Transcript 1 - SD43 Teacher Sites

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Ch 5 Rate, Ratio and
Percent
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5.1 Relating Fractions,
Decimals and Percents
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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.
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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= 0.4375


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
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Workbook

Page 100-104

Text pg 240-241 #12-14, 19-20
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5.2 Calculating Percents
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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 3 0 = 2.30

0.09% =0 0 0 0 . 0 0 9 = 0.00009
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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 = 0.5 0 = 50%

3/2 = 1.5 = 1 . 5 0 = 150%

3/200 = 0.015 = 0 . 0 1 5 = 1.5%
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Complete the chart
Percent
Decimal
Fraction
1%
0.01
1/
100
5%
1/
10
0.125
20%
1/
4
0.333...
50%
3/
4
0.8
90%
99/
100
1
125%
3/
2
2
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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%
0.333...
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
2
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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
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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
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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 = 0.6666 =
66.66%
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Workbook

Page 105 – 106
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5.3 Solving Percent Problems
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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 100. This also works if you want 85%, 115%, etc.

100% = 4 x 100 = 400

155% = 4 x 155 = 620
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You Try

6% of a number is 9





28% of a number is 56





6% = 9
1% = 1.5
100% = 150
350% = 525
28% = 56
1% = 2
100% = 200
350% = 700
150% of a number is 36



150% = 36
1% = 0.24
100% = 24
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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


Step 2 – express the difference over the original (a fraction)


300 – 50 = 250
250/50
Step 3 – calculate the decimal and than percent

5 x 100 = 500%

She had a 500% increase after her birthday.
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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


Step 2 – express the difference over the original (a fraction) and make it
equal to your unknown % over 100%


300 – 50 = 250
250 = x__
50
100
Step 3 – cross multiply and divide to solve
50(x) = 250(100)
50x = 25000
50
50
x = 500 The percent increase is 500%
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You Try

The width of the rectangle increased from 8cm to 12cm

Step 1 – calculate the difference between the two numbers


Step 2 – express the difference over the original (a fraction)


12 – 8 = 4
4/8
Step 3 – calculate the decimal and than percent

0.5 x 100 = 50%
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Cross Multiply

The width of the rectangle increased from 8cm to 12cm

Step 1 – calculate the difference between the two numbers


Step 2 – express the difference over the original (a fraction) and
make it equal to your unknown % over 100%


12 – 8 = 4
4 = _x_
8 100
Step 3 – cross multiply and divide to solve
4(100) = 8(x)
400 = 8x
8
8
x = 50 The percent increase is 50%
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Another one

The price of a hotel room increased from $90 to $120

Step 1 – calculate the difference between the two numbers


Step 2 – express the difference over the original (a fraction)


120 – 90 = 30
30/90
Step 3 – calculate the decimal and than percent

0.333 x 100 = 33.33%
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Cross Multiply

The price of a hotel room increased from $90 to $120

Step 1 – calculate the difference between the two numbers


Step 2 – express the difference over the original (a fraction) and
make it equal to your unknown % over 100%


120 – 90 = 30
30 = _x_
90 100
Step 3 – cross multiply and divide to solve
30(100) = 90(x)
3000 = 90x
90
90
x = 33.33 The percent increase is 33.33%
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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


Step 2 – express the difference over the original (a fraction)


56 – 12 = 44
44/56
Step 3 – calculate the decimal and than percent

0.7857 x 100 = 78.57%

The pitched decreased in volume for 78.57%
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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


Step 2 – express the difference over the original (a fraction) and make it
equal to your unknown % over 100%


56 – 12 = 44
44 = _x_
56 100
Step 3 – cross multiply and divide to solve
44(100) = 56(x)
4400 = 56x
56
56
x = 78.57 The percent decrease is 78.57%
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You Try

The volume of water in the tank decreased from 40L to 32L.

Step 1 – calculate the difference between the two numbers


Step 2 – express the difference over the original (a fraction)


40L – 32L = 8L
8/40
Step 3 – calculate the decimal and than percent

0.2 x 100 = 20%
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Cross Multiply

The volume of water in the tank decreased from 40L to 32L.

Step 1 – calculate the difference between the two numbers


Step 2 – express the difference over the original (a fraction) and
make it equal to your unknown % over 100%


40L – 32L = 8L
8_ = _x_
40 100
Step 3 – cross multiply and divide to solve
8(100) = 40(x)
800 = 40x
40 40
x = 20 The percent decrease is 20%
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You Try

The number of students in the class decreased from 30 – 27

Step 1 – calculate the difference between the two numbers


Step 2 – express the difference over the original (a fraction)


30 – 27 = 3
3/30
Step 3 – calculate the decimal and than percent

0.1 x 100 = 10 %
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Cross Multiply

The number of students in the class decreased from 30 – 27

Step 1 – calculate the difference between the two numbers


Step 2 – express the difference over the original (a fraction) and
make it equal to your unknown % over 100%


30 – 27 = 3
3_ = _x_
30 100
Step 3 – cross multiply and divide to solve
3(100) = 30(x)
300 = 30x
30 30
x = 10 The percent decrease is 10%
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Workbook

Try questions 5 – 10 on page 108 - 109
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5.4 Sales Tax and Discount
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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
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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
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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
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Another Example – Both Methods
Shown

Calculate the sale price on a $92 watch, 30% off

Method 1

0.3 x $92 = $27.60

$92 - $27.60 = $64.40

Method 2

0.7 x $92 = $64.40
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Note

Only use method 2 if you are calculating the sale price – not
if you are asked to calculate the discount only.
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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
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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
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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
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Another Example – Both Methods
Shown

Calculate the sale price on a $92 watch, 12%

Method 1

0.12 x $92 = $11.04

$92 + $11.04 = $103.04

Method 2

1.12 x $92 = $103.04
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Note

Only use method 2 if you are calculating the final price – not
if you are asked to calculate the tax only.
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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 $404.60 = $453.15

0.12 x $404.6 = $48.55

$404.6 + $48.55 = $453.15

Or

0.85 x 1.12 x $476 = $453.15
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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

$210 x 0.85 = $178.5

$178.5 x 0.9 = $160.65

$300 x 0.45 = $135

$160.65 ≠ $135
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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%.
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Workbook

Page 110 - 111
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5.5 Exploring Ratios
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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
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Write each ratio

A pencil case contains 7 yellow, 3 red, 1 black and 5 green pencil
crayons. Write Each Ratio

Red: green


Black: total pencil crayons


7:3:5
Yellow: red


1:16
Yellow: red: green


3:5
7:3
Yellow: total pencil crayons

7:16
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Workbook

Page 112 – 114 together
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5.6 Equivalent Ratios
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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
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Give it a try

Write three ratios that are equivalent to each ratio

4:5

Original
x2
x3
4
8
12
5
10
15
x4
x5
16
20
20
25
16:28
Original
x2
x3
16
32
48
28
56
84
x5
80
140
÷2
8
14
÷4
4
7
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Workbook

Pg 116 # 1-6
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5.7 Comparing Ratios
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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?
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
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
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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
White
Brown
2
3
5
3
1
4
6
10
6
6
Total
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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
White
Brown
Total
2
3
5
3
1
4
4
6
10
6
2
8
6
9
15
9
3
12
8
12
20
12
4
16
10
15
25
15
5
20
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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
White
Brown
Total
2
3
5
3
1
4
4
6
10
6
2
8
6
9
15
9
3
12
8
12
20
12
4
16
10
15
25
15
5
20
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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!
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Workbook

Pg 119 – 121
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5.8 Solving Ratio Problems
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
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
36
48
x12
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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)
6 t
t = 6 x 24 = 144
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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_
225
x
3(x) = 225(2)
3x =450
3
3
x = 150
225 ÷ 3 = 75
Corn
Wheat
Rice
3
4
2
225
? 150
2 x 75 = 150
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5.9 Exploring Rates
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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
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Try These

Express as a unit rate
4km/h

Serena walks 4 km in 1 h =

Sanjit reads 3 books in 1week = 3 books/week
The tap drips 25 drops in 1 min = 25 drops/min


Those were easy now try

Express as a unit rate

Betty drives 150km in 2 h. = 75km/h

The helicopter travels 180km in 3 h. = 60km/h
Gerald walks 1 km in 15min = 4km/h

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Ratio or Rate

The cost of pecans is $10.89 for each kilogram Rate

Three out of every seven people are wearing glasses Ratio

Mr. Thompson travelled 620km in 6 h

Each block of a quilt has 5 red patches, 4 yellow patches, and
6 blue patches
Ratio

In 7 games, the team scored a total of 23 points
Rate
Rate
Remember that Rates compare two different units and rations compare the
same units
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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.
Canadian
US
Australian
European Union
1.00 dollar
0.8857 dollars
1.1527 dollars
0.6940 euros

What was the value of $600 Canadian in euros?
0.6940 = _x_
1
600

416.4 = x
What was the value of $375 Canadian in US dollars?
0.8857 = _x_
1
375

600(0.6940) = 1(x)
375(0.8857) = 1(x)
332.14 = x
What was the value of $450 Canadian in Australian dollars?
1.1527 = _x_
1
450
450(1.1527) = 1(x)
518.72 = x
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Workbook

Page 125 – 126 #4 - 7
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5.10 Comparing Rates
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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
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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
+
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

60(x) = 20(24)
24
60x = 480
60
8km
60
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__
15
60(8) = 15(x)
60
He will have to ride 12km/h faster.
480 = 15x
15
15
32km
+
WB

Complete pages 127 - 129
+
The Skinny Of It
+
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
+
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
+
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
+
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
+
Definitions
+
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.
+
Sales Tax

A tax levied on the retail price of merchandise and collected
by the retailer.

In BC we currently have HST at 12%
+
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
0.25 are boys (by dividing 1 by 4)
25% are boys (0.25 as a percentage)
+
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.
+
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.
+
Proportion

comparative relation between things or magnitudes as to
size, quantity, number, etc.; ratio.