UNIT FOUR: Prealgebra in a Technical World
1.
Solve proportions using cross multiplication.
2.
Identify proportional situations and solve proportion problems.
365
Most of us solve proportion problems daily by using unit rates or scaling without calling these problems “proportions.” But they are proportion problems. Not all proportion problems can be solved using mental math, though. When numbers do not scale easily or when problem situations are more difficult to understand, we need another method.
In this section we write and solve proportions using cross multiplication.
Proportions and Cross Multiplication
Proportions are equal ratios, and all of these are proportions:
3
4
=
75
100
,
0.7
0.28
=
5
2
,
35
49
=
5
7
, and 𝑥
10
=
25
.
6
The last of these proportions has a missing value, which we can label with any symbol; often we use " 𝑥 .”
We can find the missing value x
in 𝑥
10
=
25
by working with equivalent fractions. We
6 can start by rewriting these fractions so that they have least common denominators. We find 𝑥 that
10
∙ (
3
3
) =
25
6
∙ (
5
5
) that is,
3𝑥
30
=
125
. To keep these fractions equal, the numerators
30 must be equal, so we know 3𝑥 = 125 . Because x is a missing factor, we divide 125 by 3 to find this value. We complete the division: = 125 ÷ 3 = 41
2
3
.
When using equivalent fraction ideas with proportions, we do not find the least common denominator; instead, we find the common denominator created by multiplying the two original denominators together. This allows us to look only at numerators.

366 SECTION 4.6 Solving Proportion Problems 𝑥
For our problem
10
=
25
6
, we have
6𝑥
60
=
250
. This shows that 6𝑥 = 250 . This time
60 we find the value of x by dividing 250 by 6. We complete the division: 250 ÷ 6 = 41
2
3
.
𝑥
Most importantly, notice that if
10
=
25 then 6𝑥 = 250 . This idea gives us an
6 important method for solving proportion problems: 𝑥
10
=
25
6
Given a proportion problem,
First multiply each numerator by the denominators from the other fraction.
6𝑥 = 250 The new denominators must be equal. 𝑥 = 250 ÷ 6 = 41
2
3
Because x is a missing factor, we find its value by dividing.
FORMULA: If 𝑎 𝑏
= 𝑐 𝑑
, then for all 𝑎, 𝑏, 𝑐 and 𝑑 where 𝑏 and 𝑑 do not equal zero, the cross multiplication, 𝑎 ∙ 𝑑 = 𝑏 ∙ 𝑐 , is equal.
For example, if
3
4
=
75
100
, then 3 ∙ 100 = 4 ∙ 75 ,
and if 𝑥
5
=
42
70
, then 70 ∙ 𝑥 = 5 ∙ 42,
Not only can we solve for missing values, we can also determine whether two ratios are proportional by checking their cross products.
Example 1: For these problems, determine if the ratios are equal by using cross multiplication. a.
Are
39.6
11 and
7.2
2 proportional? Is the statement
39.6
11
=
7.2
true?
2
Think it through: If
𝟑𝟗.𝟔
𝟏𝟏 and
𝟕.𝟐
𝟐 are proportional, then using cross multiplication
𝟑𝟗. 𝟔 ∙ 𝟐 = 𝟏𝟏 ∙ 𝟕. 𝟐 would be true. We find (𝟑𝟗. 𝟔 ∙ 𝟐) = 𝟕𝟗. 𝟐 𝐚𝐧𝐝
(𝟏𝟏 ∙ 𝟕. 𝟐) = 𝟕𝟗. 𝟐 , and
𝟑𝟗.𝟔
𝟏𝟏
=
𝟕.𝟐
.
𝟐
ANSWER: The ratios
𝟑𝟗.𝟔 and
𝟏𝟏
𝟕.𝟐
𝟐 are proportional.

UNIT FOUR: Prealgebra in a Technical World
Example 1 : (continued) b.
Are
17
5 and
52.8
16 proportional? Is the statement
17
5
=
52.8
true?
16
Think it through: If
𝟏𝟕
𝟓 is proportional to
𝟓𝟐.𝟖
, then cross multiplication 𝟏𝟕 ∙ 𝟏𝟔 = 𝟓 ∙ 𝟓𝟐. 𝟖
𝟏𝟔 would be true. But 𝟏𝟕 ∙ 𝟏𝟔 = 𝟐𝟕𝟐 𝒘𝒉𝒊𝒍𝒆 𝟓 ∙ 𝟓𝟐. 𝟖 = 𝟐𝟔𝟒 .
ANSWER: The ratios
𝟏𝟕
𝟓 and
𝟓𝟐.𝟖
𝟏𝟔 are not proportional:
𝟏𝟕
𝟓
≠
𝟓𝟐.𝟖
.
𝟏𝟔

Check Point 1
Use cross multiplication to determine whether these ratios are proportions.
Cross multiplication is most often used to solve for missing quantities in proportions.
Example 2: Solve for x :
16
5
=
88
. 𝑥
Think it through: Use cross products.
ANSWER: 𝒙 = 𝟐𝟕. 𝟓
367

368 SECTION 4.6 Solving Proportion Problems
Example 3: Solve for n : 𝑛
5.4
=
1
6
. (Notice that we can use any symbol for a missing value.)
Think it through: Use cross multiplication 𝑛
5.4
=
1
6
Write the original proportion.
6 ∙ 𝑛 = 5.4 Use cross multiplication. 𝑛 = 5.4 ÷ 6 Since 𝟔 ∙ 𝒏 = 𝟓. 𝟒, find n by dividing 5.4 by 6. 𝑛 = 5.4 ÷ 6 = 0.9
Divide.
0.9
5.4
9
=
54
=
1
6
ANSWER: 𝒏 = 𝟎. 𝟗
This time we can check by scaling, and the results check.

Check Point 2
Solve for 𝑥 :
450
5
= 𝑥
60
As we have seen, proportions are part of our daily lives. When solving these proportion problems, we can use (1) unit rates, (2) scaling, or (3) cross multiplication to find exact results.
Use the easiest method for any given problem, and check with a second method.
Solve Problems Using Proportions
When solving proportion problems, we write two ratios and set them equal. The most important part of this process is writing these two correct ratios.
To analyze problems that can be solved using proportions, we use one of our problemsolving strategies: drawing a diagram. The diagram we draw shows us ratios that we know are equal.

UNIT FOUR: Prealgebra in a Technical World
Example 4: Rosemarie is going to paint the outside of her house. She finds the perimeter of the house and multiplies by the height. She ignores windows and doors since she will be using a sprayer. When sprayed on siding, a gallon of the paint she selects covers 625 square feet.
Rosemarie measures and calculates that the area she will be painting is a little less than 2,300 square feet. How many gallons should Rosemarie buy?
Think it through: This is complex so we use the four step problem solving plan.
Understand: Rosemarie needs to find how many gallons for the whole house, and she realizes that this is a proportion problem.
Plan: Rosemarie will write a proportion and use 𝑥 for the missing gallons of paint. She sketches a vertical scale. Rosemarie puts a zero at the bottom, and uses units to analyze what she knows and what she is missing.
She needs to know the missing number of gallons to use to paint the total area of 2,300 square feet. These go together at the top of her scale, but they go together on opposite sides of the line.
Next she knows that 1 gallon of paint covers 625 square feet.
Again she puts these on opposite sides of the line.
She makes sure that both gallon measures are on one side of the line and that both feet measures are on the other side of the line.
Solve: Reading her diagram, she writes her proportion correctly: 𝑥
1
=
2,300
.
625
Rosemarie, using 24 ÷ 6, estimates that this is about four gallons. To solve for 𝑥, Rosemarie divides 2,300 by 625. The result is 3.68 gallons.
Check: Use cross multiplication. 3.68 x 625 = 2,300. The result checks.
ANSWER: Rosemarie buys 4 gallons, and she knows she will have extra paint. She
saves the remaining paint for any touch up she may need.
369

370 SECTION 4.6 Solving Proportion Problems
Example 5: A physician orders 20 mg of aprobarbital delivered through the IV line to begin sedating a patient. The medication is delivered at a rate of 32 mg/5mL. How many mL should the anesthesia nurse draw up in the syringe?
Think it through: This is complex so we use the four step problem solving plan.
Understand: The nurse needs to determine the dosage for this patient, and she will use a proportion.
Plan: Use 𝑥 for the unknown amount of aprobarbital in milliliters. Sketch a vertical scale. The 32 mg is the amount delivered per 5 mL. We need 20 mg, and we need to find the number of mL of solution the nurse should draw into the syringe.
Write the proportion.
Use cross products.
Divide.
Check to see that the ratios are equal.
Divide, use a calculator if needed.
ANSWER: The nurse needs to draw up 3.125 mL of the aprobarbital solution.

Check Point 3
Linda has discovered the food of the Incas, quinoa. She finds that quinoa has 8 g of protein for every 222 calories. Linda reads that at her height she should eat 30 g of protein a day. If she gets all of this protein from quinoa, how many calories of quinoa will she consume?

UNIT FOUR: Prealgebra in a Technical World
Many problems can be drawn so that we can see similar figures. In Section 4.4 we looked at many similar figures. Blueprints, maps, architectural drawings, and sketches are used by many trades’ professionals on the job.
DEFINITION: Similar figures are figures that have the same shape, but not necessarily the same size. Similar figures can be created when one figure is enlarged or reduced by a single factor.
PROPERTY: If two figures are similar figures, all lengths are proportional.
Example 6: Contractors, carpenters, and “do-it-yourselfer” home owners use similar triangles when they work with rooflines. A common roof slope is a 5:12. This roof goes up 5 units for every 12 units it goes over. If support studs are centered every 2 feet, how many inches long is the stud labeled “ 𝑥 ”? (See the drawing below.)
Think it through: This is complex so we use the four step problem solving plan.
Understand: Carpenters keep a labeled drawing to check their calculations against. In the drawing we see 6 similar triangles. We will use the one triangle that has both legs labeled, and determine another ratio that compares this triangle to the similar triangle with length x
.
371

372 SECTION 4.6 Solving Proportion Problems
Plan: For the 5:12 triangle, 𝑥 corresponds to the 5 ft leg. A 10 ft leg in the same triangle with 𝑥 corresponds to the 12 ft leg. Use the vertical scale to check whether the proportion you write is accurate. Solve this proportion to find the length, x.
Solve:
5
12
= 𝑥
10
Write a proportions using 𝑥 .
12𝑥 = 5 ∙ 10 Use cross multiplication.
𝑥 =
50
12
𝑓𝑡 𝑥 is the missing factor, so we solve using division.
Use unit analysis to solve for the length in inches.
60
12
=
50
10
Check by letting 5 ft = 60 inches in our original proportion. Using cross multiplication, these two ratios, both in inches to feet, are equivalent.
ANSWER: The missing stud needs to measure 50 inches.

Check Point 4
Find the lengths (in inches) of the studs “ 𝑦 ” and “ 𝑧 ” in the diagram from Example 6.
Many, many problems in life can be solved using proportions.
Example 7: In summer 2009, Ashland, Oregon’s, water consumption rate reached 6.7 million gallons of water a day. The population of Ashland was about 21,800 people that summer.
The population of Medford, Oregon, was about 76,850 people during summer 2009. If
Medford residents average the same amount of water use individually as do Ashland residents, about how much water was consumed daily in Medford during summer 2009?

UNIT FOUR: Prealgebra in a Technical World
Think it through: Since Medford has a bit less than four times the population of Ashland, expect that the water usage will be around
𝟔 ∙ 𝟒 = 𝟐𝟒 million gallons. Sketch a vertical scale to help think through the proportion
needed.
Write cross multiplication, and solve for the missing consumption rate. 𝑥
6.7
=
76,850
21,800
Write a proportion using x.
21,800𝑥 = 6.7 ∙ 76,850 Use cross multiplication.
𝑥 = 23.6
Find the missing factor .
23.6
6.7
≈
76,850
21,800
Check your answer by checking the cross products. In
this case both ratios are equal to the tenths place, 3.5 .
ANSWER: If individual consumption rates are about the same for the two cities, we expect that Medford, Oregon, was consuming 23.6 million gallons of water
a day during summer 2009.

Check Point 5
The population of Grants Pass, Oregon, was about 34,000 during Summer 2009. If their individual water consumption rates are similar to Ashland’s, then about how much water did the residents of Grants Pass consume each day during Summer 2009?
Estimate: ___________________________
Answer: ____________________________

Check Point 6
What was the average rate of water consumption per day for the average Ashland resident during Summer 2009?
373

374 SECTION 4.6 Solving Proportion Problems

UNIT FOUR: Prealgebra in a Technical World
Name _______________________________
Circle "yes" if the two ratios are proportional. Circle "no" if they are not.
1. 7
8
,
21
24
Yes No
2. 1
6
,
7
36
Yes No
3. 5
7
,
30
42
5. 8
15
,
144
360
7. 108
280
,
27
56
Yes No
Yes No
Yes No
4. 7
24
,
21
96
6.
1
15
,
21
315
8. 297
,
364
27
28
Yes No
Yes No
Yes No
Solve for the missing value, 𝑥 , that makes each proportion true. Round to the hundredth.
9. 8
15
= 𝑥
45
11. 3
=
7 𝑥
49
13. 11
50
=
88 𝑥
15. 𝑥
392
=
27
49
17. 𝑥
15
=
−3
4
10. 4
5
=
28 𝑥
12. 7
27
=
21 𝑥
14. 𝑥
48
=
5
3
16. 297 𝑥
=
33
40
18. −4
𝑥
=
7
32
UPS 
Write the proportions you use, then write a sentence to answer each question. Number and show your written work neatly using as much paper as you need. Round answers so that they are reasonable. (People do not come in fractions.)
19. Traveling at 45 miles per hour, how long will it take to travel 162 miles? Answer in hours and minutes.
375

376 SECTION 4.6 Solving Proportion Problems
20. If a plane goes 400 miles in 50 minutes, how far will it travel in one hour?
21. Find the stud lengths D and E. (See Example 6 and Check Point 4 on pages 369-370.)
5 ft
12 ft
22. In August 2010, 1 U.S. dollar (USD) was worth 1.05 CAD (CAD is the
Canadian dollar). Naomi traveled to Vancouver, BC, and exchanged $225
USD for Canadian currency. How many Canadian dollars and cents did she receive?
23. The Euro is the currency of the European Union. In August 2010, 10 USD were worth $7.74 Euros. If a European traveler wished to exchange
2,000 Euros for USD, how many U.S. dollars would he or she get?
24. Albert has a job washing cars at a car dealership every Saturday. He can wash 1 car every
15 minutes. If there are 27 cars to be washed, how many hours will it take him to finish the job?
25. A college survey was given to 360 students, and the survey found that 120 students were taking math classes. If there are 3,713 students at this school, use the survey results to estimate how many students are taking math classes at this college.
26. The Columbia River salmon run is endangered. Scientists are now studying how to bring the salmon run back, or at least how to keep it from collapsing to nothing. In 2006 the NW
Marine Fisheries Service proposed releasing 104,060 tagged salmon from McNary Dam on the Columbia River to check return rates of juvenile salmon. The last research showed 3 out of 10 salmon did return. If this rate stays the same, how many of the 104,060 tagged salmon would we expect to return?

UNIT FOUR: Prealgebra in a Technical World
27. To help entertain her daughter, Jennifer brought a ruler on a flight to Denver. Together
Jennifer and her daughter Sarah measured the flight paths on the air map in a travel magazine. The legend gave 75 miles per centimeter as a scale. The daughter measures
13.2 cm as the distance of the path. According to her measurement, how many air miles is it from Portland to Denver?
377
28. Next, the daughter measures the flight path from Denver to Chicago. She finds it is 9 cm.
According to her measurement, how many air miles is it from Denver to Chicago?
29. A student survey was given to 260 community college students, and the survey found that
177 of these students were planning to transfer to a four-year college after completing their two-year programs. If there are 4,148 students at this college, estimate the total number of students planning to transfer to a four-year college.
30. Rudy's favorite cookie recipe calls for 2 1 /
4
cups of flour and 1 1 /
2
cups of sugar. Rudy wants to make as many of these cookies as he can, but he only has 4 1 /
3
cups of flour left.
He has plenty of the other ingredients. About how many recipes can he make? Round down to the fourth of a recipe.
31. Jake and Judy did yard work on Saturday. Jake worked for 4.25 hours and Mom paid him
$21.25. Mom paid Judy at the same rate, and Judy has worked 6.25 hours. How much did
Mom pay Judy?
32. If 3 cans of tuna cost $7.49 and there are 5 ounces in a can, how much does the tuna cost per ounce?

378 SECTION 4.6 Solving Proportion Problems
33. A new school bond will raise property taxes 0.45 cents per thousand dollars of assessed property value. If your home is assessed at $239,000, how much will the school bond increase your taxes?
34. The U.S. is experiencing a shortage of nurses. If Oregon must cut funding for 3 nursing students for every $500,000 reduction in state tax revenues, how many nursing student positions will be cut in Oregon if tax revenues drop by $2.2 million. (Round up to a whole person.)
35. In August, 2010, Juanita went to Puerto Vallarta for her vacation. Before returning home, she wanted to buy a birthday gift for her sister. She found a beautiful sundress that cost
550 pesos. She wondered if this was a fair price, but she couldn't remember the USD -
Peso exchange rate. She did, however, have a receipt that showed she had exchanged
$250 US dollars for 3,232 pesos.
a. What was the equivalent U.S. dollar price for the sundress?
b. What was one dollar worth in pesos when Juanita was in Mexico?
36. Male Rufous hummingbirds weigh about 3.2 grams. They consume an average of about 10.5 calories per day. Use the conversion fact that
1 pound equals 453.6 grams, and round answers to the nearest whole number.
a. An average adult male weighs 190 lbs. If a man ate a proportional amount of food as
a male Rufous, how many calories would he consume a day?
b. An average adult female weighs 163 lbs. If a woman ate a proportional amount of
food as a male Rufous, how many calories would she consume a day?
c. A Big Mac® has 540 calories. How many Big Macs® would a man have to eat to keep
up with the hummingbird?