale price for each of the following items.

a. b.

3. Copy and then complete the table for each amount.

Price Rate of discount

Amount of discount

Sale price

$124.97 25 %$49.99 ¼ off$97.00 30 %$279.98 ½ off



Key Question #12

1. Calculate the amount of the discount and then the sale price for each of the following items.

a. b.

2. Copy and then complete the table for each amount.

Price Rate of discount

Amount of discount

Sale price

$135.99 15 %$249.99 1/3 off$57.97 75 %$1099.99 40 %

3. The Photo Pit is selling a camera for 20% off the regular price of $347.97. Camera City is selling the exact same camera for 25% off its regular price of $384.99. Which store offers the better deal?

4. At the end of the summer a clothing store reduced the original price of its jackets by 10% the first week, 25% the second week and 50% the third week. Calculate the price of a $74.99 jacket during each week.

5. Suppose an item was 35% off. Explain with words only the steps you would take to calculate the sale price.


Sales Tax

Lesson 13


Lesson Thirteen Concepts

solve problems involving the most commonly used equivalences between fractions and percentages

round decimal values appropriately in solving problems drawn from everyday situations

solve problems involving fractions and percentages in practical situations explain their reasoning used in problem solving and in judging reasonableness

Sales Tax

Sales tax uses the percent of a value to calculate the additional money that needs to be paid when purchasing an item. Sales tax varies from province to province.

Example

a. If the sales tax rate is 8%, what would be the cost of the tax if the item was $24.99?

Answer: Sales tax = rate (%) x cost of the item

= 8% of $24.99

= (8 100) x 24.99

= 0.08 x 24.99

= 1.9992

= $2.00

b. What is the total cost of a $24.99 after a sales tax of 5% is included?

Answer: Sales tax = 24.99 x 0.05= $1.25

Total Cost = 24.99 + 1.25= $26.24


First convert the percentage to a decimal.

Multiply by the amount of the item.

Round the total to the nearest cent.


c. What is the total cost of a $45.00 item after a Provincial Sales Tax (PST) of 8% and a Goods and Services Tax (GST) of 5% is included?

Answer: PST = 0.05 x 45.00= $2.25

GST = 0.08 x 45.00= $3.60

Total Cost = Price + PST + GST= 45.00 + 2.25 + 3.60= $50.85

Support Questions

1. Calculate the total cost including 5% Goods and Services Tax (GST).

a. Shampoo costing $2.99b. Watch costing $42.50c. Jacket costing $149.97d. Bicycle costing $425.97e. Stereo costing $229.00

2. Calculate the total cost including 8%

Provincial Sales Tax (PST).

a. Shirt costing $25.00b. Car costing $34 000c. Book costing $9.50d. Boots costing $74.97e. Clock costing $124.99

3. How much more GST do you pay on an item costing $425.00 than on an item costing $380.00?

4. A golf club costs $79.95. How much do you save when the store advertises that it will pay the GST?

5. A pair of shoes is on sale for 25% off the regular price of $124.99. How much do you save in taxes (PST and GST combined) by purchasing the shoes on sale?



Key Question #13

1. Calculate the total cost including 5% Goods and Services Tax (GST).

a. Phone costing $23.97b. Calculator costing $19.99c. Hat costing $23.99d. Bed costing $499.94e. Socks costing $7.19

2. Calculate the total cost including 8% Provincial Sales Tax (PST).

a. Pants costing $43.00b. Motorcycle costing $9 750c. Dinner costing $41.25d. Drill costing $71.96e. Perfume costing $64.95

3. How much more GST do you pay on an item costing $300.00 than on an item costing $295.00? (GST = 5%)

4. A home theatre system costs $829.97. How much do you

save when the store advertises that it will pay the PST? (PST = 8%)

5. A business suit is on sale for 30% off the regular price of

$399.99. How much do you save in taxes (PST (8%) and GST (5%) combined) by purchasing the suit on sale?

6. If you multiply the cost of an item by 1.15 the price of item with taxes included is calculated. Assuming that GST is 5% and PST is 8%, what do you think the number 1 in 1.13 represents and what do you think the .13 in 1.13 represents?


Ratio and Proportion

Lesson 14


Lesson Fourteen Concepts

write ratios describing relationships on the school environment describe the effects of changing the parts of a given ratio proportionately and

disproportionately, in activities in which the results are observable solve problems using proportions read, interpret, and explain, orally and in writing,

data displayed in tables and graphs

Ratios

A Ratio is a comparison of two like quantities.

To make orange juice from concentrate, combine 4 cans of water with every 1 can of concentrate.

The ratio of the number of cans of water to the number of cans of concentrate is 4 to 1, 8 to 2, 12 to 3 and so on.

4 to 1, 8 to 2 and 12 to 3 represent the same comparison.

These ratios are called “equivalent ratios”.

When ratios are equivalent, they can be written as a proportion.

4 : 1 = 8 : 2 or

Equivalent ratios are made by multiplying or dividing the values in the ratio by the same non-zero number.

When one ratio can be made by dividing or multiplying the other ratio’s numerator and denominator by the same number then the two ratios are equivalent.



Example

a. Express the following diagram as a ratio of O’s to X’s.

Answer:

4 : 3 or

b. Express as a ratio. $30 to 45 kg.

Answer: 30 : 45

Support Questions

2. Express each as a ratio.

a. 8 s to 4 min b. 3 cm to 10 km

c. 9 trees to 5 shrubs d. 1 apples to 6 oranges

e. 13 men to 7 women f. 8 dogs to 11 cats



Support Questions (continued)

3. What is the missing value in each of the pairs of equivalent ratios?

4. To make lemonade from concentrate, combine 3 cans of water with every 1 can of concentrate. Copy and complete the table to show equivalent ratios.

Concentrate (cans) 3 7 1Water (cans) 18 21

Proportion

Proportion is a statement that two ratios are equivalent.

Below are two examples of ratios that are equivalent.



Support Questions

5. Prove whether or not the following ratios are proportionate.

a. 3 : 5 = 21: 35 b.

c. d. 4 : 12 = 24 : 74

e. 50 : 25 = 2 : 1 f.

Proportionate objects

The following rectangles are proportionate to each other because the ratio of lengths to widths for each can make equivalent fractions.



Support Questions

7. A rectangle has a width of 3 cm and a length of 4 cm. Draw this triangle using exact measurements then draw a proportionate rectangle that has a width of 9 cm.



Key Question #14

2. Express each as a ratio.

a. 3 s to 5 min b. 4 cm to 13 km

c. 8 trees to 2 shrubs d. 5 apples to 1 oranges

e. 4 men to 9 women f. 12 dogs to 7 cats

3. What is the missing value in each of the pairs of equivalent ratios?

4. To make lemonade from concentrate, combine 3 cans of water with every 1 can of concentrate. Copy and complete the table to show equivalent ratios.

Concentrate (cans) 1.5 4 10Water (cans) 24 1.5


Rate

Lesson 15


Lesson Fifteen Concepts

solve problems involving the calculations of rates drawn from a variety of everyday contexts and from familiar social issues

Rate

Rate is the comparison of quantities with different units.

Example

a. $8.50/hr.

The two quantities are dollars ($) and time (hours).

b. 30 m/s

The two quantities are distance (metres) and time (seconds).

Unit Rate is the rate that has one as its second term.

Example

a. $8.50/hr. Second term is 1 hour which makes this a unit rate.

Both of these examples are unit rates because the second term is one.

b. 30 m/s Second term is 1 second which makes this a unit rate.



Calculating the Unit Rate



Support Questions

1. Express each as a unit rate.

a. 326 words typed in 4 min b. 208 heartbeats in 3 min

c. 500 km driven in 5.5 hours d. $168.50 in 6.5 hours worked 2. The human heart beats approximately 70 times/min. How many beats are there

for each of the time periods?

a. 5 min b. 2 hours c. 1 hour 15 min

3. Kristen ran 10 km in 44 min. At this rate, how many km will she run in 2.5 hours?

4. Noah is paid an hourly rate. When he works a 40 h week, he receives $475.50. How much does he receive if he works a 37.5 h week?



Key Question #15

1. Express each as a unit rate.

a. 400 words typed in 6 min b. 192 heartbeats in 3 min

c. 725 km driven in 6.5 hours d. $208.50 in 7.5 hours worked 2. The human heart beats approximately 70 times/min. How many beats are there

for each of the time periods?

a. 4 min b. 2.25 hours c. 1 hour 10 min

3. Kristen walked 3 km in 32 min. At this rate, how many km will she walk in 1.5 hours?

4. Noah is paid an hourly rate. When he works a 40 h week, he receives $625.75. How much does he receive if he works a 34.5 h week?

5. According to Statistics Canada, 7 out of every 100 employable Canadian people are unemployed. If there are 32 000 000 people in Canada, how many people are jobless?

6. Canadians throw away 21 000 000 000 kg of waste every year. How much does the average Canadian throw away in kg per day? (hint 365 days per year)

7. British Columbia is currently cutting down trees at a rate of 15 logging trucks loads per hour. At this rate, how many logging truck loads of trees are cut down in one year?

8. List 3 situations in your life where a rate is involved? Explain.


MAT2L – Mathematics Support Question Answers

Answers to Support Questions: Unit Three

Lesson Eleven

1. a. 25% b. 91% c. 49%

2. a. 3 100 = 0.03 b. 9 100 = 0.09 c. 0.5 100 = 0.005

d. 23 100 = 0.23 e. 112 100 = 1.12

3. a. 1 5 = 0.2 b. 17 50 = 0.34 c. 5 10 = 0.5

d. 3 100 = 0.03 e. 3 8 = 0.375

4. a. 1 3 = 0.33 b. 5 11 = 0.45 c. 5 8 = 0.63

d. 37 1000 = 0.04 e. 4 6 = 0.67 5. a. b. c.

d. e.

6. a. b. c.

d. e. f.

Lesson Twelve

1. a. b. c.

d. e. f.



2. a. b.

3.Price Rate of

discountAmount of discount Sale price

$124.97 25 % 0.25 x 124.97 = $31.24 124.97 – 31.24 = $93.73$49.99 ¼ off 0.25 x 49.99 = $12.50 49.99 – 12.50 = $37.50$97.00 30 % 0.30 x 97.00 = $29.10 97.00 – 29.10 = $67.90$279.98 ½ off 0.50 x 279.98 = $139.99 279.98 – 139.99 = $139.99

Lesson Thirteen

1. a. b. c.

d. e.

2. a. b.

c. d.

e.



3.

4.

5.

Lesson Fourteen

1. 4:3

2. a. 8:4 b. 3:10 c. 9:5 d. 1:6

e. 13:7 f. 8:11

3. a. 10 b. 7 c. 5

4.Concentrate (cans) 3 7 18 3 =

6213 =7

1

Water (cans) 3 x 3 = 9 7 x 3 = 21 18 21 1 x 3 = 3

5. a.

; proportionate

b.

; proportionate



5. c.

; not proportionate

d.

; not proportionate

e.

; proportionate

f.

; not proportionate

6. a.

; proportionate

b.

; not proportionate

7. Proportionate triangle should have a width of 9 cm and a length of 12 cm.

Lesson Fifteen

1. a.

; 81.5 wds/min

b.

; approximately 69.3 beats/min

c.

; approximately 90.9 km/hr

d.

; approximately $25.92/hr



2. a.

; 350 beats in 5 min

b.

; 8 400 beats in 2 hours

c.

; 5 250 beats in 75 min or 1 hour and 15 min

3.

; 0.23 km/min; ;

Kristen runs 34.5 km in 2.5 hours.

4.

; $11.89/hr; ;

Noah earns $445.88 in 37.5 hours


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