1. Ratio is a comparison of two numbers by division.
Proportion is an equation stating that two ratios are equal.
In a proportion, the cross products are equal.

2. If a proportion contains a variable, you can cross multiply to solve for the variable. When you set the cross products equal, you create a linear equation that you can solve by using the skills that you learned in Lesson 2-1.

3. Reading Math
In a ÷ b = c ÷ d, b and c are the means, and a and d are the extremes. In a proportion, the product of the means is equal to the product of the extremes.

4. Check It Out! Example 1
Solve each proportion. y A. = B. = x y x = =
Set cross products equal.
924 = 84y
2.5x =105 y = =
2.5x
Divide both sides.
11 = y
x = 42

5. Because percents can be expressed as ratios, you can use the proportion to solve percent problems.
Percent is a ratio that means per hundred.
For example:
30% = 0.30 =
Remember! 30
100

6. Check It Out! Example 2
At Clay High School, 434 students, or 35% of the students, play a sport. How many students does Clay High School have?
You know the percent and the number of students that play a sport, so you are trying to find the number of students that Clay High School has.

7. Check It Out! Example 2 Continued
Method 1 Use a proportion.
Method 2 Use a percent equation.
Divide the percent by 100.
35% = 0.35
Percent (as decimal) × whole = part
0.35x = 434
Cross multiply.
100(434) = 35x
x = 1240
Solve for x.
x = 1240
Clay High School has 1240 students.

8. A rate is a ratio that involves two different units.
You are familiar with many rates, such as miles per hour (mi/h), words per minute (wpm), or dollars per gallon of gasoline. Rates can be helpful in solving many problems.

9. Example 3: Fitness Application
Ryan ran 600 meters and counted 482 strides. How long is Ryan’s stride in inches? (Hint: 1 m ≈ in.)
Use a proportion to find the length of his stride in meters.
Write both ratios in the form
meters
strides
600 m
482 strides
x m
1 stride =
600 = 482x
Find the cross products.
x ≈ 1.24 m

10. Example 3: Fitness Application continued
Convert the stride length to inches.
is the conversion factor.
39.37 in.
1 m
× ≈
1.24 m
1 stride length
39.37 in.
1 m
49 in.
Ryan’s stride length is approximately 49 inches.

11. Similar figures have the same shape but not necessarily the same size
Similar figures have the same shape but not necessarily the same size. Two figures are similar if their corresponding angles are congruent and corresponding sides are proportional.
The ratio of the corresponding side lengths of similar figures is often called the scale factor.
Reading Math

12. Example 5: Nature Application
A 12-foot flagpole casts a 10 foot-shadow. At the same time, a nearby building casts a 48-foot shadow. How tall is the building?
Sketch the situation. The triangles formed by using the shadows are similar, so you can use a proportion to find h the height of the building.
12 ft
10 ft = 10 12 h 48 =
Shadow of pole
Height of pole
Shadow of building
Height of building
h ft
48 ft
10h = 576
h = 57.6
The building is 57.6 feet high.

13. Example 5: Nature Application
The tree in front of Luka’s house casts a 6-foot shadow at the same time as the house casts a 22-fot shadow. If the tree is 9 feet tall, how tall is the house?
Sketch the situation. The triangles formed by using the shadows are similar, so Luka can use a proportion to find h the height of the house.
9 ft
6 ft = 6 9 h 22 =
Shadow of tree
Height of tree
Shadow of house
Height of house
h ft
22 ft
6h = 198
h = 33
The house is 33 feet high.

14. Check It Out! Example 5
A 6-foot-tall climber casts a 20-foot long shadow at the same time that a tree casts a 90-foot long shadow. How tall is the tree?
Sketch the situation. The triangles formed by using the shadows are similar, so the climber can use a proportion to find h the height of the tree.
6 ft
20 ft = 20 6 h 90 =
Shadow of climber
Height of climber
Shadow of tree
Height of tree
h ft
90 ft
20h = 540
h = 27
The tree is 27 feet high.

15. Lesson Quiz: Part I
Solve each proportion. 2.
3. The results of a recent survey showed that 61.5% of those surveyed had a pet. If 738 people had pets, how many were surveyed?
4. Gina earned $68.75 for 5 hours of tutoring. Approximately how much did she earn per minute?
g = 42
k = 8
1200
$0.23

16. Lesson Quiz: Part II
5. ∆XYZ has vertices, X(0, 0), Y(3, –6), and Z(0, –6). ∆XAB is similar to ∆XYZ, with a vertex at B(0, –4). Graph ∆XYZ and ∆XAB on the same grid. Y Z A B X

17. Lesson Quiz: Part III
6. A 12-foot flagpole casts a 10 foot-shadow. At the same time, a nearby building casts a 48-foot shadow. How tall is the building?
57.6 ft