4-1 RATIOS & PROPORTIONS

A ratio is a comparison of two quantities.

Ratios can be written in several ways.

7 to 5, 7:5, and name the same ratio.75

Notes

Example 1: Writing Ratios in Simplest Form

Write the ratio 15 bikes to 9 skateboards in simplest form.

159

53

The ratio of bikes to skateboards is , 5:3, or 5 to 3.

=

15 ÷ 39 ÷ 3

Write the ratio as a fraction.

= = Simplify.

53

bikesskateboards

Check It Out! Example 2

Write the ratio 24 shirts to 9 jeans in simplest form.

249

83

The ratio of shirts to jeans is , 8:3, or 8 to 3.

=shirtsjeans

24 ÷ 39 ÷ 3

Write the ratio as a fraction.

= = Simplify.

83

Practice

15 cows to 25 sheep

24 cars to 18 trucks

30 Knives to 27 spoons

When simplifying ratios based on measurements, write the quantities with the same units, if possible.

Write the ratio 3 yards to 12 feet in simplest form.

First convert yards to feet.

9 feet12 feet

=3 yards12 feet

34

=9 ÷ 312 ÷ 3

=

There are 3 feet in each yard.

Example 3: Writing Ratios Based on Measurement

3 yards = 3 ● 3 feet

= 9 feet Multiply.

Now write the ratio.

Simplify.

The ratio is , 3:4, or 3 to 4.34

Write the ratio 36 inches to 4 feet in simplest form.

First convert feet to inches.

36 inches48 inches

=36 inches4 feet

34

=36 ÷ 1248 ÷ 12

=

There are 12 inches in each foot.

Check It Out! Example 3

4 feet = 4 ● 12 inches

= 48 inches Multiply.

Now write the ratio.

Simplify.

The ratio is , 3:4, or 3 to 4.34

Practice

4 feet to 24 inches

3 yards to 12 feet

2 yards to 20 inches

Ratios that make the same comparison are equivalent ratios.

To check whether two ratios are equivalent, you can write both in simplest form.

Notes

Example 4: Determining Whether Two Ratios Are Equivalent

Simplify to tell whether the ratios are equivalent.

1215

B. and 2736

327

A. and 218

Since ,

the ratios are

equivalent.

19

= 19

19

=3 ÷ 327 ÷ 3

327

=

19

=2 ÷ 218 ÷ 2

218

=

45=

12 ÷ 315 ÷ 3

1215

=

34=

27 ÷ 936 ÷ 9

2736

=

Since ,

the ratios are not

equivalent.

45

34

Practice

56

28

49

2148

16

39

13

and

and

Lesson Quiz: Part I

Write each ratio in simplest form.

1. 22 tigers to 44 lions

2. 5 feet to 14 inches

415

3.

721

4.

830

1245

Possible answer: ,

13

1442

Possible answer: ,

Find a ratio that is equivalent to each given ratio.

12307

Lesson Quiz: Part II

7. Kate poured 8 oz of juice from a 64 oz bottle. Brian poured 16 oz of juice from a 128 oz bottle. Are the ratios of poured juice to starting amount of juice equivalent?8

6416

128and ; yes, both equal 1

8

85

85

= ; yes1610

5.

3624

6.

Simplify to tell whether the ratios are equivalent.

and 32 20

and 28 18

32

149

; no

Vocabulary A proportion is an equation stating that two

ratios are equal.

To prove that two ratios form a proportion, you must prove that they are equivalent. To do this, you must demonstrate that the relationship between numerators is the same as the relationship between denominators.

Examples: Do the ratios form a proportion?

710

, 2130

x 3

x 3

Yes, these two ratios DO form a proportion, because the same relationship exists in both the numerators and denominators.

89

, 23

÷ 4

÷ 3

No, these ratios do NOT form a proportion, because the ratios are not equal.

Example

3

40=

7

÷ 5

÷ 5

8

Cross Products

When you have a proportion (two equal ratios), then you have equivalent cross products.

Find the cross product by multiplying the denominator of each ratio by the numerator of the other ratio.

Example: Do the ratios form a proportion? Check using cross products.

412

, 39

12 x 3 = 369 x 4 = 36

These two ratios DO form a proportion because their cross products are the same.

Example 2

58

, 23

8 x 2 = 163 x 5 = 15

No, these two ratios DO NOT form a proportion, because their cross products are different.

Solving a Proportion Using Cross Products

Use the cross products to create an equation.

Solve the equation for the variable using the inverse operation.

Example 1: Solve the Proportion

k17

=2068

Start with the variable.

=68k 17(20)

Simplify.

68k = 340

Now we have an equation. To get the k by itself, divide both sides by 68.

68 68

k = 5

Example 2: Solve the Proportion

Start with the variable.

=2x(30) 5(3)

Simplify.

60x = 15

Now we have an equation. Solve for x.

60 60

x = ¼

Example 3: Solve the Proportion

Start with the variable.

=(2x +1)3 5(4)

Simplify.

6x + 3 = 20

Now we have an equation. Solve for x.

x =

=

Example 4: Solve the Proportion

Cross Multiply.

=3x 4(x+2)

Simplify.

3x = 4x + 8

Now we have an equation with variables on both sides. Solve for x.

x = -8

=