Unit 5 Percent, Ratio and Rate Section 5.1 Relating...

Unit 5 – Percent, Ratio and Rate Grade 8 Mathematics

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Unit 5 – Percent, Ratio and Rate

Section 5.1 – Relating Fractions, Decimals and Percents

Percent: means “out of 100”.

For example, when you get back a quiz and you score 92%

that means you received 92 marks out of a possible 100.

Percents are used frequently in everyday life. For example:

test / quiz marks (ie- scoring 78% on your Math test)

sales tax (ie- 13% on purchases in the province of NL)

discount (ie- 30% off everything at West 49)

probability (ie- 60% chance of flurries tomorrow)

athletic statistics (ie- scored 25% of shots on goal)

We can say, “80% of professional basketball players are males”.

Percents Greater than 100

Think about what percent a student would earn on a test containing bonus questions if the student got all of the questions correct.

o What percent has the cost of a can of soda increased when today’s cost is compared to the cost 20 years ago.

Representing Percents on Grid Paper

We can use a hundred chart to represent a whole number percent. The hundred chart represents one whole, or 100% and each small square represents 1%.



For example, we can represent 37% by shading 37 blocks out of 100.

Try these:

a) Represent 13% on the grid below: b) Represent 66% on the grid below:

We can also use a hundreds chart to represent fractional percents when the fraction is easily recognizable. For example: Represent 23.5% on the grid below.

This should be easy to do

as we can easily “eyeball”

0.5 or half of a block.



Represent 46.25% on the grid below.

For fractional percents that would be harder to recognize, we can use a small hundredths chart to the side of the original chart. Example 1: Represent 64.38% on grid paper.

The hundreds chart shows the 64%, but 0.38 of a block is harder to recognize. We must use an additional small hundredths chart to enlarge the hundredths partitions of the 65th square.

Try these:

a) Represent 11.92% on the grids below:

This should be easy to do

as we can easily “eyeball”

0.25 or a quarter of a

block.



b) Represent 83.54% on the grids below:

What percents are represented by the shaded region on the following grids?

A) B)

_____________________________ ____________________________

Expressing Percents as Fractions

Examples: a) 50%

RULE: To write a percent as a fraction... Write the number as a fraction out of 100. Reduce if possible!



b) 64% c) 35%

Examples:

a) 141

2%

b) 8.25%

c) 0.7%

d) 1.237%

If the number is not a whole number: Write the number as a fraction over 100. Multiply the numerator and denominator by the appropriate power of 10 to get rid of the decimal. Reduce if possible.



Expressing Percents as Decimals

Examples: Write the following percents as decimals. a) 15% = b) 63% = c) 36.7% = d) 8 3/4% = e) 0.28% =

Expressing Decimals as Percents

Example: Write the following decimals as percents

a) 0.34

To write percents as decimals... Divide by 100 (move the decimal two places to the left).

To write decimals as percents... Multiply the decimal by 100 (move the decimal 2 places to the right).



b) 2.5

c) 9 1/4

Expressing Decimals as Fractions To write decimals as fractions...

Examples: Write the following decimals as fractions. a) 0.17 b) 0.035 c) 2.28 d) 0.319

Write the fraction over a power of ten depending on the number of decimal places. Reduce if possible.



Expressing Fractions as Decimals To write fractions as decimals...

Examples: Write the following fractions as decimals

a) 1

4

b) 24

25

c) 7

6

d) 5

6

e) 2

3

f) 11

4

change to an equivalent fraction over 10, 100 or 1000 divide the denominator into the numerator



Expressing Fractions as Percents To write fractions as percents...

Examples: Write the following fractions as percents:

a) 1

4

b) 2

5

c) 24

25

d) 15

4

e) 13

8

Express the fraction as a decimal, and multiply by 100 (move the decimal 2 places to the right)



Lesson 5.1 Worksheet: Relating Fractions, Decimals & Percents

1. Write each percent as a fraction and as a decimal.

a) 24.5% b) 245% c) 73.25% d) 99

34%

2. Write each as a percent. a) 0.3 b) 0.55 c) 0.04 d) 0.9

e) 0.335 f) 0.5525 g) 0.0475 h) 15

3. Write each fraction as a decimal and as a percent.

a) 5

200 b) 3

150 c) 12

500 d) 9

300

e) 16

400 f) 12

250 g) 15

600 h) 28

800



4. Write each percent as a fraction and as a decimal. a) 0.7% b) 0.44% c) 0.15% d) 0.9% e) 0.92% f) 0.27% g) 0.55% h) 0.36% 5. Write each decimal as a fraction and as a percent. a) 0.221 b) 0.003 c) 0.2225 d) 0.0095 e) 0.016 f) 0.375 g) 0.1875 h) 0.0031 6. Erica scored 19 out of 24 on her science test. Michelle had 81.25% on the same

test. Who did better? How do you know?

7. During a school tournament, Team A had 10 of its 12 team members present. Team B had 13 of its 15 players present. Which team had the lesser percent of its team present at the tournament?



Section 5.2 – Calculating Percents

Percents between 0% and 1%:

These percents can be represented on the hundredths chart since no full blocks

would be shaded on the hundreds chart.

Example: Use a diagram to show 0.28%

Try the following:

a) Show 0.37% on the grid(s) below b) Show 0.95% on the grid(s) below



You may also use a number line to represent a percent between 0-1.

Example: Use a number line to represent 0.2%

Try the following: a) Show 0.35% on the number line. b) Show 11.6% on the number line.



Percents greater than 100

These percents can be represented by using more than one hundreds chart.

Example: Use a diagram to show 240%.

You may also use a number line to represent a percent greater than 100.

Example: Use a number line to represent 110%.

Try the following: a) Show 157% on the grids below:



b) Show 381% on the grids below: Use the number lines provided to show the following: a) 185% b) 205%



We can use number lines to help estimate answers to problems. Example: If the cost of a coat was $80 in 2009 and increased by 230% in 2012, what

is the new price?

Example: If a shirt was selling for $50 and has been marked up by 180%, what is the new cost?



percent

portion

whole

Section 5.3 – Solving Percent Problems

When solving problems involving percent, it can be helpful to remember:

Example 1: There are 35 students in Mr. Bennett’s class. 20% of them have paid for their field trip. How many students have paid? How many have not paid?

Sometimes you will be given the “whole” and the percent and asked to find the

“part”:

write the percent as a decimal

multiply the decimal by the “whole”

part = whole × percent whole = part / percent percent = part / whole

Part : _______

%: _______

Whole: _______



Example 2: The cost price of a coat is $80 (cost price is the price the store has to

pay for the coat). The selling price of the coat is 230% of the cost price. What is the

selling price of the coat? Show your answer on a number line.

Example 3: A tree farm in rural Alberta has 58 782 trees. The owner has

discovered that 0.28% of his trees have been infected with a deadly fungus. About

how many trees have been hit by this fungus? Show your answer on a number line.

Part : _______

%: _______

Whole: _______

Part : _______

%: _______

Whole: _______



Example 1: 30 is 60% of what number?

Example 2: 35% of what number is 14?

Example 3: 132 is 110% of what number?

Sometimes you will be given the “part” and the percent and asked to find the

“whole”.

determine 1% of the number by dividing the “part” by the percent.

multiply by 100 to find 100% of the number (the number itself).

Part : _______

%: _______

Whole: _______

Part : _______

%: _______

Whole: _______

Part : _______

%: _______

Whole: _______



Example 1: Brad had 6 hits in 20 at bats. On what percent of his at-bats did he get a

hit?

Example 2: Michelle ate 3 pieces of a 12 slice pizza. What percentage of the pizza

did she eat?

Example 3: Mrs. White has 32 students in her class and 8 of them are boys. What

percentage of her students are girls?

Sometimes you will be given the “part” and the “whole” and asked to find the

percent.

write the “part” and “whole” as a fraction. convert the fraction to a decimal and then to a percent.

Part : _______

%: _______

Whole: _______

Part : _______

%: _______

Whole: _______

Part : _______

%: _______

Whole: _______



Percent Increase

Percent increase is calculated when the price of an item goes up.

*Calculate the amount of increase by subtracting the original price from the new

higher price.

Example 1: The price of a can of soda increased from $0.95 to $1.25. What is the

percent increase in price? Illustrate your answer on a number line.

Example 2: Mr Janes bought a mining stock for $35. Two weeks later he sold it for

$105. What was the percent increase?

Percent Increase = difference

original × 100

Difference: ________

Original: ________


Original: ________



Percent Decrease

Percent decrease is calculated when the price of an item goes down. We calculate it

the same way as the percent increase.

Example 1: At the Keg Restaurant, the price of a ceasar salad decreased from $2.50

to $1.95. What is the percent of the decrease? Illustrate your answer on a number

line.

Example 2: The value of a stock was $2.50. On Monday, the value dropped to $1.20.

What was the percent decrease?

Percent Decrease = difference

original × 100


Original: ________


Original: ________



Section 5.4 – Sales Tax and Discounts

Discounts are often described in terms of percent, for example, “20% off sale.”

Tax is always calculated as a percent. In Newfoundland and Labrador, we have a

harmonized sales tax which combines both the PST and GST. Currently, the

percentage of HST we pay is 13%.

We have already dealt with solving similar kinds of problems. When we calculate

tax or discount, we have to find the portion of the original amount and add or

subtract it to or from the original amount.

Example 1: Bootlegger at the Avalon Mall has a pair of jeans retailing for $39.99.

Calculate the amount of sales tax you would need to pay if you bought these jeans.

To calculate sales tax:

Write the percent as decimal.

Multiply the cost of the item by this decimal.



Example 2: At a local restaurant, you purchase an entrée for $19.99 and a beverage

for $2.25. How much would your meal cost including HST?

Example 3: Calculate the total cost of a DVD with a price tag of $15.99.

Example 4: Calculate the total cost of a t-shirt retailing for $18.99 and a sweater

retailing for $34.99.



Discount

When an item is on sale for 15% off, we say there is a discount of 15%.

A discount of 20% means that you will pay:

100% - 20% = 80% of the regular price.

Example 1: An mp3 player normally sells for $200. It has been discounted 25%.

How much money will you save on the mp3 player?

Example 2: A sweater which retails for $34.99 is now 15% off. What is the amount

the discount?

To calculate the amount of the discount:

Write the percent as a decimal. Multiply this decimal by the regular price of the item.



Example 3: A television retails for $400 and it has now been discounted by 15%.

What is the discounted price of the television?

Example 4: Sally’s Salon usually charges $65 for a shampoo and conditioner combo

pack. Right now the salon is offering the pack for 20% off. Calculate the new price

of the combo pack.

Using More than One Percent Calculation

Sometimes you must apply a percentage calculation more than once before an answer

is found.

The “No Tax” sale at the Avalon Mall before Christmas is a good example of this.

Stores are required by law to charge tax, so during this kind of sale, they must first

discount the regular price by the tax rate, then calculate the tax owing on this new

amount.

Since they are calculating the tax on a smaller amount, the final price will be a little less

than the original sticker price.



Example 1: Brenda finds a new winter coat at the mall on the day they are having their

No Tax sale. The price tag on the coat says $125. How much will Brenda pay for the

coat including HST?

Example 2: Chapter’s bookstore has a hardcover world atlas which normally retails for

$79.99. The book has now been placed in the “50% OFF” section of the store. One

particular copy is slightly damaged and has been discounted a further 30%. If I use my

Chapter’s card, I am eligible for a further 20% reduction today only. What would the

discounted price of the damaged copy be if I were to purchase it today?



Section 5.5 – Exploring Ratios

Ratio: _________________________________________________________________________________________

Example:

We can say that the ratio of light to dark circles is 1 to 2. This means that for every light circle, there are 2 dark circles.

The ratio 1 to 2 can be written as

1 : 2

The number 1 in the example above is called the first term. The number 2 in the example above is called the second term.

Two term ratio: A ratio with two terms. i.e. 3:2

Three term ratio: A ratio with three terms. i.e. 3:2:5

Most ratios can be written in three ways:

1 to 2 1 : 2

2

1

Not all ratios can be written as fractions however.

The numerator of a fraction represents the “part” that is being represented and the

denominator represents the “whole”.



Part to whole ratio: A ratio that compares a portion of something to its maximum

amount.

Example: There are 5 chairs, 2 of which are blue. The ratio of the blue chairs to the

total number is 2:5.

Part to part ratio: A ratio that compares a portion of something to another portion.

Example: There are 5 chairs, 2 blue and 3 white. The ratio of blue to white is 2:3.

Part to whole ratios can be written as a percent.

Consider the following scenario:

A survey of 100 Canadians was conducted to determine whether

chocolate, vanilla, or Neapolitan was the most popular flavor. 41

Canadians said that their favorite flavor of ice cream was

chocolate, 53 Canadians named vanilla as their favorite, and only 6

said they preferred Neapolitan.

1. Write a ratio to represent the number of people who prefer chocolate to

the number of people who were surveyed.

This is a ____________________________________ratio.



2. Write a ratio to represent the number of people who prefer vanilla to the number of

people who were surveyed.

This is a _____________________________________ ratio.

3. Write a ratio to represent the number of people who prefer chocolate to vanilla.

This is a _______________________ ratio therefore it cannot be written as a fraction.

4. Write a ratio to represent the number of people who prefer chocolate, to the number

of people who prefer vanilla, to the number of people who prefer Neapolitan.

This is a _________________________ ratio.

Writing Part-to-Part Ratios as Part-to-Whole Ratios / Fractions.

Example 1: Bill has only hockey and baseball cards in his collection. He has 100

hockey cards and 133 baseball cards. Write a fraction to compare the number of

hockey cards to the total number of cards in his collection.



Example 2: The ratio of boys to girls in Mrs. Billard’s class is 2 to 3.

a) Write two part-to-whole ratios for this information. (ratio of boys to all students

and ratio of girls to all students).

b) Write both of these ratios as percents.



Section 5.6 – Equivalent Ratios

Equivalent ratios: _____________________________________________________________________.

If a parking lot has 6 trucks and 9 cars, the ratio of trucks to cars is 6:9.

In words we could say, “For every six trucks there are nine cars.”

An equivalent ratio of trucks to cars is 2:3. We find this by dividing both terms by

3.

We can make equivalent ratios by multiplying or dividing all terms by the same

number.

Example:

2

3 =

6

9 =

12

18 =

18

27

To put a ratio in simplest form, divide all terms by the greatest common factor.

Example: 6 ∶ 9

The largest number that divides evenly into 6 and 9 is 3. Therefore, if we divide 6

and 9 by 3, we get the reduced equivalent ratio, 2 : 3.

To write equivalent ratios, multiply or divide each term by the same non-zero number.

× 2

× 3

÷ 3



Example 1: Write two equivalent ratios for the following.

a) 1 : 2

b) 8 : 10

c) 1 : 3 : 5

*Note that there are an infinite number of equivalent ratios that can be written for each

example above.

Example 2: Write 4 equivalent ratios for 2:3 and use a table to show your work.

Example 3: Find the missing number.

a) 3 : 4 = 6 : _____ We can use cross multiplication to solve.

b) 4 : _____ = 12 : 15

c) 4 : 6 = _____ : 9

First Term 2

Second Term 3



Writing Ratios in Simplest Form

Example 1: Write the following ratios in lowest terms.

a) 30:24 the GCF is ______

b) 15:20:30 the GCF is ______

c) 21:49 the GCF is ______

To write a ratio in lowest terms, divide the terms by their greatest common factor.



Section 5.7 – Comparing Ratios

Cleary puts 2 scoops of coffee in 5 scoops of water. Sarah Jane puts 3 scoops of

coffee in 7 cups of water. Whose coffee is strongest?

For each scoop:

Cleary: Sarah Jane:

Sarah Jane used 15 scoops for 35 cups of water as compared to Cleary who used 14

scoops for 35 cups of water. Therefore, Sarah Jane’s coffee is stronger. We used

equivalent ratios to find the same amount of water, and therefore determining who

used more scoops.

Example 2: When Laurie makes a jug of Kool-Aid, she uses 3 scoops of powder and 4

cups of water. When her sister makes Kool-Aid, she uses 5 scoops of powder and 6

cups of water. Which jug of Kool-Aid is sweeter?

2: 5

4: 10

6: 15

8: 20

10: 25

12: 30

14: 35

3: 7

6: 14

9: 21

12: 28

15: 35

18: 42



Example 3: Compare 20:5 to 140:20. Which one is the greater?



Section 5.8 – Solving Ratio Problems

You can often set up a proportion to help you solve a word problem.

Proportion: a statement which says two ratios are equal.

i.e. 1:2 = 2:4

Or

2

1 =

4

2

Example 1: Mike and Nathan agree to share the profits from their lawn mowing

business in a ratio of 5:2. Mike’s profit is $75. What is Nathan’s share of the profit?

Example 2: For every 24 phone call to Open Line supporting John as a candidate for

mayor, there were 31 calls not supporting him. If there were 216 phone calls in

support of John, how many calls did not support his bid for mayor?



Example 3: Michelle has 40 students in her class.

The ratio of boys to girls is 2:3. How many boys are

in the class?

Example 4: Joan has 135 stamps in her collection. The ratio of mint stamps to used

stamps is 2:7. How many of Joan’s stamps are mint?

Example 5: A map has a scale of 1 cm : 250 000cm. If two towns are 20 cm apart on

the map, how far apart are they in real life?



Section 5.9 – Exploring Rates

A rate is a kind of ratio that compares an amount of one unit to a different kind of

unit.

Example: kilometers per hour

48 𝑘𝑚

4 ℎ =

12 𝑘𝑚

1 ℎ

Rate Unit Rate

Example:

- walking 6 km in two hours.

- swimming 400 m in 8 minutes

You must always include the units when referring to rates otherwise the rate has no meaning.

Since the units in a rate are different, a rate is not a part-to-whole comparison. Therefore it cannot be expressed as a fraction or as a percent.

Unit Rate: A unit rate is a kind of rate that is reduced so the denominator is 1.

Example:

- jumping rope 80 times in one minute - 80 skips / minute

- $2.80 for one kilogram of tomatoes - $2.80 / kg



Example 1: Express each as a unit rate.

a) Typing 180 words in 3 minutes.

b) Earning $72 for 9 hours of work.

c) Paying $10.50 for 7 packs of Kraft dinner.

d) Delivering 112 flyers in 4 hours.

e) Paying $9.95 for 5 pounds of mussels.

Example 2: A human walks at an average speed of 5km/h. What is this speed in

meters per second? Round your answer to the nearest tenth.

First convert km to m. ( K H D m d c m)

5 km = ___________ m (move the decimal 3 places to the right)

Next convert 1 hour to seconds.

There are 60 minutes in 1 hour and 60 seconds in 1 minute

As a unit rate:



Example 3: A squirrel can run at a top speed of about 5 m/s.

What is his speed in kilometers per hour?

There are 60 seconds in 1 minute, and 60 minutes in one hour.

Convert m to km.

18 000 m = _________________ km (move the decimal 3 places to the left).



Section 5.10 – Comparing Rates

Example 1: At Shopper’s Drug Mart, a case of macaroni and cheese costs $8.99 and

there are 12 boxes in each case. The Dominion flyer says you can buy 6 boxes of

macaroni and cheese for $5. Which one is the better buy?

We can use equivalent rates to solve this problem.

Drugstore: Grocery Store:

Example 2: Mary earns $24 for 3 hours of work. George earns $102 for 12 hours of

work. Who has the better rate of pay?

George: Mary:

For some comparisons, it may be easiest to use equivalent rates.



Example 1: Marcia works at Dominion after school. For 4 hours of work, she earns

$30. At this rate, how much will Marcia earn in 6 hours? 10 hours?

Unit rate of pay:

For 6 hours of work:

For 10 hours of work:

Example 2: At Wal-Mart, Mitchell buys 6 packs of gum for $3. How much would 20

packs of gum cost?

Unit cost:

For 20 packs:

For other problems, it is best to use unit rate.



Example 3: Sarah can swim 70 meters in 2

minutes. At this rate, how far can she swim in

5 minutes?

Unit rate:

Distance in 5 minutes:

Example 4: Pat paid $9 for 4 packages of No Name batteries. How much would 3

packages cost?

Unit cost:

Cost for 3:



Example 5: A car trip covered 747km in 9 hours. What

was the average speed of the car during the trip?

Unit rate of speed:

Other problems involving speed, distance, and time.

Example 1: If the average speed of a high powered scooter is 30km/h and it travels for 3

hours, what distance would the scooter travel?

Example 2: A car travels 210km at an average speed of 70km/h.

How long does the trip take?

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