solving proportions

Solving Proportions

Warm UpLesson

Warm UpSolve each equation.

1.

Multiply.

3.

5. 7.

3.6 48

7

2. 5m = 18

Change each percent to a decimal.

4. 10

Change each fraction to a decimal.

9.

6. 112% 8. 1% 73% 0.6%

10.

0.73 0.0061.12

0.01

0.5 0.3

ratio scale rate scale modelcross productsscale drawingproportion unit ratepercent

Vocabulary

A ratio is a comparison of two quantities. The ratio of a to b can be written as a:b or , where b ≠ 0.

A statement that two ratios are equal, such as

is called a proportion.

Additional Example 1: Using Ratios

Write a ratio comparing bones in ears to bones in skull.

Write a proportion. Let x be the number of bones in ears.

Since x is divided by 22, multiply both sides of the equation by 22.

There are 6 bones in the ears.

The ratio of the number of bones in a human’s ears to the number of bones in the skull is 3:11. There are 22 bones in the skull. How many bones are in the ears?

Your Turn! Example 1

The ratio of red marbles to green marbles is 6:5. There are 18 red marbles. How many green marbles are there?

greenred

56

Write a ratio comparing green to red marbles.

15 = x

Write a proportion. Let x be the number green marbles.

Since x is divided by 18, multiply both sides by 18.

There are 15 green marbles.

A common application of proportions is rates. A

rate is a ratio of two quantities with different

units, such as Rates are usually written as

unit rates. A unit rate is a rate with a second

quantity of 1 unit, such as or 17 mi/gal. You

can convert any rate to a unit rate.

Additional Example 2: Finding Unit Rates

Ralf Laue of Germany flipped a pancake 416 times in 120 seconds to set the world record. Find the unit rate. Round your answer to the nearest hundredth.

Write a proportion to find an equivalent ratio with a second quantity of 1.

3.47 ≈ x Divide on the left side to find x.

The unit rate is approximately 3.47 pancake flips per second.

Your Turn! Example 2a

Cory earns $52.50 in 7 hours.

Find the unit rate. Round to the nearest hundredth if necessary.

7.50 = x


Divide on the left side to find x.

The unit rate is $7.50 per hour.

Your Turn! Example 2b

Find the unit rate. Round to the nearest hundredth if necessary.

A machine seals 138 envelopes in 23 minutes.

6 = x


Divide on the left side to find x.

The unit rate is 6 envelopes seals per minute.

In the proportion the products a d and b c are called cross products. You can solve a proportion for a missing value by using the Cross Products Property

Additional Example 3A: Solving Proportions

Solve the proportion.

3m = 45

3(m) = 9(5)

m = 15

Use cross products.

Divide both sides by 3.

Additional Example 3B: Solving Proportions

Solve the proportion.

Use cross products.

6(7) = 2(y – 3)

42 = 2y – 6 +6 +648 = 2y

24 = y

Add 6 to both sides.


Solve the proportion. Check your answer.


–5(8) = 2(y)

–40 = 2y

–20 = y

Use cross products.


Solve the proportion. Check your answer.

4g + 12 = 35

4(g + 3) = 5(7)

g = 5.75

Use cross product.


–12 –12 4g = 23

Subtract 12 from both sides.


Another common application of proportions is

percents. A percent is a ratio that compares a

number to 100. For example, 25% =

You can use the proportion to

find unknown values.

Additional Example 4A: Percent Problems

Find 30% of 80.

Method 1 Use a proportion.

100x = 2400

x = 24

30% of 80 is 24.

Use the percent proportion.

Let x represent the part.

Find the cross product. Since x is multiplied by 100, divide both sides to undo the multiplication.

Additional Example 4B: Percent Problems

230 is what percent of 200?

Method 2 Use an equation.

230 = x 200

230 = 200x

1.15 = x

115% = x

230 is 115% of 200.

Write an equation. Let x represent the percent.

Since x is multiplied by 200, divide both sides by 200 to undo the multiplication.

The answer is a decimal.

Write the decimal as a percent. This answer is reasonable; 230 is more than 100% of 200.

Additional Example 4C: Percent Problems

20 is 0.4% of what number?


2000 = 0.4x


Let x represent the whole.

5000 = x

Cross multiply.

Since x is multiplied by 0.4, divide both sides by 0.4.

20 is 0.4% of 5000.


Find 20% of 60.


100x = 1200

x = 12

20% of 60 is 12.


Let x represent the part.

Find the cross product. Since x is multiplied by 100, divide both sides to undo the multiplication.


48 is 15% of what number?


4800 = 15x

x = 320

48 is 15% of 320.


Let x represent the whole.

Find the cross product. Since x is multiplied by 15, divide both sides by 15 to undo the multiplication.

Proportions are used to create scale drawings and scale models. A scale is a ratio between two sets of measurements, such as 1 in.:5 mi. A scale drawing, or scale model, uses a scale to represent an object as smaller or larger than the actual object. A map is an example of a scale drawing.

Additional Example 5A: Scale Drawings and Scale Models

A contractor has a blueprint for a house drawn to the scale 1 in.:3 ft.

A wall on the blueprint is 6.5 inches long. How long is the actual wall?

x 1= 3(6.5)x = 19.5

The actual length is 19.5 feet.

Write the scale as a fraction.

Let x be the actual length.

Use cross products to solve.

Additional Example 5B: Scale Drawings and Scale Models


Let x be the blueprint length.

x 3 = 1(12)x = 4

The blueprint length is 4 inches.


A contractor has a blueprint for a house drawn to the scale 1 in.:3 ft.

A wall in the house is 12 feet long. How long is the wall on the blueprint?

A scale written without units, such as 32:1, means that 32 units of any measure corresponds to 1 unit of that same measure.

Reading Math


The actual distance between North Chicago and Waukegan is 4 mi. What is the distance between these two locations on the map?

18x = 4

x ≈ 0.2

The distance on the map is about 0.2 in.


Let x be the map distance.



A scale model of a human heart is 16 ft long. The scale is 32:1 How many inches long is the actual heart that the model represents?

32x = 16

x = 0.5

The actual heart is 0.5 feet or 6 inches.


Let x be the actual distance.


solving proportions

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