5.4
Solving Proportions
5.4
OBJECTIVE
1. Solve a proportion for an unknown value
A proportion consists of four values. If three of the four values of a proportion are known,
you can always find the missing or unknown value.
a
10
In the proportion , the first value is unknown. We have chosen to represent the
3
15
unknown value with the letter a. Using the proportion rule, we can proceed as follows.
? 10
is a proportion
3 15
in which the first value is
unknown. Our work in this
section will be learning how to
find that unknown value.
NOTE
a
10
3
15
15 a 3 10
or
15 a 30
The equals sign tells us that 15 a and 30 are just different names for the same number.
This type of statement is called an equation.
Definitions: Equation
An equation is a statement that two expressions are equal.
One important property of an equation is that we can divide both sides by the same
nonzero number. Here let’s divide by 15.
15 a 30
15 a
30
15
15
1
NOTE We will always divide by
the number multiplying the
variable. This is called the
coefficient of the variable.
2
15 a
30
15
15
1
Divide by the coefficient of the variable. Do you
see why we divided by 15? It leaves our unknown
a by itself in the left term.
1
a2
© 2001 McGraw-Hill Companies
You should always check your result. It is easy in this case. We found a value of 2 for a.
Replace the unknown a with that value. Then verify that the fractions are proportional. We
10
a
started with and found a value of 2 for a. So we write
3
15
NOTE Replace a with 2 and
multiply.
2
3
10
15
3 10 2 15
30 30
The value of 2 for a is correct.
The procedure for solving a proportion is summarized as follows.
437
438
CHAPTER 5
RATIOS AND PROPORTIONS
Step by Step: To Solve a Proportion
Step 1 Use the proportion rule to write the equivalent equation a d b c.
Step 2 Divide both terms of the equation by the coefficient of the variable.
Step 3 Use the value found to replace the unknown in the original
proportion. Check that the ratios or the rates are proportional.
NOTE This gives us the
unknown value.
Now check the result.
Example 1
Solving Proportions for Unknown Values
Find the unknown value.
(a)
8
6
x
9
Step 1 Using the proportion rule, we have the following.
6x89
NOTE You are really using
or
6x 72
Step 2 Locate the coefficient of the variable, 6, and divide both sides of the equation by
that coefficient.
1
12
6x
72
6
6
1
1
x 12
Step 3 To check, replace x with 12 in the original proportion.
8 6
12
9
Multiply:
12 6 8 9
72 72
(b)
The value of 12 for x checks.
3
b
5
25
Step 1 Use the proportion rule.
5 b 3 25
or 5 b 75
Step 2 Locate the coefficient of the variable, 5, and divide both sides of the equation by
that coefficient.
1
15
75
5b
5
5
1
1
b 15
© 2001 McGraw-Hill Companies
algebra to solve these
proportions. In algebra, we
write the product 6 x as 6x,
omitting the dot. Multiplication
of the number and the variable
is understood and doesn’t need
to be written.
SOLVING PROPORTIONS
SECTION 5.4
439
Step 3 To check, replace b with 15 in the original proportion.
3 15
5
25
Multiply:
5 15 3 25
75 75
The value of 15 checks for b.
CHECK YOURSELF 1
Solve the proportions for n. Check your result.
(a)
4
n
5
25
(b)
42
7
9
n
In solving for a missing term in a proportion, we may find an equation involving fractions or decimals. Example 2 involves finding the unknown value in such cases.
Example 2
Solving Proportions for Unknown Values
(a) Solve the proportion for x.
1
4
4
3
x
1
x 12
4
1
x
12
4
1
1
4
4
x
© 2001 McGraw-Hill Companies
NOTE
x 12 4
48
1
12
1
4
x 48
We divide by the coefficient of x.
1
In this case it is .
4
Remember:
1
12
is 12 .
4
1
4
Invert the divisor and multiply.
To check, replace x with 48 in the original proportion.
1
4 4
3
48
1
3 4 48
4
12 12
440
CHAPTER 5
RATIOS AND PROPORTIONS
(b) Solve the proportion for a.
0.5
3
2
a
0.5a 6
by 0.5 to find the unknown
value. The steps of that division
are shown below for review.
1 2.
0.5 B6.0
^ ^
5
10
10
0
6
0.5a
0.5
0.5
Divide by the coefficient, 0.5.
a 12
We will leave it to you to confirm that 0.5 12 2 3.
CHECK YOURSELF 2
(a) Solve for a.
(b) Solve for x.
1
2
3
5
a
0.4
2
x
30
CHECK YOURSELF ANSWERS
1. (a) 5n 100
5n
100
5
5
n 20
To check:
4 20
5 25
5 20 4 25
100 100
(b) 7n 42 9
7n
42 9
7
7
n69
n 54
To check:
7 42
9 54
7 54 9 42
378 378
2. (a) 30; (b) 6
© 2001 McGraw-Hill Companies
NOTE Here we must divide 6
Name
5.4
Exercises
Section
Date
Solve for the unknown in each of the following proportions.
1.
x
6
3
9
2.
x
3
6
9
3.
10
15
n
6
ANSWERS
1.
2.
4.
4
8
3
n
5.
4
y
7
14
6.
7
14
m
8
3.
4.
5
a
7.
8
16
5
x
8.
7
35
8
6
9.
p
3
5.
6.
7.
10.
4
8
15
n
11.
11
2
a
44
12.
5
15
x
9
8.
9.
10.
13.
35
7
40
n
14.
x
15
8
24
15.
a
5
42
7
11.
12.
16.
7
m
12
24
17.
18
12
12
p
18.
x
7
32
8
13.
14.
15.
19.
x
64
18
72
20.
20
100
15
a
21.
75
6
n
100
16.
17.
18.
© 2001 McGraw-Hill Companies
36
8
22.
x
6
25.
28.
12
3
100
x
5
20
x
88
5
a
23.
35
28
26.
b
21
7
49
1
2
3
29.
2
a
20
p
24.
24
18
27.
30.
p
25
24
120
x
2
5
1
3
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
441
ANSWERS
31.
1
4
m
31.
12
96
32.
32.
33.
34.
2
5
1.2
33.
8
n
35.
12
108
1
y
3
2
4
5
34.
a
10
36.
37.
35.
0.2
1.2
2
a
36.
n
6
3
0.5
37.
p
8
7
0.7
38.
5
y
12
0.6
39.
x
1.1
3.3
6.6
40.
1.25
0.5
a
5
38.
39.
40.
Answers
1. 9x 18; x 2
3. 4
5. 8
7. 10
13. 35n 280; n 8
15. 30
17. 18p 144; p 8
19. 16
21. 8
23. 35a 140; a 4
25. 25
27. 5
35. 12
37. 80
29. 12
11. 242
31. 2
39. 0.55
© 2001 McGraw-Hill Companies
33. 24
9. 4
442