5-2 Identifying and Writing Proportions
Learn to find equivalent ratios and to
identify proportions.
Course 2
Vocabulary
equivalent ratios
proportion
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.
These ratios can be written as the
25
fractions 10
and
. Since both simplify
6
15
to 5 , they are equivalent. Equivalent
3
ratios are ratios that name the same
comparison.
An equation stating that two ratios are
equivalent is called a proportion. The
equation, or proportion, below states that
the ratios 10 and 25 are equivalent.
6
15
10 = 25
15
6
Reading Math
Read the proportion 10 = 25 by saying “ten is to six
6
15
as twenty-five is to fifteen.”
If two ratios are equivalent, they are
said to be proportional to each other,
or in proportion.
Additional Example 1A: Comparing Ratios in
Simplest Forms
Determine whether the ratios are proportional.
A. 24 , 72
51 128
8
24 ÷ 3
=
17
51 ÷ 3
Simplify 24 .
51
72 ÷ 8
9
=
16
128 ÷ 8
Simplify 72 .
128
Since
9
8
=
, the ratios are not proportional.
17 16
Additional Example 1B: Comparing Ratios in
Simplest Forms
Determine whether the ratios are proportional.
B. 150 , 90
105 63
150 ÷ 15 10
=
7
105 ÷ 15
90 ÷ 9
10
=
7
63 ÷ 9
Since
Simplify 150 .
105
Simplify
90
.
63
10 10
=
, the ratios are proportional.
7
7
Try This: Example 1A
Determine whether the ratios are proportional.
A. 54 , 72
63 144
6
54 ÷ 9
=
7
63 ÷ 9
Simplify 54 .
63
72 ÷ 72
1
=
144 ÷ 72 2
Simplify 72 .
144
Since
1
6
= , the ratios are not proportional.
2
7
Try This: Example 1B
Determine whether the ratios are proportional.
B. 135 , 81
75 45
9
135 ÷ 15
=
5
75 ÷ 15
81 ÷ 9
9
=
5
45 ÷ 9
Since
Simplify 135 .
75
Simplify
81
.
45
9
9
=
, the ratios are proportional.
5
5
Additional Example 2: Comparing Ratios Using a
Common Denominator
Use the data in the table to determine whether the ratios
of rice to water are proportional for both servings of rice.
Servings of Rice
Cups of Rice
Cups of Water
12
3
6
40
10
19
Write the ratios of rice to water for 12 servings and for 40 servings.
3
Ratio of rice to water, 12 servings: 6
10
Ratio of rice to water, 40 servings:
19
3 = 3 · 19 = 57
6
6 · 19
114
10 = 10 · 6 = 60
19 19 · 6 114
Write the ratio as a fraction.
Write the ratio as a fraction
Write the ratios with a
common denominator,
such as 114.
Since 57 = 60 , the two ratios are not proportional.
114
114
Try This: Example 2
Use the data in the table to determine whether the ratios of
beans to water are proportional for both servings of beans. Servings of Beans
Cups of Beans
Cups of Water
8
4
3
35
13
9
Write the ratios of beans to water for 8 servings and for 35 servings.
4
Write the ratio as a fraction.
Ratio of beans to water, 8 servings: 3
13
Write the ratio as a fraction
Ratio of beans to water, 35 servings:
9
4 = 4 · 9 = 36
Write the ratios with a
3
3·9
27
common denominator,
13 = 13 · 3 = 39
such as 27.
9
27
9·3
Since 36 = 39 , the two ratios are not proportional.
27
27
You can find an equivalent ratio by
multiplying or dividing the numerator
and the denominator of a ratio by the
same number.
Insert Lesson Title Here
Additional Example 3: Finding Equivalent Ratios and
Writing Proportions
Find a ratio equivalent to each ratio. Then use
the ratios to find a proportion.
A. 3
B.
5
3 = 3 · 2 =6
5
5 · 2 10
3 = 6
10
5
28
16
28 = 28 ÷ 4 7
=
16 16 ÷ 4 4
28 = 7
16 4
Multiply both the numerator and
denominator by any number such as 2.
Write a proportion.
Divide both the numerator and
denominator by any number such as 4.
Write a proportion.
Insert Lesson Title Here
Try This: Example 3
Find a ratio equivalent to each ratio. Then use
the ratios to find a proportion.
A. 2
B.
3
2 = 2 · 3 =6
3
3·3 9
2 = 6
9
3
16
12
16 = 16 ÷ 4 = 4
3
12 12 ÷ 4
16 = 4
12 3
Multiply both the numerator and
denominator by any number such as 3.
Write a proportion.
Divide both the numerator and
denominator by any number such as 4.
Write a proportion.
Insert Lesson Title Here
Lesson Quiz
Determine whether the rates are proportional
by writing them in simplest form and comparing
them.
9 , 12
1. 30
40
3 , 3 ; proportional
10 10
, 10
2. 12
21 15
4 , 2 ; not proportional
7 3
Determine if the ratios are proportional
by finding a common denominator.
, 2
3. 3
8 5
15 , 16 ; not proportional
40 40
4. 3 , 9
7 21
9 , 9 ; proportional
21 21
Insert Lesson Title Here
Lesson Quiz
5. In a local pre-school, there are 5 children for every
teacher. Write an equivalent ratio to show how
many children there would be if there were 4
teachers.
5 , 20
1 4
5-3 Solving Proportions
Learn to solve proportions by using cross
products.
Course 2
Insert Lesson Title Here
Vocabulary
cross product
The tall stack of Jenga® blocks is
25.8 cm tall. How tall is the shorter
stack of blocks? To find the answer
you will need to solve a proportion.
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.
2 = 6
5 15
5 · 6 = 30
2 · 15 = 30
CROSS PRODUCT RULE
In the proportion a = c , the cross products,
b
d
a · d and b · c are equal.
You can use the cross product rule to solve
proportions with variables.
Additional Example 1: Solving Proportions Using
Cross Products
Use cross products to solve the proportion.
9 = m
15
5
15 · m = 9 · 5
15m = 45
15m = 45
15
15
m=3
The cross products are equal.
Multiply.
Divide each side by 15 to
isolate the variable.
Insert Lesson Title Here
Try This: Example 1
Use cross products to solve the proportion.
6 = m
7
14
7 · m = 6 · 14
The cross products are equal.
7m = 84
Multiply.
7m = 84
7
7
Divide each side by 7 to
isolate the variable.
m = 12
When setting up a proportion to solve a
problem, use a variable to represent the
number you want to find. In proportions that
include different units of measurement, either
the units in the numerators must be the same
and the units in the denominators must be the
same or the units within each ratio must be
the same.
16 mi = 8 mi
4 hr
x hr
16 mi = 4 hr
8 mi
x hr
Additional Example 2: Problem Solving Application
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?
1
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.
Additional Example 2 Continued
2
Make a Plan
Set up a proportion using the given
information.
3 volumes = 26 volumes
4 inches
x
Let x be the
unknown space.
Additional Example 2 Continued
3
Solve
3 = 26
Write the proportion.
4
x
3 · x = 4 · 26 The cross products are equal.
3x = 104
Multiply.
Divide each side by 3 to isolate
3x = 104
the variable.
3
3
x = 34 2
3
She needs 34 2 inches for all 26 volumes.
3
Additional Example 2 Continued
4
Look Back
3 =
4
26
34 23
4 · 26 = 104
3 · 34 23 = 104
The cross products are equal, so 34 23 is the
answer
1
Try This: Example 2
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?
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.
Try This: Example 2 Continued
2
Make a Plan
Set up a proportion using the given
information.
6 pints
10 inches
=
x
25 inches
Let x be the
unknown amount.
Try This: Example 2 Continued
3
Solve
6 = x
Write the proportion.
10 25
10 · x = 6 · 25 The cross products are equal.
10x = 150
Multiply.
10x = 150
10
10
x = 15
Divide each side by 10 to isolate
the variable.
15 pints of coolant will fill the radiator to the 25
inch level.
Try This: Example 2 Continued
4
Look Back
6 =
10
15
25
10 · 15 = 150
6 · 25 = 150
The cross products are equal, so 15 is the answer.
Insert Lesson Title Here
Lesson Quiz: Part 1
Use cross products to solve the proportion.
1. 25 = 45 t = 36
t
20
2. x = 19 x = 3
9 57
3. 2 = r
r = 24
3 36
4. n = 28 n = 35
10
8
Insert Lesson Title Here
Lesson Quiz: Part 2
5. Carmen bought 3 pounds of bananas for $1.08.
June paid $ 1.80 for her purchase of bananas.
If they paid the same price per pound, how
many pounds did June buy?
5 pounds