Section 4.2
PRE-ACTIVITY
PREPARATION
Proportions
In the photo at right, you can see that the top teapot is a smaller version of
the lower one. In fact, if you were to compare the ratios of each teapot’s
width to its height you would find that they are equal. For this reason,
you can say that the two are in proportion to each other.
Building upon your knowledge of equivalent fractions, you will learn in
this section how to determine if two ratios are equal (in proportion) to
each other and how to determine the value of the missing numerator or
denominator in a pair of equal ratios.
These are the important skills to master before you explore, in the
next section, the many and varied contexts in which you can use a
proportion.
LEARNING OBJECTIVES
• Validate proportions to see if they are true statements.
• Solve for an unknown quantity in a proportion.
• Test the validity of the solution to a proportion.
TERMINOLOGY
PREVIOUSLY USED
NEW TERMS
TO
LEARN
cross-multiply
approximately equals (represented by the symbol ≈)
cross-product
equate
denominator
equation
multiplier
expression
numerator
in proportion
ratio
proportion
reduce
solution
solve an equation
solve for the unknown
solve for the variable
unknown
variable
381
Chapter 4 — Ratios and Proportions
382
BUILDING MATHEMATICAL LANGUAGE
Equations
An expression is a mathematical symbol or combination of symbols that represents a value.
An equation is a mathematical statement of equality of two expressions. When you write an equation,
you equate two values.
You have already written many equations in this course. Whenever you wrote a statement with an equal
sign (=) and the statement was true, you presented an equation. Up until now, the equations you presented
contained only numbers and operation signs. For example:
24 = 2 × 2 × 2 × 3
12 × 2
24
=
5× 2
10
12
2
=
18
3
34 ÷ 5 = 6.8
12 × 10 = 120
36 = 22 × 32
However, an equation might be written with one or both expressions containing unknown quantities. For
example, an equation might ask,
14 × what number = 42?
In mathematics, the common way to express “what number” is with a letter to represent the unknown
number (any letter of the alphabet you choose)—that is, to represent the unknown number with a
variable.
When you substitute a variable (say, n) for the unknown quantity (the unknown), the same equation
becomes
14 × n = 42
You might assign any value to n, but only one value for n will make this equation a true statement.
Only if n = 3, will the equation be true (14 × 3 = 42). Any other value for n (for example, n = 12.8)
will result in a false statement (14 × 12.8 ≠ 42)
To solve an equation containing one variable, is to determine the value(s) for the variable that will
make the equation a true statement—that is, to determine its solution(s). This process is referred to as
solving for the unknown or solving for the variable.
To maintain the equality of the values represented on both sides of an equation, whenever you make a
change to the expression on one side of the statement, you must do the same to the other side.
For example,
3 × 4 = 12
3 × 4 + 2 = 12 + 2
3 × 4 − 2 = 12 − 2
3 × 4 × 2 = 12 × 2
3 × 4 12
=
2
2
⇒
⇒
⇒
⇒
12 = 12
14 = 14
10 = 10
24 = 24
⇒
6= 6
Section 4.2 — Proportions
383
Proportions
A proportion is a special type of equation that states the equality of two ratios.
For example,
3
39 The proportion may be read “Three to five equals thirty-nine to sixty-five.”
=
.
5
65 However, it is more commonly read, “Three is to five as thirty-nine is to sixty-five.”
The parts of a proportion are not always whole numbers.
When two ratios are equal, we say they are in proportion to each other.
Determining if a Given Proportion is True or False
A true (valid) proportion is a proportion whose ratios are equal to each other. If the ratios are not equal, the
proportion is false.
When unit labels are included, the first thing to verify is that the comparison of units is in the same order for
both ratios. For example,
miles
gallons
miles
gallons
annd
and
;
, or
gallons
miles
gallons
miles
not
g
miles
gallons
and
gallonss
miles
If the units do not match numerator to numerator and denominator to denominator, the
proportion is false, no matter what the numbers are.
Once you have verified that the units match in their comparisons, you can test the validity of the proportion
(determine whether it is true or false) by using any one of the following three techniques presented on the next
several pages. They will be familiar to you from your knowledge of fractions.
Chapter 4 — Ratios and Proportions
384
TECHNIQUES
Testing a Proportion by Applying the Equality Test for Fractions
Technique
Apply the Equality Test for Fractions and cross-multiply to determine if the proportion is true.
MODELS
A
►
Is this proportion true or false?
12 156
=
19 247
?
12 × 247 = 156 × 19
2964 = 2946 9
Cross-multiply to determine the equality of the ratios.
The proportion is true.
B
►
Is this proportion true or false?
57 miles
21.5 miles
=
5 gallons 1.9 gallons
Verify that the units are compared in the same order.
THINK
miles per gallon in both ratios of the proportion 9
?
57 × 1.9 = 21.5 × 5
108.3 ≠ 107.5
Cross-multiply.
The proportion is false.
C
►
Is this proportion true or false?
Cross-multiply.
3
13
1
1
8
4 = 8
8
20
3
1
? 1
× 20 = 8 × 8
4
8
5
1
20 ? 65
8
= 1 ×
1
1
1
4
8
65 = 65 9
The proportion is true.
×
Section 4.2 — Proportions
385
Testing a Proportion by Building Up One of the Ratios
Technique
If it is obvious by inspection that you can multiply the numerator and denominator of one
of the ratios by the same multiplier to make it equal to the other ratio, then the proportion
is true.
MODELS
A
►
Is this proportion true or false?
5
15
=
36 108
It is apparent that multiplying the numerator 5 by 3 will equal 15.
Is the denominator 36 multiplied by 3 equal to 108?
Yes.
B
►
5×3
15
=
36 × 3 108
The proportion is true.
Is this proportion true or false?
84 males
42 males
=
60 females
30 females
Verify that the units are compared in the same order.
THINK
males to females on both sides of the proportion 9
It is apparent that multiplying both 42 and 30 by the multiplier 2 in the ratio on the right
will yield the ratio on the left.
84
42 × 2
The proportion is true.
=
60
30 × 2
C
►
Is this proportion true or false?
2 23
=
5 50
You can see that the denominator of the first ratio (5) times 10 will yield 50.
Is the numerator 2 times the same multiplier (10) equal to 23?
No. (2 × 10 ≠ 23) The proportion is false.
Chapter 4 — Ratios and Proportions
386
Testing a Proportion by Comparing the Reduced Forms of the Ratios
Technique
If easily done, reduce both ratios to their simplest forms. If the reduced ratios are equal,
then the proportions are true.
MODELS
A
►
Is this proportion true or false?
84 cats
98 cats
=
60 dogs
70 dogs
Verify that the units are compared in the same order.
THINK
cats to dogs in both ratios 9
Reduce both sides:
B
►
84 ÷ 12 7
=
60 ÷ 12 5
Is this proportion true or false?
Reduce both sides:
10 ÷ 2
5
=
12 ÷ 2
6
and
98 ÷ 2
49 ÷ 7 7
The proportion is true.
=
=
70 ÷ 2
35 ÷ 7
5
10
35
=
12
45
and
35 ÷ 5
7
=
45 ÷ 5 9
The proportion is false.
Solving for a Variable in a Proportion
A variable can take the place of any one of the four components of the proportion. When the other three
components are known quantities, you can determine the value for the unknown that will make the proportion
true.
For example, in the proportion 12 = 64 , the task is to determine the value for n (solve for n) that will make
5
n
the proportion true.
Occasionally, when the solution for the variable is a decimal number and the answer is not exact, you will find
it necessary to round your answer to a specified decimal place. When this is the case, use the symbol ≈ for “is
approximately equal to” or “approximately equals,” as in n ≈ 26.67 for the example proportion above.
That is, “n is approximately equal to 26.67.”
Also, when the solution has been rounded and you cross-multiply to test the equality of the two ratios, use the
≈ symbol to indicate that the cross-products are close but not exactly equal.
Section 4.2 — Proportions
387
TECHNIQUE
If the relationship between the two ratios of a proportion is easily recognizable, that is, if one ratio is a multiple
of the other, you can use the following technique to solve for the unknown quantity.
Solving a Proportion when One Ratio is a Multiple of the Other
Technique
Step 1:
If one numerator is a multiple of the other numerator (or one denominator is a
multiple of the other denominator), use the multiplier to determine the value of
the variable.
Step 2:
Validate that the solution is correct by substituting the answer into the original
proportion. Apply the Equality Test for Fractions by cross-multiplying.
MODELS
A
►
Solve for n in the proportion
Step 1
Step 2
B
►
THINK
26 × 2 = 52
Then 9 × 2 = 18 = n
9 ? 18
=
26
52
Validate:
Solve for n in the proportion
Step 1
Step 2
THINK
Validate:
9
n
=
26 52
Answer : n = 18
?
9 × 52 = 18 × 26
468 = 468 9
0.3 1.2
=
5
n
0.3 × 4 = 1.2
Then 5 × 4 = 20 = n
0.3 ? 1.2
=
5
20
Answer : n = 20
?
0.3 × 20 = 1.2 × 5
6.0 = 6.0 9
Chapter 4 — Ratios and Proportions
388
METHODOLOGY
If the relationship between the two ratios of a proportion is not easily recognizable, use the following
methodology to solve for the unknown quantity.
Solving a Proportion
►
Example 1:
11 is to 15 as what number is to 37.5?
►
Example 2:
n
5
=
39
6
Try It!
Steps in the Methodology
Step 1
Set up the
proportion
with a
variable.
Step 2
Equate the
cross products.
Write the given proportion. If it is not
already set up in proportion form, set
up the proportion using a variable for
the unknown component.
Divide by the
multiplier of
the variable.
Example 2
11
n
=
15 37.5
Cross multiply and equate the cross-
products.
???
Why can you do this?
Shortcut #1
Step 3
Example 1
Reduce the known ratio
first
(see page 390, Model 1)
Divide both sides of the equation by
the multiplier of the variable.
???
Why do you do this?
Step 4
Calculate the value for the variable.
Solve for the
variable.
Compute the numerator and divide the
product by the denominator.
Round, if necessary to the specified
place value.
Shortcut #2
11 × 37.5 = n × 15
Cancel common factors
before dividing
(see page 391, Model 2)
1
11 × 37.5
n × 15
= 1
15
15
37.5
× 11
375
+3750
412.5
27.5
15 412.5
−30
)
112
−105
75
−75
0
Section 4.2 — Proportions
Steps in the Methodology
Step 5
Present the
answer.
State your answer, the value for the
unknown.
Step 6
Validate your answer.
Validate your
answer.
In the original proportion, replace the
variable with your answer and use
one of the Techniques for Testing a
Proportion.
Note: When the answer is rounded,
the cross-products will be close but not
exactly equal. (See Model 1 on page
390.)
389
Example 1
Example 2
n = 27.5
11 ? 27.5
=
15
37.5
?
11 × 37.5 = 27.5 × 15
412.5 = 412.5 9
37.5
× 11
375
+3750
1375
+2750
412.5
412.5
27.5
× 15
???
Why can you do Step 2?
You are determining the value for the variable that will make a true proportion—the value that will make the
two ratios equal. The Equality Test for Fractions says that, if the ratios are equal, then their cross-products
will have to be equal.
???
Why do you do Step 3?
You want your end result to be an equation that either states, “n = the solution number” or “the solution number
= n,” as that statement will provide the missing quantity of your proportion.
Because the Special Property of Division Involving One tells you that any number divided by itself = 1, and
the Identity Property of Multiplication tells you that 1 times any number = that same number, you divide by the
multiplier of the variable n to yield 1 as its multiplier:
1
15 × n
= 1× n = n
1
15
Recall that to maintain the equality of the statement, however, you must also do the same to the expression
on the other side of the equal sign.
Chapter 4 — Ratios and Proportions
390
MODELS
Model 1
A
►
Shortcut #1: Reduce the Known Ratio First
108 25
=
45
n
Solve for the unknown. Round to the nearest hundredth place:
Step 1
108 25
=
45
n
Shortcut (optional):
To simplify the next computations, reduce the
known ratio before equating the cross-products.
Step 2
108 × n = 25 × 45
1
Step 3
108 × n
1
108
=
25 × 45
108
Step 2
108 ÷ 9 12
12 25
=
⇒
=
⇒ 12 × n = 25 × 5
45 ÷ 9
5
5
n
1
Step 3
12 × n
1
12
=
25 × 5
12
25
× 45
25
× 5
125
1000
Step 4
n=
1125
108
1125
Step 4
n=
125
12
125
10.416 ≈ 10.42
12 125.000
)
10.416 ≈ 10.42
108 1125.000
)
−12
−108
50
−48
20
−12
450
−432
180
−108
80
−72
8
720
−648
72
Step 5 n ≈ 10.42
Step 6
Validate:
108 ?
25
=
45
10.42
?
108 × 10.42 = 25 × 45
1125.36 ≈ 1125 9
Section 4.2 — Proportions
B
►
Solve for n:
Step 1
Step 2
391
75
9
=
100
n
75
9
=
100
n
75 ÷ 25
3
=
100 ÷ 25
4
Use the shortcut and reduce:
⇒
3
9
=
4
n
At this point, there is no need to cross-multiply.
Use the Technique and skip Steps 2-4.
THINK
3×3 = 9
4 × 3 = 12 = n
Step 5
n = 12
Step 6
Validate:
Model 2
? 9 × 100
75 × 12 =
900 = 900 9
75 ? 9
=
100
12
Shortcut #2: Cancel Common Factors before Dividing
64 is to 5.52 as 4 is to what number?
Step 1
64
4
=
5.52
n
Step 2
64 × n = 4 × 5.52
1
Step 3
5.52
× 4
Step 4
22.08
n=
64
0.345
64 22.080
)
22.08
Step 5
64 × n
1
64
=
4 × 5.52
64
Shortcut: Cancel common factors before
dividing to simplify the computation.
1
Step 4
n=
4 × 5.52
16
64
=
5.52
16
0.345
16 5.520
)
n = 0.345
−48
−192
288
−256
320
−320
0
Step 6
Validate:
64
4
?
=
5.52
0.345
?
64 × 0.345 = 4 × 5.52
22.08 = 22.08 9
72
−64
80
−80
0
Chapter 4 — Ratios and Proportions
392
Model 3
Solve for the unknown. Round to the nearest hundredth place:
Step 1
48
64
=
20
n
Use shortcut #1.
48 ÷ 4 12
=
20 ÷ 4
5
Reduce:
Step 2
12 × n
1
12
16
=
12
64
=
5
n
64 × 5
3
Use shortcut #2. Cancel common factors.
12
26.66
3 80.00
)
−6
80
3
Step 4
n=
Step 5
n ≈ 26.67
Step 6
⇒
12 × n = 64 × 5
1
Step 3
48
64
=
20
n
20
−18
Validate:
20
−18
20
−18
?
48 × 26.67 = 64 × 20
1280.16 ≈ 1280 9
48 ?
64
=
20
26.67
2
Model 4
Solve: What number is to 35 as 1½ is to 12?
Step 1
1
1
n
= 2
35
12
Step 2
1
n × 12 = 1 × 35
2
1
Step 3
Step 4
n × 12
1
12
1
1 × 35
= 2
12
Compute the numerator:
1
3 35 105
1 × 35 = ×
=
2
2 1
2
Divide by the denominator: 105 ÷ 12 =
2
Step 5
Step 6
n=4
35
105
1
35
3
×4
=
=4
2
8
8
12
3
8
Validate:
3
8
35
4
?
=
1
2
12
4
1
35
2
8
3
? 1 1 × 35
× 12 =
8
2
3
×
12 ? 3 35
= ×
1
2 1
105 105
=
9
2
2
Section 4.2 — Proportions
393
How Estimation Can Help
When solving for the unknown quantity in a proportion, an effective practice is to estimate the size of the
solution relative to its position in the proportion. This can go a long way in preventing decimal placement
errors in your answers.
Example 1:
n
37
=
15 12.8
Before solving, think about the fact that the two ratios must be in proportion to one another. This means
that, minimally, you can predict that your solution for n will be greater than 15 because 37 is greater than
12.8 in the ratio on the right.
You can also predict the value for n that will make this a true proportion.
THINK
Estimate:
On the right side, estimate that 37 is approximately three times 12.8, so n will be
approximately three times 15.
n = 45
Actual answer: n ≈ 43.4, reasonably close to the estimate
Example 2:
9
18.7
=
1.2
n
Here is a case in which you might estimate in either of two ways:
THINK
Estimate:
Since numerator 18.7 is about two times numerator 9, denominator n will be about two
times denominator 1.2.
n = 2.4
OR
THINK
Estimate:
Since numerator 9 is about nine times its denominator 1.2, numerator 18.7 will be about
nine times its denominator n.
n=2
Actual answer: n ≈ 2.49, reasonably close to the estimates
Example 3:
THINK
Estimate:
320
6
=
2.8
n
n will be considerably less than 6 because 2.8 is considerably less than 320.
320 is about one hundred times as large as 2.8, so 6 will be about one hundred times as
large as n.
n = 0.06
Actual answer: n ≈ 0.53, reasonably close to the estimate
Go back and estimate the answers for the previous Models. Were the answers reasonable as compared to your
estimates?
Chapter 4 — Ratios and Proportions
394
ADDRESSING COMMON ERRORS
There are no additional new common errors to address. However, the errors that do most frequently occur are
those addressed in previous sections of this book.
Issue
Resolution
Incorrectly rounding the answer
when solving for the unknown in
a proportion
Review the Methodology for
Rounding a Decimal Number
on page 136.
Incorrect placement of the
decimal point when multiplying
decimal numbers
Review the Methodology for
Multiplying Decimal Numbers
on page 167.
Errors in the quotient when
dividing decimal numbers
Review the Methodology for
Dividing Decimal Numbers
on page 186.
PREPARATION INVENTORY
Before proceeding, you should have an understanding of each of the following:
the terminology and notation associated with proportions
how to determine if a proportion is true
how to solve a proportion for the unknown component
how to validate the solution to a proportion
Section 4.2
ACTIVITY
Proportions
PERFORMANCE CRITERIA
• Validating a proportion
• Solving a proportion
– answer rounded to the specified place
– validation of the answer
CRITICAL THINKING QUESTIONS
1. What are two different ways the following proportion can be read?
8
12.8
=
15
2
•
•
2. Why must the comparison order of the units be verified before determining if a proportion is true?
3. What are the ways you can validate that a proportion is true?
395
396
Chapter 4 — Ratios and Proportions
4. When should you use the Methodology versus the Technique for solving a proportion?
5. Why can you cross-multiply and equate the cross-products in Step 2 of the Methodology for Solving a
Proportion?
6. Why do you divide each side of the equation by the multiplier of the variable in Step 3 of the Methodology
for Solving a Proportion?
7. How do you assure that your answer for the unknown value in a proportion is correct?
Section 4.2 — Proportions
TIPS
FOR
397
SUCCESS
•
To assure a correct answer when solving a proportion, do not skip steps. Write the equation for the crossmultiplication followed by the equation showing the division by the multiplier of the variable. In this way, the
numbers you must multiply and divide will be clearly presented.
•
Use your skills of reducing and building up fractions to shortcut the process of validating proportions and
the process of solving a proportion.
•
When validating a proportion with decimal components, be attentive to the placement of the decimal point
in the cross-products.
•
To check for exact computational accuracy, validate the decimal division (see Methodology for Dividing
Decimal Numbers in Section 2.4).
DEMONSTRATE YOUR UNDERSTANDING
In problems 1-5, determine if the given proportions are true or false.
Proportion
1)
9
16
=
15 25
2)
6.4 gallons
4 gallons
=
11.2 acres
7 acres
3)
28 SUVs
16 SUVs
=
91 total vehicles 52 total vehicles
Worked Solution
True or
False?
Chapter 4 — Ratios and Proportions
398
Proportion
Worked Solution
True or
False?
2
3
3
4 = 3
12
9
4) 2
5)
8.8
88
=
55
550
6) The directions on a well-known brand of parboiled rice say to combine the rice and water in the ratio of
1/2 cup rice to 2 1/4 cups water for four servings, and in the ratio of 1 1/2 cups rice to 3 1/3 cups water
for six servings. Are the ratios in proportion to each other?
For the following proportions, solve for the unknown. Round to the nearest tenth, if necessary.
Proportion
7)
n
21
=
13 39
Worked Solution
True or
False?
Section 4.2 — Proportions
Proportion
8)
25
n
=
7
10
9)
4
n
=
7
73.5
10)
39 13
=
n
72
11)
0.3
7
=
0.5
n
399
Worked Solution
True or
False?
Chapter 4 — Ratios and Proportions
400
Proportion
True or
False?
Worked Solution
12) 5 is to 8, as
42 is to what
number?
IDENTIFY
AND
CORRECT
THE
ERRORS
In the second column, identify the error(s) you find in each of the following worked solutions. If the answer
appears to be correct, validate it in the second column and label it “Correct.” If the worked solution is incorrect,
solve the problem correctly in the third column and validate your answer in the last column.
Worked Solution
What is Wrong Here?
1) Is this a true
proportion?
0.752
7.52
=
9
90
Identify Errors
or Validate
The decimal point was
placed improperly in
the product of 0.752
and 90.
(Should have been 3
decimal places.)
Correct Process
0.752
× 90
000
67680
7.52
× 9
67.68
67.680
TRUE
or
0.752 ? 7.52
=
9
90
0.752 × 10 7.52
=
9 × 10
90
TRUE
Validation
Validation is not
necessary for this
problem.
Section 4.2 — Proportions
Worked Solution
What is Wrong Here?
2) Solve for the
unknown.
44
20
=
n
28
3) 11 is to 9 as 33 is to
what number?
401
Identify Errors
or Validate
Correct Process
Validation
Chapter 4 — Ratios and Proportions
402
Worked Solution
What is Wrong Here?
4) Solve for n. Round
to the nearest tenth.
25
9.1
=
n
11
5) Solve for n. Round
to the nearest tenth.
2.6
6
=
48
n
Identify Errors
or Validate
Correct Process
Validation
Section 4.2 — Proportions
403
ADDITIONAL EXERCISES
Determine if the following proportions are true or false.
1.
16 feet
24 feet
=
10 seconds 15 seconds
2.
44
96.8
=
6.1 13.2
3.
12
8
=
56
84
4.
0.7
77
=
3
33
5.
7 cups sauce
=
39 servings
9
1
cups sauce
3
52 servings
Solve for the unknown in each of the following proportions. Round to the nearest tenth if necessary.
Validate your solutions.
6.
6
15
=
n 13.8
7.
136
n
=
9
10.51
8.
1
n
= 3
12
5
9.
7
44.75
=
8
n
10.
n
24
=
100
56
11.
12.
1
1
4= n
9
24
2
44 is to 50 as what number is to 850?