4.2 PROPORTIONS
Jake has a part-time job walking dogs for an elderly neighbor. He is paid $20 for every ninety minutes he
works. He has walked Boris, Buster, and Bonnie a total of 315 minutes this week.
How much can Jake expect
to be paid for his work?
20
n
=
90 315
70.00
$_____________________
Assess your readiness to complete this activity. Rate how well you understand:
Not
ready
Almost
ready
Bring
it on!
• 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
• Validating a proportion
• Solving a proportion
– answer rounded to the specified place
– validation of the answer
181
Chapter 4 — Ratios and Proportions
Technique
Apply the Equality Test for Fractions and cross-multiply to determine if the proportion is true.
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.
miles per gallon in both ratios of the proportion
?
57 × 1.9 = 21.5 × 5
108.3 ≠ 107.5
Cross-multiply.
The proportion is false.
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.
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.
182
5×3
15
=
36 × 3 108
The proportion is true.
Activity 4.2 — Proportions
Technique
If easily done, reduce both ratios to their simplest forms. If the reduced ratios are equal,
then the proportion is true.
Is this proportion true or false?
Reduce both sides:
10 ÷ 2
5
=
12 ÷ 2
6
10
35
=
12
45
and
35 ÷ 5
7
=
45 ÷ 5 9
The proportion is false.
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.
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.
Solve for n (a variable with an unknown value) in the proportion
0.3 × 4 = 1.2
Then 5 × 4 = 20 = n
Step 1
Step 2
Validate:
0.3 ? 1.2
=
5
20
0.3 1.2
=
5
n
Answer : n = 20
?
0.3 × 20 = 1.2 × 5
6.0 = 6.0
183
Chapter 4 — Ratios and Proportions
If the relationship between the two ratios of a proportion is not easily recognizable, use the following
methodology to solve for the variable.
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
Example 1
Write the given proportion. If it is not
already set up in proportion form, set
Set up the
up the proportion using a variable for
proportion
with a variable. the unknown component.
11
n
=
15 37.5
Step 1
Step 2
Equate the
cross products.
Cross multiply and equate the crossproducts.
Divide by the
multiplier of
the
11 × 37.5 = n × 15
Remember that the Equality Test for
Fractions says that, if the ratios are equal,
then their cross-products will be equal.
Shortcut
Step 3
Example 2
Reduce the known ratio
first (see Model 2)
Divide both sides of the equation by
the multiplier of the variable.
This results in an equation that will
provide the missing quantity of your
proportion.
Step 4
Calculate the value for the variable.
Solve for the
variable.
Compute the numerator and divide
the product by the denominator.
Round to the specified place value, if
necessary.
1
11 × 37.5
n × 15
= 1
15
15
37.5
× 11
375
+3750
27.5
15 412.5
−30
)
112
−105
412.5
75
−75
0
Step 5
Present the
answer.
184
State your answer, the value for the
unknown.
n = 27.5
Activity 4.2 — Proportions
Steps in the Methodology
Example 2
Example 1
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 11 × 37.5 = 27.5 × 15
Proportion.
412.5 = 412.5
11 ? 27.5
=
15
37.5
Note: When the answer is rounded,
the cross-products will be close but
not exactly equal.
37.5
×11
375
+3750
412.5
27.5
× 15
1375
+2750
412.5
Model 1
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
3
8
Validate:
3
8
35
4
?
=
1
2
12
4
1
35
2
8
35
105
1
35
3
×4
=
=4
2
8
8
12
3
? 1 1 × 35
× 12 =
8
2
3
×
12 ? 3 35
= ×
1
2 1
105 105
=
2
2
185
Chapter 4 — Ratios and Proportions
Model 2
Solve for n:
Step 1
Step 2
Shortcut: Reduce the Known Ratio First
75
9
=
100
n
75
9
=
100
n
Shortcut (optional): To simplify computations, reduce
the known ratio before equating the cross-products.
Use the shortcut and reduce:
75 ÷ 25
3
=
100 ÷ 25
4
⇒
3
9
=
4
n
At this point, there is no need to cross-multiply.
Use the Technique and skip Steps 2-4.
3×3 = 9
4 × 3 = 12 = n
Step 5
Step 6
n = 12
Validate:
Make Your Own Model
75 ? 9
=
100
12
? 9 × 100
75 × 12 =
900 = 900
Either individually or as a team exercise, create a model demonstrating
how to solve the most difficult problem you can think of.
Answers will vary.
Problem: _________________________________________________________________________
186
Step 1
Step 4
Step 2
Step 5
Step 3
Step 6
Activity 4.2 — Proportions
1. Why must the comparison order of the units be verified before determining if a proportion is true?
The numerators should both reflect the same unit and likewise the denominators of both ratios should reflect the
same unit. Therefore the comparison will both be the same, like dollars per share or miles per hour.
2. What are the ways you can validate that a proportion is true?
You can accomplish this by cross multiplication. If the answers are the same, then the proportion is true. You could
also reduce or build both ratios to show that the fractions are equivalent.
Corresponding parts can be reduced: numerator with denominator on the same side of the equal sign or numerator
to numerator or denominator to denominator on opposite sides of the equal signs.
3. When should you use the Methodology versus the Technique for solving a proportion?
If the relationship between the two ratios is easily recognizable, then use a Technique, otherwise use the
Methodology.
4. Why can you cross-multiply and equate the cross-products in Step 2 of the Methodology for Solving a
Proportion?
This can be used because of the Equality Test for Fractions.
5. 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?
Because the Special Properties of Division Involving One tell us that any number divided by itself equals one, and
the Identity Property of Multiplication tells us that one times any number equals that same number.
6. How do you ensure that your answer for the unknown quantity (the variable) in a proportion is correct?
Replace the unknown with the answer that you found then cross-multiply to show that the cross products are
equivalent. This will validate your answer.
187
Chapter 4 — Ratios and Proportions
7. What aspect of the model you created is the most difficult to explain to someone else? Explain why.
Answers will vary.
1. Determine if the given proportions are true or false.
Proportion
Worked Solution
True or
False?
a)
9
16
=
15 25
False
b)
6.4 gallons
4 gallons
=
11.2 acres
7 acres
True
c)
28 SUVs
16 SUVs
=
91 total vehicles 52 total vehicles
True
188
Activity 4.2 — Proportions
Proportion
d)
True or
False?
Worked Solution
2
3
3
4 = 3
12
9
2
True
2. 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?
3. For the following proportions, solve for the variable. Round to the nearest tenth, if necessary.
Proportion
a)
n
21
=
13 39
b)
25
n
=
7
10
Worked Solution
Validation
189
Chapter 4 — Ratios and Proportions
Worked Solution
Proportion
c)
4
n
=
7
73.5
d)
0.3
7
=
0.5
n
Determine if the following proportions are
true or false.
1.
2.
3.
4.
16 feet
24 feet
=
10 seconds 15 seconds
44
96.8
=
6.1 13.2
12
8
=
56
84
0.7
77
=
3
33
FALSE
6.
6
15
=
n 13.8
7.
136
n
=
9
10.51
1
1
n
= 3
8.
12
5
FALSE
1
9 cups sauce
7 cups sauce
5.
= 3
39 servings
52 servings
190
Solve for the variable in each of the following
proportions. Round to the nearest tenth if necessary.
Validate your solutions.
TRUE
FALSE
9.
TRUE
Validation
7
44.75
=
8
n
5.52
158.8
3
1
or 3.2
5
51.1
Activity 4.2 — Proportions
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
Validation
0.752
7.52
× 90 × 9
000 67.68
67680
67.680
Validation is not
necessary for this
problem.
TRUE or
TRUE
2) Solve the proportion
for the variable.
44
20
=
n
28
You cannot cancel
across the equal (=)
sign.
191
Chapter 4 — Ratios and Proportions
Worked Solution
What is Wrong Here?
3) 11 is to 9 as 33 is to
what number?
Identify Errors
or Validate
The problem is set up
incorrectly.
The correct set up is:
11
33
=
9
n
192
Correct Process
Validation