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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.

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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.


Practice
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.


3 yards quantities with the same units, if possible.

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 quantities with the same units, if possible.

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.


Practice1
Practice quantities with the same units, if possible.

  • 4 feet to 24 inches

  • 3 yards to 12 feet

  • 2 yards to 20 inches


Notes quantities with the same units, if possible.

Ratios that make the same comparison are equivalent ratios.

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


1 quantities with the same units, if possible.

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


Practice2
Practice quantities with the same units, if possible.


8 quantities with the same units, if possible.

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 quantities with the same units, if possible.

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?


Vocabulary
Vocabulary quantities with the same units, if possible.

  • 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
Examples: Do the ratios form a proportion? quantities with the same units, if possible.

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


Example
Example quantities with the same units, if possible.

÷ 5

3

7

=

8

40

÷ 5


Cross products
Cross Products quantities with the same units, if possible.

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

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


Example 2
Example 2 products.

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


Solving a proportion using cross products
Solving a Proportion Using Cross Products products.

  • Use the cross products to create an equation.

  • Solve the equation for the variable using the inverse operation.


Example 1 solve the proportion
Example 1: products.Solve the Proportion

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

=


Example 2 solve the proportion
Example 2: products.Solve the Proportion

Start with the variable.

Simplify.

Now we have an equation. Solve for x.

2x(30)

5(3)

=

60x

=

15

60

60

x

¼

=


Example 3 solve the proportion
Example 3: products.Solve the Proportion

=

Start with the variable.

Simplify.

Now we have an equation. Solve for x.

(2x +1)3

5(4)

=

6x + 3

=

20

x

=


Example 4 solve the proportion
Example 4: products.Solve the Proportion

=

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

=


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