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5-5 Solving Proportions
Course 2
Warm UpWarm Up
Problem of the DayProblem of the Day
Lesson PresentationLesson Presentation
Warm UpFind the unit rate.
1. 18 miles in 3 hours
2. 6 apples for $3.30
3. 3 cans for $0.87
4. 5 CD’s for $43
Course 2
5-4 Identifying and Writing Proportions
6 mi/h
$0.55 per apple
$0.29 per can
$8.60 per CD
Course 2
5-4 Identifying and Writing Proportions
Students in Mr. Howell’s math class are measuring the width w and the length l of their heads. The ratio of l to w is 10 inches to 6 inches for Jean and 25 centimeters to 15 centimeters for Pat.
Course 2
5-4 Identifying and Writing Proportions
106
2515
These ratios can be written as the
fractions and . Since both simplify
to , they are equivalent. Equivalent
ratios are ratios that name the same comparison.
53
Course 2
5-4 Identifying and Writing Proportions
An equation stating that two ratios are equivalent is called a proportion. The equation, or proportion, below states that the ratios and are equivalent.
106
2515
106
= 2515
Read the proportion by saying “ten is to six
as twenty-five is to fifteen.”
Reading Math
106
=2515
Determine whether the ratios are proportional.
Additional Example 1A: Comparing Ratios in Simplest Forms
Course 2
5-4 Identifying and Writing Proportions
,2451
72128
72 ÷ 8
128 ÷ 8= 9
16
24 ÷ 351 ÷ 3
=8
17Simplify .24
51
Simplify . 72128
817
Since = , the ratios are not proportional.9
16
Determine whether the ratios are proportional.
Check It Out: Example 1A
Course 2
5-4 Identifying and Writing Proportions
,5463
72144
72 ÷ 72
144 ÷ 72 = 1
2
54 ÷ 963 ÷ 9
=67
Simplify .5463
Simplify . 72144
67
Since = , the ratios are not proportional.12
Determine whether the ratios are proportional.
Check It Out: Example 1B
Course 2
5-4 Identifying and Writing Proportions
, 135 75
94
9 4
135 ÷ 15 75 ÷ 15
= 9 5
Simplify .135 75
is already in simplest form. 9 4
95
Since = , the ratios are not proportional.94
Lesson Quiz:
Insert Lesson Title Here
Course 2
5-4 Identifying and Writing Proportions
In pre-school, there are 5 children for everyone teacher. In another preschool there are 20 children for every 4 teachers. Determine whether the ratios of children to teachers are proportional in both preschools.
20
4
5
1
=
Course 2
5-5 Solving Proportions
Quick checkDetermine whether the ratios are proportional.
1. 58
, 1524
2. 1215
, 1625
3. 1510
, 2016
4. 1418
, 4254
yes
no
yes
no
Course 2
5-5 Solving Proportions
For two ratios, the product of the numerator in one ratio and the denominator in the other is a cross product. If the cross products of the ratios are equal, then the ratios form a proportion.
5 · 6 = 30
2 · 15 = 30=2
56
15
Course 2
5-5 Solving Proportions
You can use the cross product rule to solve proportions with variables.
CROSS PRODUCT RULE
In the proportion = , the cross products,
a · d and b · c are equal.
ab
cd
Use cross products to solve the proportion.
Additional Example 1: Solving Proportions Using Cross Products
Course 2
5-5 Solving Proportions
15 · m = 9 · 5
15m = 45
15m15
= 4515
m = 3
The cross products are equal.
Multiply.
Divide each side by 15 to isolate the variable.
= m5
915
Check It Out: Example 1
Insert Lesson Title Here
Course 2
5-5 Solving Proportions
Use cross products to solve the proportion.
7 · m = 6 · 14
7m = 84
7m7
= 847
m = 12
The cross products are equal.
Multiply.
Divide each side by 7 to isolate the variable.
67
= m14
Additional Example 2: Problem Solving Application
Course 2
5-5 Solving Proportions
If 3 volumes of Jennifer’s encyclopedia takes up 4 inches of space on her shelf, how much space will she need for all 26 volumes?
11 Understand the Problem
Rewrite the question as a statement.• Find the space needed for 26 volumes of
the encyclopedia.
List the important information:• 3 volumes of the encyclopedia take up 4
inches of space.
Course 2
5-5 Solving Proportions
22 Make a Plan
Set up a proportion using the given information. Let x represent the inches of space needed.
3 volumes4 inches
= 26 volumesx
volumesinches
Additional Example 2 Continued
Course 2
5-5 Solving Proportions
Solve33
34
= 26x
3 · x = 4 · 26
3x = 104
3x3
= 1043
x = 34 23
She needs 34 23
inches for all 26 volumes.
Write the proportion.
The cross products are equal.
Multiply.
Divide each side by 3 to isolate the variable.
Additional Example 2 Continued
Course 2
5-5 Solving Proportions
Look Back44
34
= 2634 2
3
4 · 26 = 104
3 · 34 23
= 104
The cross products are equal, so 34 23
is theanswer
Additional Example 2 Continued
Check It Out: Example 2
Course 2
5-5 Solving Proportions
John filled his new radiator with 6 pints of coolant, which is the 10 inch mark. How many pints of coolant would be needed to fill the radiator to the 25 inch level?
11 Understand the Problem
Rewrite the question as a statement.• Find the number of pints of coolant required
to raise the level to the 25 inch level.
List the important information:• 6 pints is the 10 inch mark.
Course 2
5-5 Solving Proportions
22 Make a Plan
Set up a proportion using the given information. Let p represent the pints of coolant.
6 pints10 inches
= p25 inches
pints inches
Check It Out: Example 2 Continued
Course 2
5-5 Solving Proportions
Solve33
610
= p25
10 · x = 6 · 25
10x = 150
10x10
= 15010
p = 15
15 pints of coolant will fill the radiator to the 25 inch level.
Write the proportion.
The cross products are equal.
Multiply.
Divide each side by 10 to isolate the variable.
Check It Out: Example 2 Continued
Course 2
5-5 Solving Proportions
Look Back44
610
= 1525
10 · 15 = 150
6 · 25 = 150
The cross products are equal, so 15 is the answer.
Check It Out: Example 2 Continued
Lesson Quiz: Part I
Insert Lesson Title Here
Course 2
5-5 Solving Proportions
Use cross products to solve the proportion.
1. 2520
= 45t
2. x9
= 1957
3. 23
= r36
4. n10
=288
t = 36
x = 3
r = 24
n = 35