+ Ch 5 Rate, Ratio and Percent. + 5.1 Relating Fractions, Decimals and Percents

+

Ch 5 Rate, Ratio and Percent

+5.1 Relating Fractions, Decimals and Percents

+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.

+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

+Workbook

Page 100-104

Text pg 240-241 #12-14, 19-20

+

5.2 Calculating Percents

+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

+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%

+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

+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

+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

+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

+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%

+Workbook

Page 105 – 106

+5.3 Solving Percent Problems

+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

+You Try

6% of a number is 9 6% = 9 1% = 100% = 350% =

28% of a number is 56 28% = 56 1% = 100% = 350% =

150% of a number is 36 150% = 36 1% = 100% =

1.5150525

2200700

0.2424

+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.

+Cross Multiply

Mary had $50 before her birthday in her account. After her birthday she had $300. Calculate the percent increase.


Step 2 – express the difference over the original (a fraction) and make it equal to your unknown % over 100% 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%

+You Try

The width of the rectangle increased from 8cm to 12cm



Step 3 – calculate the decimal and than percent 0.5 x 100 = 50%

+Cross Multiply

The width of the rectangle increased from 8cm to 12cm


Step 2 – express the difference over the original (a fraction) and make it equal to your unknown % over 100% 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%

+Another one

The price of a hotel room increased from $90 to $120



Step 3 – calculate the decimal and than percent 0.333 x 100 = 33.33%

+Cross Multiply

The price of a hotel room increased from $90 to $120



90 100


3000 = 90x

90 90

x = 33.33 The percent increase is 33.33%

+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 3 – calculate the decimal and than percent 0.7857 x 100 = 78.57% The pitched decreased in volume for 78.57%

+Cross Multiply

Susie made a pitcher of punch that was 56L, after her party she had 12L left. Calculate the percent decrease.



56 100


4400 = 56x

56 56

x = 78.57 The percent decrease is 78.57%

+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 3 – calculate the decimal and than percent 0.2 x 100 = 20%

+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_

40 100


800 = 40x

40 40

x = 20 The percent decrease is 20%

+You Try

The number of students in the class decreased from 30 – 27



Step 3 – calculate the decimal and than percent 0.1 x 100 = 10 %

+Cross Multiply

The number of students in the class decreased from 30 – 27


Step 2 – express the difference over the original (a fraction) and make it equal to your unknown % over 100% 3_ = _x_

30 100


300 = 30x

30 30

x = 10 The percent decrease is 10%

+Workbook

Try questions 5 – 10 on page 108 - 109

+5.4 Sales Tax and Discount

+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

+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

+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

+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

+Note

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

+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

+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

+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

+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

+Note

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

+Another Example – Both Methods Shown with discount and tax Calculate the sale price on a $476 TV, 15% off, 12% tax

Method 1 0.15 x $476 = $71.40 $476 - $71.4 = $404.6 0.12 x $404.6 = $48.55 $404.6 + $48.55 = $453.15

Method 2 0.85 x $476 = $404.60 1.12 x $404.60 =

$453.15

Or

0.85 x 1.12 x $476 = $453.15

+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

+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%.

+Workbook

Page 110 - 111

+

5.5 Exploring Ratios

+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

+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

+Workbook

Page 112 – 114 together

+

5.6 Equivalent Ratios

+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

+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

Original

x2 x3 x5 ÷2 ÷4

16

28

16

20

20

25

32

56

48

84

80

140

8

14

4

7

+Workbook

Pg 116 # 1-6

+

5.7 Comparing Ratios

+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?

+

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

+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

2 3 5

4 6 10

6

White Brown

Total

3 1

6

+Try Another


The total number of mice in each cage would be 20

The total number of mice would be 40.

White Brown

Total

2 3 5

4 6 10

6 9 15

8 12 20

10 15 25

White Brown

Total

3 1 4

6 2 8

9 3 12

12 4 16

15 5 20

+Try Another


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

2 3 5

4 6 10

6 9 15

8 12 20

10 15 25

White Brown

Total

3 1 4

6 2 8

9 3 12

12 4 16

15 5 20

+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!

+Workbook

Pg 119 – 121

+

5.8 Solving Ratio Problems

+

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

x12

36

+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

+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?

Corn Wheat Rice

3 4 2

225 ? 225 ÷ 3 = 75 2 x 75 = 150

150

3_ = _2_225 x

3(x) = 225(2)

3x =450 3 3 x = 150

+

5.9 Exploring Rates

+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

+Try These

Express as a unit rate 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

Express as a unit rate 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

+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

+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? 0.6940 = _x_ 600(0.6940) = 1(x) 416.4 = x

1 600

What was the value of $375 Canadian in US dollars?0.8857 = _x_ 375(0.8857) = 1(x) 332.14 = x

1 375

What was the value of $450 Canadian in Australian dollars?1.1527 = _x_ 450(1.1527) = 1(x) 518.72 = x

1 450

Canadian US Australian European Union

1.00 dollar 0.8857 dollars 1.1527 dollars 0.6940 euros

+Workbook

Page 125 – 126 #4 - 7

+

5.10 Comparing Rates

+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

+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(x) = 20(24) 60x = 480 8km

60 24 60 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__ 60(8) = 15(x) 480 = 15x 32km

15 60 15 15

He will have to ride 12km/h faster.

+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 girls0.25 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.

Documents

+ Ch 5 Rate, Ratio and Percent. + 5.1 Relating Fractions, Decimals and Percents