1 - ratios & proportions

TENNESSEE ADULT EDUCATION

RATIOS & PROPORTIONS

LESSON 1

This curriculum was written with funding of the Tennessee Department of Labor and Workforce Development and may not be reproduced in any way without written permission. ©

RATIOS

For example:

There are eight girls and seven boys in a class.

Ratios are comparisons made between two sets of numbers.

The ratio of girls to boys is 8 to 7.

80 miles to 1 hour = 80mph

Ratios are used everyday. They are used for:

Miles per hourThe cost of items per pound, gallon, etc.Hourly rate of pay

THERE ARE 3 WAYS TO WRITE RATIOS.

For example: There are 8 girls and 5 boys in my class.

What is the ratio of girls to boys?

1. Write the ratio using the word “to” between the two numbers being compared.

The ratio is: 8 girls to 5 boys

8 to 5

2. Write a ratio using a colon between the two numbers being compared.

For example: There are 3 apples and 4 oranges in the basket. What is the ratio of apples to oranges?

The ratio is: 3 apples to 4 oranges.

3 : 4

3. Write a ratio as a fraction.

For example: Hunter and Brandon were playing basketball. Brandon scored 5 baskets and Hunter scored 6 baskets. What was the ratio of baskets Hunter scored to the baskets Brandon scored?

The ratio of baskets scored was:

6 baskets to 5 baskets

65

GUIDED PRACTICE:

There are 13 boys and 17 girls in sixth grade.

Find the ratio of boys to the girls in sixth grade.

Directions: Write the ratio in three different ways.

13 to 17131713 : 17

RULES FOR SOLVING RATIO PROBLEMS.

1. When writing ratios, the numbers should be written in the order in which the problem asks for them.

For example: There were 4 girls and 7 boys at the birthday party.

What is the ratio of girls to boys?

4 girls to 7 boys 4 girls : 7 boys 4 girls 7 boys

Hint: The question asks for girls to boys; therefore, girls will be listed first in the ratio.

GUIDED PRACTICE:

Directions: Solve and write ratios in all three forms.

1. The Panthers played 15 games this season. They won 13 games. What is the ratio of games won to games played?

The questions asks for Games won to Games played.

13 to 15

13:151315

THE QUESTION ASKS FOR GAMES LOST TO GAMES WON. THEREFORE, THE NUMBER OF GAMES LOST SHOULD BE WRITTEN FIRST, AND THE GAMES WON SHOULD BE WRITTEN SECOND.

2. Amanda’s basketball team won 7 games and lost 5. What is the ratio of games lost to games won?

Games lost = 5 to Games won = 7

5 to 7 5 7

5:7

REDUCING RATIOSRatios can be reduced without changing their relationship.

2 boys to 4 girls =

1 boy to 2 girls =

REDUCING RATIOSIs this relationship the same?

2 boys to 4 girls =

1 boy to 3 girls =

2. ALL RATIOS MUST BE WRITTEN IN LOWEST TERMS.

For example: You scored 40 answers correct out of 45 problems on a test. Write the ratio of correct answers to total questions in lowest form. Step 1: Read the problem. What does it want to know?

Steps:1. Read the word problem.

2. Set up the ratio.

40 : 4540 to 45 4045

3. Reduce the ratio if necessary.

Reduce means to break down a fraction or ratio into the lowest form possible.

40 to 45

HINT: When having to reduce ratios, it is better to set up the ratio in the vertical form. (Fraction Form)

=4045

Reduce = smaller number; operation will always be division.

4045

Look at the numbers in the ratio. What ONE number can you divide BOTH numbers by?

÷÷

55

==

89

Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40Factors of 45: 1, 3, 5, 9, 15, 45

Guided Practice:

Directions: Solve each problem. Remember to reduce.

1. There are 26 black cards in a deck of playing cards. If there are 52 cards in a deck, what is the ratio of black cards to the deck of cards?

Step 1: Read the problem. (What does it want to know?)Step 2: Set up the ratio.

26 black cards to 52 cards

Step 3: Can the ratio be reduced? If so, set it up like a fraction.

2652 What is the largest number that will go

into both the top number and the bottom number evenly? (It can not be the number one!)

÷÷

2626

==

12

2. Kelsey has been reading Hunger Games for class. She read 15 chapters in 3 days. What is the ratio of chapters read to the number of days she read?

15 chapters to 3 days

15 3

÷÷

33

==

51

Hint: When a one is on the bottom, it must remain there. If the one is dropped, there is no longer a ratio.

PROPORTIONSProportions are two ratios of equal value.

1 girl 4 girls

4 boys 16 boys

Are these ratios saying the same thing?

PROPORTIONSProportions are two ratios of equal value.

1 girl 5 girls

4 boys 16 boys

Are these ratios proportions?

DETERMINING TRUE PROPORTIONS:To determine a proportion true, cross multiply.

45

2025

=

4 x 2520 x 5 =

If the cross products are equal, then it is a true proportion.

100 = 100

The cross products were equal, therefore 4 And 20 makes a true proportion.

5 25

Directions: Solve to see if each problem is a true proportion.

Guided Practice:

3 155 25=1

.

2. 68

= 5776

3. 712

= 3760

Directions: Solve to see if each problem is a true proportion.

Guided Practice:

3 155 25=1. 2

.

68 =

5776 3

.

712

= 3760

15 x 5 = 3 x 25

75 = 75

true

57 x 8 = 6 x 76

456 = 456

true

7 x 60 = 37 x 12

420

= 444

false

SOLVING PROPORTIONS WITH VARIABLESWhat is a variable?

Eric rode his bicycle a total of 52 miles in 4 hours. Riding at this same rate, how far can he travel in 7 hours?

Look for the two sets of ratios to make up a proportion.

You have 52 miles in 4 hours. This is the first ratio.

52 miles4 hours

Next, the problem states “how far can he travel in 7 hours. The problem is missing the miles. Therefore, the miles becomes the variable.

n miles7 hours

Set 1 Set 2

The proportion should be set equal to each other.

52 4

=n7

HINT: The order of the ratio does matter!

A variable is any letter that takes place of a missing number or information.

SOLVING THE PROPORTION:

3. Check the answer to see if it makes a true proportion.

Problem:

52 x 74 x n =

Which number is connected to the variable?

4n = 364

Since the 4 is connected to the variable, DIVIDE both sides by the 4.

4 4 ÷ 4 = 1;

therefore you are left with “n” on one side.

When solving proportions, follow these rules:

1. Cross multiply.

2. Divide BOTH sides by the number connected to the variable.

524

= n 7

n =

364 ÷ 4 =

91

91 miles4

Check your answer!

52 4 =

917

52 x 7 = 91 x 4 364 = 364

If it comes out even, then the answer is correct.

GUIDED PRACTICE

Directions: Solve each proportion.

1.For every dollar Julia spends on her Master Card, she earns 3 frequent flyer miles with American Airlines. If Julia spends $609 dollars on her card, how many frequent flyer miles will she earn?

GUIDED PRACTICE

Directions: Solve each proportion.

1. For every dollar Julia spends on her Master Card, she earns 3 frequent flyer miles with American Airlines. If Julia spends $609 dollars on her card, how many frequent flyer miles will she earn?

Step 1: Set up the proportion.$1.00 3 miles

= $609.00 d miles

Step 2: Cross multiply. 1d = 1827 Step 3: Divide

1 1Step 4: Check answer. d = 1827

1. Justin’s car uses 40 gallons of gas to drive 250 miles. At this rate, approximately, how many gallons of gas will he need for a trip of 600 miles.

2. If 3 gallons of milk cost $9, how many jugs can you buy for $45?

3. On Thursday, Karen drove 400 miles in 8 hours. At this same speed, how far can she drive in 12 hours?

1. Justin’s car uses 40 gallons of gas to drive 250 miles. At this rate, approximately, how many gallons of gas will he need for a trip of 600 miles.

40 gal250 mi

x gal 600mi

=

250x 24000=

250x250

24000 250

=

x = 96

40 250

x 600

=

40 250

96600

=

2400024000

Check:

=

2. If a 3 gallon jug of milk cost $9, how many 3 gallon jugs can be purchased for $45?

5 jugs of milk can be purchased for $45

19

n 45

=

19

n 45

=

1x45 9n=

45 9

9n 9

=

5 = n

19

5 45=

Check:

45 = 45

3. On Thursday, Karen drove 400 miles in 8 hours. At this same speed, how far can she drive in 12 hours?

400 8

x_ 12

=400 miles 8 hours

x miles12 hours

=

400 8

x_ 12

=

8x 4800=

x 600 miles=

4. Susie has two flower beds in which to plant tulips and daffodils. She wants the proportion of tulips to daffodils to be the same in each bed. Susie plants 10 tulips and 6 daffodils in the first bed. How many tulips will she need for the second bed if she plants 15 daffodils?

10 6

x_ 15

=10 tulips 6 daffodils

x tulips15 daffodils

=

10 6

x15

=

6x 150=

6x 6

150 6

=

x 25=

x 25 tulips=

10 6

25 15

=

150 150=