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Holt McDougal Algebra 2
2-2 Proportional Reasoning 2-2 Proportional Reasoning
Holt Algebra 2
Warm UpWarm Up
Lesson PresentationLesson Presentation
Lesson QuizLesson Quiz
Holt McDougal Algebra 2
Holt McDougal Algebra 2
2-2 Proportional Reasoning
Warm UpWrite as a decimal and a percent.
1.
2.
0.4; 40%
1.875; 187.5%
Holt McDougal Algebra 2
2-2 Proportional Reasoning
Warm Up Continued
Graph on a coordinate plane.
3. A(–1, 2)
4. B(0, –3) A(–1, 2)
B(0, –3)
Holt McDougal Algebra 2
2-2 Proportional Reasoning
Warm Up Continued
5. The distance from Max’s house to the park is 3.5 mi. What is the distance in feet? (1 mi = 5280 ft)
18,480 ft
Holt McDougal Algebra 2
2-2 Proportional Reasoning
Apply proportional relationships to rates, similarity, and scale.
Objective
Holt McDougal Algebra 2
2-2 Proportional Reasoning
ratioproportionratesimilarindirect measurement
Vocabulary
Holt McDougal Algebra 2
2-2 Proportional Reasoning
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.
Holt McDougal Algebra 2
2-2 Proportional Reasoning
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.
Holt McDougal Algebra 2
2-2 Proportional Reasoning
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.
Reading Math
Holt McDougal Algebra 2
2-2 Proportional Reasoning
Solve each proportion.
Example 1: Solving Proportions
A.
206.4 = 24p Set cross products equal.
=
=16 24 p 12.9
16 24 p 12.9
206.4 24p 24 24 Divide both sides.
8.6 = p
14 c 88 132
=
=
=
B.
14 c 88 132
88c = 1848
=88c 184888 88
c = 21
Holt McDougal Algebra 2
2-2 Proportional Reasoning
Solve each proportion.
A.
924 = 84y Set cross products equal.
=
=
y 77 12 84
y 77 12 84
Divide both sides.
11 = y
15 2.5 x 7
=
924 84y 84 84
=
=
B.
15 2.5 x 7
2.5x =105
=2.5x 1052.5 2.5
x = 42
Check It Out! Example 1
Holt McDougal Algebra 2
2-2 Proportional Reasoning
Percent is a ratio that means per hundred.
For example:
30% = 0.30 =
Remember!
30100
Because percents can be expressed as ratios, you can use the proportion
to solve percent problems.
Holt McDougal Algebra 2
2-2 Proportional Reasoning
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?
Example 2: Solving Percent Problems
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).
Holt McDougal Algebra 2
2-2 Proportional Reasoning
Example 2 Continued
Method 1 Use a proportion.
Cross multiply.
Solve for x.
So 405 voters are planning to vote for that candidate.
Method 2 Use a percent equation.22.5% 0.225 Divide the percent
by 100.
Percent (as decimal) whole = part
0.225 1800 = x
405 = x
x = 405
22.5(1800) = 100x
Holt McDougal Algebra 2
2-2 Proportional Reasoning
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 total number of students, so you are trying to find the part of the whole (the number of students that Clay High School has).
Check It Out! Example 2
Holt McDougal Algebra 2
2-2 Proportional Reasoning
Check It Out! Example 2 Continued
Method 1 Use a proportion.
Cross multiply.
Solve for x.
Clay High School has 1240 students.
Method 2 Use a percent equation.
Divide the percent by 100.
0.35x = 434
35% = 0.35
x = 1240
Percent (as decimal) whole = part
x = 1240
100(434) = 35x
Holt McDougal Algebra 2
2-2 Proportional Reasoning
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.
Holt McDougal Algebra 2
2-2 Proportional Reasoning
Ryan ran 600 meters and counted 482 strides. How long is Ryan’s stride in inches? (Hint: 1 m ≈ 39.37 in.)
Example 3: Fitness Application
Use a proportion to find the length of his stride in meters.
Find the cross products.
600 m 482 strides
x m 1 stride=
600 = 482x
x ≈ 1.24 m
Write both ratios in the form .
metersstrides
Holt McDougal Algebra 2
2-2 Proportional Reasoning
Example 3: Fitness Application continued
Convert the stride length to inches.
Ryan’s stride length is approximately 49 inches.
is the conversion factor. 39.37 in.1 m
≈ 1.24 m1 stride length
39.37 in.1 m
49 in.1 stride length
Holt McDougal Algebra 2
2-2 Proportional Reasoning
Use a proportion to find the length of his stride in meters.
Check It Out! Example 3
Luis ran 400 meters in 297 strides. Find his stride length in inches.
x ≈ 1.35 m
400 = 297x Find the cross products.
400 m 297 strides
x m 1 stride=
Write both ratios in the form .
metersstrides
Holt McDougal Algebra 2
2-2 Proportional Reasoning
Convert the stride length to inches.
Luis’s stride length is approximately 53 inches.
Check It Out! Example 3 Continued
is the conversion factor. 39.37 in.1 m
≈ 1.35 m1 stride length
39.37 in.1 m
53 in.1 stride length
Holt McDougal Algebra 2
2-2 Proportional Reasoning
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
Holt McDougal Algebra 2
2-2 Proportional Reasoning
Example 4: 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.
Holt McDougal Algebra 2
2-2 Proportional Reasoning
Example 4 Continued
= height of ∆XAB width of ∆XAB
height of ∆XYZ width of ∆XYZ
=3 x
9 6
9x = 18, so x = 2
Step 2 To find the width of ∆XAB, use a proportion.
Holt McDougal Algebra 2
2-2 Proportional Reasoning
Example 4 Continued
The width is 2 units, and the height is 3 units, so the coordinates of A are (–2, 3).
BA
X
YZ
Step 3 To graph ∆XAB, first find the coordinate of A.
Holt McDougal Algebra 2
2-2 Proportional Reasoning
∆DEF has vertices D(0, 0), E(–6, 0) and F(0, –4).
∆DGH is similar to ∆DEF with a vertex at G(–3, 0).
Graph ∆DEF and ∆DGH on the same grid.
Check It Out! Example 4
Step 1 Graph ∆DEF. Then draw DG.
Holt McDougal Algebra 2
2-2 Proportional Reasoning
= width of ∆DGH height of ∆DGH
width of ∆DEF height of ∆DEF
Check It Out! Example 4 Continued
Step 2 To find the height of ∆DGH, use a proportion.
6x = 12, so x = 2
=36 4
x
Holt McDougal Algebra 2
2-2 Proportional Reasoning
The width is 3 units, and the height is 2 units, so the coordinates of H are (0, –2).
Check It Out! Example 4 Continued
G(–3, 0)D(0, 0)
H(0, –2)
● ●
●
●●
●
E(–6, 0)
F(0,–4)
Step 3
To graph ∆DGH, first find the coordinate of H.
Holt McDougal Algebra 2
2-2 Proportional Reasoning
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.
=6
9 h22
=Shadow of tree Height of tree
Shadow of house Height of house
6h = 198
h = 33
The house is 33 feet high.
9 ft
6 ft
h ft
22 ft
Holt McDougal Algebra 2
2-2 Proportional Reasoning
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.
= 20 6 h
90=
Shadow of climber Height of climber
Shadow of treeHeight of tree
20h = 540
h = 27
The tree is 27 feet high.
6 ft
20 ft
h ft
90 ft
Check It Out! Example 5
Holt McDougal Algebra 2
2-2 Proportional Reasoning
Lesson Quiz: Part ISolve each proportion.
1. 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?
k = 8 g = 42
$0.23
1200
Holt McDougal Algebra 2
2-2 Proportional Reasoning
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.
YZ
AB
X
Lesson Quiz: Part II
Holt McDougal Algebra 2
2-2 Proportional Reasoning
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
Lesson Quiz: Part III