1. Ch 5 Rate, Ratio and Percent

2. 5.1 Relating Fractions, Decimals and Percents

3. Hundred Grid
To represent a percent, you can shade squares on a grid of 100 squares, called a hundred grid. One completely shaded grid represents 100%.
To represent a percent greater than 100%, shade more than one grid.
To represent a fraction percent between 0% and 1%, shade part of one square.
To represent a fractional percent greater than 1%, shade squares from a hundred grid to show the whole number and part of one square from the grid to show the fraction.

4. Key Ideas
Fractions, decimals and percents can be used to represent numbers in various situations
Percents can be written as fractions and decimals
½ % = 0.5% = 0.5/100 = 5/1000= 0.005
150% = 150/100 = 1.5 or 1 ½
43 ¾ % = 43.75% = 43.75/100 = 4375/10000=
When we have a decimal percent, we express it in fraction form. You add as many zeros as there are decimal places. The example above had two decimal places, so we added 2 zeros.
To get the decimal – remember we divide the numerator by the denominator

5. Workbook
Page
Text pg #12-14, 19-20

6. 5.2 Calculating Percents

7. Converting Percents to Decimals
Remember that 1% = 1/100 = 0.01
So 175% = 175/100 = 1.75
0.5% = 0.5/100 = 5/1000 = 0.005
Notice that the decimal is in the place value of the denominator
Another way to look at it is how to move the decimal when converting from percent to decimal move the decimal 2 spots to the left
230% = = 2.30
0.09% = =

8. How to convert a fraction to percent
First convert the fraction to a decimal. Once you have a decimal the properties are similar to as converting a percent to a decimal – instead move the decimal 2 spots to the right to get percent.
½ = 0.5 = = 50%
3/2 = 1.5 = = 150%
3/200 = = = 1.5%

9. Complete the chart Percent Decimal Fraction 1% 0.01 1/100 5% 1/10
0.125
20%
1/4
50%
3/4
0.8
90%
99/100 1
125%
3/2 2

10. Answers to Chart Percent Decimal Fraction 1% 0.01 1/100 5% 0.05 1/20
10%
0.1
1/10
12½%
0.125
1/8
20%
0.2
1/5
25%
0.25
1/4
331/3%
1/3
50%
0.5
1/2
75%
0.75
3/4
80%
0.8
4/5
90%
0.9
9/10
99%
0.99
99/100
100% 1 1 
125%
1.25
5/4
150%
1.5
3/2
200% 2

11. Calculating percent of a number
Take the percent and convert it to a decimal, than multiply by the number you are calculating the percent of
200% of 40 = 2.0 x 40 = 80
20% of 40 = 0.2 x 40 = 8
2% of 40 = 0.02 x 40 = 0.8

12. Word Problems – Give this a Try
A marathon had 618 runners registered. Of these runners, about 0.8% completed the race in under 2h 15min. How many runners completed the race in under 2h 15min?
0.8% of 618 runners
0.008 of 618 runners
4.94 = 5 runners

13. Word Problems – Try This One
Twenty boys signed up for the school play. The number of girls who signed up was 195% of the number of boys. At the auditions, only 26 girls attended. What percent of the girls who signed up for the play attended the auditions?
195% of 20 = 39
26 of the girls who signed up attended = 26/39 = = %

14. Workbook
Page 105 – 106

15. 5.3 Solving Percent Problems

16. You are given a number that equals a certain percent
40% = 160
You want to find out what 100% is so first find out what 1% is.
1% = 160/40 = 4
To calculate 100% take the number you got for 1% and multiply by This also works if you want 85%, 115%, etc.
100% = 4 x 100 = 400
155% = 4 x 155 = 620

17. You Try
6% of a number is 9
6% = 9
1% =
100% =
350% =
28% of a number is 56
28% = 56
150% of a number is 36
150% = 36
1.5
150
525 2
200
700
0.24 24

18. To Calculate the Percent Increase
Mary had $50 before her birthday in her account. After her birthday she had $300. Calculate the percent increase.
Step 1 – calculate the difference between the two numbers
300 – 50 = 250
Step 2 – express the difference over the original (a fraction)
250/50
Step 3 – calculate the decimal and than percent
5 x 100 = 500%
She had a 500% increase after her birthday.

19. Cross Multiply
Mary had $50 before her birthday in her account. After her birthday she had $300. Calculate the percent increase.
Step 1 – calculate the difference between the two numbers
300 – 50 = 250
Step 2 – express the difference over the original (a fraction) and make it equal to your unknown % over 100%
250 = x__
Step 3 – cross multiply and divide to solve
50(x) = 250(100)
50x = 25000
x = The percent increase is 500%

20. You Try The width of the rectangle increased from 8cm to 12cm
Step 1 – calculate the difference between the two numbers
12 – 8 = 4
Step 2 – express the difference over the original (a fraction)
4/8
Step 3 – calculate the decimal and than percent
0.5 x 100 = 50%

21. Cross Multiply The width of the rectangle increased from 8cm to 12cm
Step 1 – calculate the difference between the two numbers
12 – 8 = 4
Step 2 – express the difference over the original (a fraction) and make it equal to your unknown % over 100%
4 = _x_
Step 3 – cross multiply and divide to solve
4(100) = 8(x)
400 = 8x
x = 50 The percent increase is 50%

22. Another one The price of a hotel room increased from $90 to $120
Step 1 – calculate the difference between the two numbers
120 – 90 = 30
Step 2 – express the difference over the original (a fraction)
30/90
Step 3 – calculate the decimal and than percent
0.333 x 100 = 33.33%

23. Cross Multiply The price of a hotel room increased from $90 to $120
Step 1 – calculate the difference between the two numbers
120 – 90 = 30
Step 2 – express the difference over the original (a fraction) and make it equal to your unknown % over 100%
30 = _x_
Step 3 – cross multiply and divide to solve
30(100) = 90(x)
3000 = 90x
x = The percent increase is 33.33%

24. To Calculate the Percent Decrease
Susie made a pitcher of punch that was 56L, after her party she had 12L left. Calculate the percent decrease.
Step 1 – calculate the difference between the two numbers
56 – 12 = 44
Step 2 – express the difference over the original (a fraction)
44/56
Step 3 – calculate the decimal and than percent
x 100 = 78.57%
The pitched decreased in volume for 78.57%

25. Cross Multiply
Susie made a pitcher of punch that was 56L, after her party she had 12L left. Calculate the percent decrease.
Step 1 – calculate the difference between the two numbers
56 – 12 = 44
Step 2 – express the difference over the original (a fraction) and make it equal to your unknown % over 100%
44 = _x_
Step 3 – cross multiply and divide to solve
44(100) = 56(x)
4400 = 56x
x = The percent decrease is 78.57%

26. You Try The volume of water in the tank decreased from 40L to 32L.
Step 1 – calculate the difference between the two numbers
40L – 32L = 8L
Step 2 – express the difference over the original (a fraction)
8/40
Step 3 – calculate the decimal and than percent
0.2 x 100 = 20%

27. Cross Multiply
The volume of water in the tank decreased from 40L to 32L.
Step 1 – calculate the difference between the two numbers
40L – 32L = 8L
Step 2 – express the difference over the original (a fraction) and make it equal to your unknown % over 100%
8_ = _x_
Step 3 – cross multiply and divide to solve
8(100) = 40(x)
800 = 40x
x = 20 The percent decrease is 20%

28. You Try The number of students in the class decreased from 30 – 27
Step 1 – calculate the difference between the two numbers
30 – 27 = 3
Step 2 – express the difference over the original (a fraction)
3/30
Step 3 – calculate the decimal and than percent
0.1 x 100 = 10 %

29. Cross Multiply
The number of students in the class decreased from 30 – 27
Step 1 – calculate the difference between the two numbers
30 – 27 = 3
Step 2 – express the difference over the original (a fraction) and make it equal to your unknown % over 100%
3_ = _x_
Step 3 – cross multiply and divide to solve
3(100) = 30(x)
300 = 30x
x = 10 The percent decrease is 10%

30. Workbook
Try questions 5 – 10 on page

31. 5.4 Sales Tax and Discount

32. Discount
When an item is sold at a reduced price – it is said to be sold at a discount.
There are 2 ways to calculate discount

33. Discount Calculations – Method 1 (A Review)
20% off $129
Step 1 – calculate how much the discount is
0.2 x $129 = $25.8
Step 2 – calculate how much the cost is after you subtract the discount
$129 - $25.8 = $103.2

34. Discount Calculations – Method 2
20% off $129 (means you are paying 80% of $129)
Step 1 – calculate the cost of what you are paying (in this case 80% of $129)
0.8 x $129 = $103.2
You are done – this method allows you to calculate in one step – you don’t have to do the subtraction – less steps, means less chance of making a silly mistake

35. Another Example – Both Methods Shown
Calculate the sale price on a $92 watch, 30% off
Method 1
Method 2
0.3 x $92 = $27.60
0.7 x $92 = $64.40
$92 - $27.60 = $64.40

36. Note
Only use method 2 if you are calculating the sale price – not if you are asked to calculate the discount only.

37. Sales Tax
Sales tax is added to the final cost of you bill – in BC we currently have HST which is 12%.
Again there are 2 methods

38. Sales Tax Calculations – Method 1 (A Review)
12% tax on $288
Step 1 – calculate how much the tax is
0.12 x $288 = $34.56
Step 2 – calculate how much the cost is after you add the tax
$288 + $34.56 = $322.56

39. Discount Calculations – Method 2
12% tax on $288 (means you are paying 112% of $288)
Step 1 – calculate the cost of what you are paying (in this case 112% of $288)
1.12 x $288 = $322.56
You are done – this method allows you to calculate in one step – you don’t have to do the addition – less steps, means less chance of making a silly mistake

40. Another Example – Both Methods Shown
Calculate the sale price on a $92 watch, 12%
Method 1
Method 2
0.12 x $92 = $11.04
1.12 x $92 = $103.04
$92 + $11.04 = $103.04

41. Note
Only use method 2 if you are calculating the final price – not if you are asked to calculate the tax only.

42. Another Example – Both Methods Shown with discount and tax
Calculate the sale price on a $476 TV, 15% off, 12% tax
Method 1
Method 2
0.15 x $476 = $71.40
0.85 x $476 = $404.60
$476 - $71.4 = $404.6
1.12 x $ = $453.15
0.12 x $404.6 = $48.55
$ $48.55 = $453.15 Or
0.85 x 1.12 x $476 = $453.15

43. Note about multiply discounts
If a company offers multiple discounts – you cannot add them together – you must calculate each one
Example
Macy’s offers 30% off all 7 jeans, because you are a Canadian citizen, you get an additional 15% off using your WOW card. If your mom sign’s up for a Macy’s card, you will get an additional 10% off. You cannot add 30% + 15% + 10%, because you get 15% off the price after the 30% is taken and the 10% off after the other two are taken
$300 x 0.7 = $210
$300 x 0.45 = $135
$210 x 0.85 = $178.5
$178.5 x 0.9 = $160.65
$ ≠ $135

44. Sports R Us vs. Sports Galore
Sports R Us offers a 2 day discount where you get 10% off on day 1 and an additional 10% off on day 2. Sports Galore is offing a one day sale of 20% off. Who has the better sale if the object that you want is $200?
Sports R Us would be $200 x 0.9 = $180 x 0.9 = $162
Sports Galore would be $200 x 0.8 = 160
Sports Galore has the better sale.
What is the total discount that Sports R Us Offers
The selling price after two 10% discounts is $162. Find the difference - $38. Express the difference over the original $38/$200. Convert to a decimal 0.19 than to a percent 19%.

45. Workbook
Page

46. 5.5 Exploring Ratios

47. Ratio Definitions
Part to Whole Ratio: How many of one item to all items
Part to Whole Ratios can be written as follows
Circles to all shapes
4 to 12 or 4:12 or 1/3 or 33.33%
Part to Part Ratio: How many of one item to another item
Part to Part Ratios can be written as follows
Circles to squares
4 to 5 or 4:5
Part to Part Ratios cannot be written in Fraction or Percent form, as it is not comparing one part to the whole.
You can do a 3 term ratio for part to part – 3 to 4 to 5 or 3:4:5

48. Write each ratio
A pencil case contains 7 yellow, 3 red, 1 black and 5 green pencil crayons. Write Each Ratio
Red: green
3:5
Black: total pencil crayons
1:16
Yellow: red: green
7:3:5
Yellow: red
7:3
Yellow: total pencil crayons
7:16

49. Workbook
Page 112 – 114 together

50. 5.6 Equivalent Ratios

51. Equivalent Ratios
These are similar to equivalent fractions – they are ratios that are equal to each other.
An equivalent ratio can be formed by multiplying or dividing the terms of a ratio by the same number.
÷ 4
÷ 2
original x2 x4 x5 1 2 4 8 16 20
0.75
1.5 3 6 12 15

52. Give it a try Write three ratios that are equivalent to each ratio 4:5
16:28
Original x2 x3 x4 x5 4 8 12 5 10 15 16 20 20 25
Original x2 x3 x5 ÷2 ÷4 16 28 32 48 80 8 4 56 84
140 14 7

53. Workbook
Pg 116 # 1-6

54. 5.7 Comparing Ratios

55. Comparing Ratios You can use equivalent ratios to compare ratios
How would you compare the following
Mr Durand makes a pitcher of iced tea with 8 scoops of crystals and 10 cups of water
Ms White makes a glass of iced tea with 1 scoop of crystals and ¾ cups of water
Who’s iced tea is stronger?

56. Mr Durand makes a pitcher of iced tea with 8 scoops of crystals and 10 cups of water
8:10
Ms White makes a glass of iced tea with 1 scoop of crystals ¾ cups of water
1:0.75
Cross Multiply and Divide
1 = 0.75
8 = x
x = 6

57. Try Another
Two cages contain white mice and brown mice. In one cage, the ratio of white mice to brown mice is 2:3. In the other cage, the ratio is 3:1. The cages contain the same number of mice.
What could the total number of mice be? Which cage contains more white mice?
White
Brown
Total 3 1 6
White
Brown
Total 2 3 5 4 6 10

58. Try Another
What could the total number of mice be? Which cage contains more white mice?
The total number of mice in each cage would be 20
The total number of mice would be 40.
White
Brown
Total 3 1 4 6 2 8 9 12 16 15 5 20
White
Brown
Total 2 3 5 4 6 10 9 15 8 12 20 25

59. Try Another
What could the total number of mice be? Which cage contains more white mice?
Number of white mice in A is 8 and the number of white mice in B is 15 so cage B has more white mice.
White
Brown
Total 3 1 4 6 2 8 9 12 16 15 5 20
White
Brown
Total 2 3 5 4 6 10 9 15 8 12 20 25

60. One More
Hamid jogs 5 laps in 6 min. Amelia jogs 8 laps in 11min. Which person jogs faster?
Laps: mins
Hamid = 5:6
Amelia = 8:11
To know who jogs faster we want to compare minutes to see who does the most laps so we need to make the minutes the same – 66 would be the LCM
Hamid = 55:66 (times by 11)
Amelia = 48:66 (times by 8)
Hamid runs faster!

61. Workbook
Pg 119 – 121

62. 5.8 Solving Ratio Problems

63. You can often solve a problem involving ratios by setting up a proportion. A proportion is a statement that two ratios are equal.
For example if the ratio of red marbles to blue marbles is 3:4 and there are 48 blue marbles we can find how many red marbles there are.
Original
Red 3
Blue 4 48 36
x12

64. You Try
A wildlife biologist wants to know how many trout are in a slough in Saskatchewan. He captures and tags 24 trout and releases them back into the slough. Two weeks later he returns and captures 30 trout and finds that 5 of them are tagged. He uses the following ratios to estimate the number of fish in the slough.
Fish recaptured with tags: total fish recaptured = fish caught and tagged: total fish in the slough
5:30 = 24:t (we can turn the first ratio into lowest terms to help us solve)
1 = 24 (Cross Multiply) t
t = 6 x 24 = 144

65. You Try
A breakfast cereal contains corn, wheat, and rice in a ratio of 3 to 4 to 2. If a box of cereal contains 225g of corn, how much rice does it contain?
3_ = _2_ x
3(x) = 225(2)
3x =450
x = 150
Corn
Wheat
Rice 3 4 2
225 ?
2 x 75 = 150
225 ÷ 3 = 75
150

66. 5.9 Exploring Rates

67. Rates When we compare two things with different units we have a rate.
We need 5 sandwiches for every 2 people
Oranges on sale are $1.49 for 12
Gina earns $4.75 per hour for baby-sitting
There are 500 sheets on one roll of paper towels.
A unit rate is rate in which the second term is one.
The most common one we know is speed
60km/h

68. Try These Express as a unit rate Those were easy now try
Serena walks 4 km in 1 h =
Sanjit reads 3 books in 1week =
The tap drips 25 drops in 1 min =
Those were easy now try
Betty drives 150km in 2 h. =
The helicopter travels 180km in 3 h. =
Gerald walks 1 km in 15min =
4km/h
3 books/week
25 drops/min
75km/h
60km/h
4km/h

69. Ratio or Rate The cost of pecans is $10.89 for each kilogram
Three out of every seven people are wearing glasses
Mr. Thompson travelled 620km in 6 h
Each block of a quilt has 5 red patches, 4 yellow patches, and 6 blue patches
In 7 games, the team scored a total of 23 points
Rate
Ratio
Rate
Ratio
Rate
Remember that Rates compare two different units and rations compare the same units

70. Word Problem
Conversion Rates among currencies vary from day to day. The numbers in the table below give the value of foreign currency of one Canadian dollar on one particular day.
What was the value of $600 Canadian in euros?
= _x_ (0.6940) = 1(x) = x
What was the value of $375 Canadian in US dollars?
= _x_ (0.8857) = 1(x) = x
What was the value of $450 Canadian in Australian dollars?
= _x_ (1.1527) = 1(x) = x
Canadian US
Australian
European Union
1.00 dollar
dollars
dollars
euros

71. Workbook
Page 125 – 126 #4 - 7

72. 5.10 Comparing Rates

73. Comparing Rates
To Compare different rates, you need to calculate their unit rates
ie. Compare
A case of 12 cartons of juice for $11.76 and
A packet of 3 cartons of the same juice for $2.88
To find the better buy, compare the unit costs of the 2 packages
$11.76 ÷ 12 = $0.98
$2.88 ÷ 3 = $0.96
So the better buy is 3 cartons at $2.88

74. Solving Problems with Rate Comparison
Shamar types 279 words in 4.5min, Tasha types 320 words in 5 min and Cody types 341 words in 5.5 min. Who has the best average typing speed.
Shamar = 279words/4.5min = 62words/1min
Tasha = 320 words/ 5min = 64words/1min
Cody = 341words/5.5min = 62words/1min
Tasha has the best typing speed

75. Another Problem
Troy rides his bike to school. He cycles at an average speed of 20km/h. It takes Troy 24 minutes to get to school.
How far is it from Troy’s home to school?
20km = __x__ 60(x) = 20(24) 60x = 480 8km
One morning, Troy is late leaving. He has 15 minutes to get to school. How much faster will Troy have to cycle to get to school on time?
8km = __x__ 60(8) = 15(x) 480 = 15x 32km
He will have to ride 12km/h faster.

76. WB
Complete pages

77. The Skinny Of It

78. Ratios A part-to-part ratio compares different parts of a group
A part-to-whole ratio compare one part of a group to the whole group
A part-to-whole ratio can be written as a fraction, decimal and percent
A three-term ratio compares three quantities measured in the same units
A two-term ratio compares two quantities measured in the same units

79. Rate
A rate is a comparison of two quantities measured in different units
A rate can be expressed as a fraction that includes the two different units. A rate cannot be expressed as a percent because a percent is a ratio that compares quantities expressed in the same units.
A unit rate is rate in which the second term is one.
A unit price is a unit that makes it easier to compare the cost of similar items

80. Proportion
A proportion is a relationship that two ratios or two rates are equal. A proportion can be expressed in a fraction form.
You can solve proportional reasoning problems using several different methods. A potato farmer can plant three potato plants per 0.5m2. How many potato plants can she plant in an area of 85 m2?
Use a unit rate 3plants:0.5m2 = 6plants: 1m2.
6 x 85 = 510 potato plants or
Use a proportion 3plants:0.5m2 = ? : 85m2.
85 ÷ 0.5 = 170
3 x 170 = 510 potato plants

81. Percents
Fractions, decimals and percents can be used to represent numbers in various situations.
Percents can be written as fractions and as decimals.
You can use mental math strategies such as halving, doubling, and dividing by ten to find the percents of some numbers.
To calculate the percent of a number, write the percent as a decimal and then multiply by the number
Review the two methods from the slides above

82. Definitions

83. Discount
A reduction in price Sometimes discounts are in percent, such as a 10% discount, and then you need to do a calculation to find the price reduction.
to offer for sale or sell at a reduced price: The store discounted all clothing for the sale.

84. Sales Tax
A tax levied on the retail price of merchandise and collected by the retailer.
In BC we currently have HST at 12%

85. Ratio
A ratio shows the relative sizes of two or more values. Ratios can be shown in different ways. Using the ":" to separate example values, or as a single number by dividing one value by the total. Example: if there is 1 boy and 3 girls you could write the ratio as: 1:3 (for every one boy there are 3 girls) 1/4 are boys and 3/4 are girls are boys (by dividing 1 by 4) 25% are boys (0.25 as a percentage)

86. Equivalent Ratios
If two ratios have the same value when simplified, then they are called Equivalent Ratios.
Equivalent ratios can be obtained by multiplying or dividing both sides by the same non-zero number.
The two ratios 8 : 24 and 4 : 12 are equivalent.
There are 10 dolls for every 40 children in a preschool. Then the ratio of the number of children to that of the dolls = 40:10 = 4:1.

87. Rate Rate is a ratio that compares two quantities of different units.
20 oz of juice for $4, kilometers per hour, cost per pound etc. are examples of rate.
Unit rate: Unit rate is a rate in which the second term is 1. For example, Jake types 10 words in 5 seconds. Jake’s unit rate is the number of words he can type in a second. His unit rate is 2 words per second.

88. Proportion
comparative relation between things or magnitudes as to size, quantity, number, etc.; ratio.