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4-1 Ratios & Proportions

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4-1 Ratios & Proportions

Notes

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.

15 ÷ 3

9 ÷ 3

bikes

skateboards

Example 1: Writing Ratios in Simplest Form

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

15

9

Write the ratio as a fraction.

=

5

3

Simplify.

=

=

5

3

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

shirts

jeans

24 ÷ 3

9 ÷ 3

Check It Out! Example 2

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

Write the ratio as a fraction.

24

9

=

8

3

Simplify.

=

=

8

3

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

- 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.

3 yards

12 feet

9 ÷ 3

12 ÷ 3

=

=

=

3

4

3

4

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

Example 3: Writing Ratios Based on Measurement

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

First convert yards to feet.

3 yards = 3 ● 3 feet

There are 3 feet in each yard.

Multiply.

= 9 feet

Now write the ratio.

9 feet

12 feet

Simplify.

36 inches

4 feet

36 ÷ 12

48 ÷ 12

=

=

=

3

4

3

4

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

Check It Out! Example 3

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

First convert feet to inches.

4 feet = 4 ● 12 inches

There are 12 inches in each foot.

= 48 inches

Multiply.

Now write the ratio.

36 inches

48 inches

Simplify.

- 4 feet to 24 inches
- 3 yards to 12 feet
- 2 yards to 20 inches

Notes

Ratios that make the same comparison are equivalent ratios.

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

1

9

1

9

12

15

3

27

27

36

2

18

Since ,

the ratios are equivalent.

B.

A.

=

and

and

2

18

3

27

2 ÷ 2

18 ÷ 2

3 ÷ 3

27 ÷ 3

=

=

=

=

4

5

3

4

Since ,

the ratios are not equivalent.

12

15

27

36

12 ÷ 3

15 ÷ 3

27 ÷ 9

36 ÷ 9

=

=

=

=

Example 4: Determining Whether Two Ratios Are Equivalent

Simplify to tell whether the ratios are equivalent.

1

9

1

9

4

5

3

4

8

30

30

7

1

2

12

45

Possible answer: ,

Possible answer: ,

4

15

7

21

3.

4.

1

3

14

42

Lesson Quiz: Part I

Write each ratio in simplest form.

1. 22 tigers to 44 lions

2. 5 feet to 14 inches

Find a ratio that is equivalent to each given ratio.

36

24

16

10

5.

6.

8

64

16

128

and ; yes, both equal

28 18

32 20

8

5

3

2

14

9

8

5

1 8

=

; yes

; no

Lesson Quiz: Part II

Simplify to tell whether the ratios are equivalent.

and

and

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?

- 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.

x 3

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

7

21

,

10

30

x 3

÷ 4

8

2

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

,

9

3

÷ 3

÷ 5

3

7

=

8

40

÷ 5

- 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.

4

3

,

12

9

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

12 x 3 = 36

9 x 4 = 36

5

2

,

8

3

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

8 x 2 = 16

3 x 5 = 15

- Use the cross products to create an equation.
- Solve the equation for the variable using the inverse operation.

Start with the variable.

20

k

=

17

68

Simplify.

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

68k

17(20)

=

68k

=

340

68

68

k

5

=

Start with the variable.

Simplify.

Now we have an equation. Solve for x.

2x(30)

5(3)

=

60x

=

15

60

60

x

¼

=

=

Start with the variable.

Simplify.

Now we have an equation. Solve for x.

(2x +1)3

5(4)

=

6x + 3

=

20

x

=

=

Cross Multiply.

Simplify.

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

3x

4(x+2)

=

3x

=

4x + 8

x

-8

=