1. Proportional Reasoning
2-2
Proportional Reasoning
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 2

2. Warm Up
Write as a decimal and a percent. 1. 2.
0.4; 40%
1.875; 187.5%

3. Graph on a coordinate plane.
Warm Up Continued
Graph on a coordinate plane.
3. A(–1, 2)
4. B(0, –3)
A(–1, 2)
B(0, –3)

4. Warm Up Continued
5. The distance from Max’s house to the park is mi. What is the distance in feet? (1 mi = 5280 ft)
18,480 ft

5. Objective
Apply proportional relationships to rates, similarity, and scale.

6. Vocabulary
ratio
proportion
rate
similar
indirect measurement

7. Recall that a ratio is a comparison of two numbers by division and a proportion is an equation stating that two ratios are equal. In a proportion, the cross products are equal.

8. 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.

9. Example 1: Solving Proportions
Solve each proportion. c = p A. B. = p c = =
206.4 = 24p
Set cross products equal.
88c = 1848 p =
88c =
Divide both sides.
8.6 = p
c = 21

10. 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

11. 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

12. Example 2: Solving Percent Problems
A poll taken one day before an election showed that 22.5% of voters planned to vote for a certain candidate. If 1800 voters participated in the poll, how many indicated that they planned to vote for that candidate?
You know the percent and the total number of voters, so you are trying to find the part of the whole (the number of voters who are planning to vote for that candidate).

13. Method 1 Use a proportion. Method 2 Use a percent equation.
Example 2 Continued
Method 1 Use a proportion.
Method 2 Use a percent equation.
Divide the percent by 100.
Percent (as decimal)  whole = part
0.225  1800 = x
Cross multiply.
22.5(1800) = 100x
405 = x
Solve for x.
x = 405
So 405 voters are planning to vote for that candidate.

14. A rate is a ratio that involves two different units
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.

15. 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

16. Example 3: Scaling Geometric Figures in the Coordinate Plane
∆XYZ has vertices X(0, 0), Y(–6, 9) and Z(0, 9).
∆XAB is similar to ∆XYZ with a vertex at B(0, 3).
Graph ∆XYZ and ∆XAB on the same grid.
Step 1 Graph ∆XYZ. Then draw XB.

17. Example 3 Continued
Step 1 Y Z
graph ∆XYZ and vertex B. B A X

18. Example 4 Continued
Step 2 To find the width of ∆XAB, use a proportion. =
height of ∆XAB width of ∆XAB
height of ∆XYZ width of ∆XYZ = x
9x = 18, so x = 2

19. To graph ∆XAB, first find the coordinate of A.
Example 4 Continued
Step 3
To graph ∆XAB, first find the coordinate of A. Y Z
The width is 2 units,
and the height is 3 units, so the
coordinates of A are (–2, 3). B A X

20. 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.

21. 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

22. 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

23. 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