Proportions: Word Problems—Explanation & Practice

Proportions: Word Problems--Explanation & Practice

Writing a proportion is helpful in solving many word problems. To write the proportion, carefully observe the word order in the problem. You will usually find two relationships in the problem. The first relationship should be between two known numbers, and the second relationship should have one unknown in it: the answer you are seeking. Write these relationships in the form of fractions (ratios). Follow these steps to solve the problem:

A. Determine what you are looking for. This is called the unknown. Give it a name like x or N.

B. Determine one relationship between two numbers in the word problem. Write them as a fraction.

C. Find a relationship between a number and the unknown you are looking for. Write them as a fraction.

D. Set these two fractions equal to each other. (This gives you a proportion). E. Check the numerators. Are they in the same units of measure? (For example,

"apples" and "apples.") How about the denominators? (For example, "dollars" and "dollars.") If so, solve the proportion. F. To solve the proportion, cross multiply and divide both sides of the equation by the number in front of x.

EXAMPLE 1. If there are 3 cups of sugar in 24 cookies, how much sugar is there in 72 cookies? Write a proportion.

3cups = x cups 24 cookies 72 cookies

Important: It is helpful to use words or abbreviations beside the numbers in your proportion. The labels in the numerators should be the same, and the labels in the denominators should be the same.

EXAMPLE 2. The word order in some problems requires special care. "A 20-pound turkey serves 28 people. How many people will a 30-pound turkey serve?"

Write a proportion. Careful labeling will result in identical words in the numerators of your proportion and identical labels in the denominators:

20lb. = 30lb. 28 people x people

Note:

Attempting to write this proportion without labels will usually result in the following INCORRECT proportion:

20 = x 28 30

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EXAMPLE 3:

Write a proportion and solve. A biologist determined that there were 42 oak trees in a 5-acre forest. How many oak trees would be expected on 40 acres of identical habitat?

42 oaks = x oaks 5acres 40 acres

(x)(5) = (42)(40)

Write the proportion starting with the known information for the first ratio.

Solve for x by using cross products.

5x = 1680

5x = 1680 55 x = 336 oak trees

EXAMPLE 4: There are 6 cups of detergent in a bottle. If 1/3 cup is need for 1 load of laundry, how many loads can be cleaned with one bottle of detergent?

Unknown: How many loads can be cleaned with one bottle? Call it X.

First ratio: 1/3 cup to 1 load.

Second ratio: one bottle to X loads.

Proportion: Notice below that the ratio numerators will not be in the same units if

we use "cup" and "bottle."

1 cup 3

=

1 bottle

1load X loads

(the units don't match!)

One way to make the units match is to change "bottle" into "cups" in the second ratio because we know that there are 6 cups of detergent in a bottle from the first sentence in the word problem. By the way, a common error is to use the first sentence "6 cups to one bottle" as the first ratio, but then it cannot be related to the second ratio which deals with LOADS, not bottles. The key is to keep looking in the problem for any two relationships which match up in the units, making sure that the unknown is always included in the second relationship. Be careful: not every piece of information in a word problem is meant to be used directly in the proportion.

Proportion:

1 3

cup

=

6 cups 1 bottle

1 load X loads

1 3

X

1X = 1 6 31

= 16

33

X= 63

X = 18 Answer: 18 loads

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EXAMPLE 5:

Write a proportion based on the following information: "The directions on a bottle of fuel additive call for pouring in 12 ounces of additive to each 16 gallons of fuel. How many gallons of fuel will 9 ounces treat?"

Write a proportion using labels: 12 oz. = 9 oz. 16 gal. x gal.

Note the following multiple choice answers:

A. 12 = x 16 9

B.

16 = 9

12 x

C. 16 = 12 9x

D. 16 = x 12 9

The original proportion is not one of the answer choices. If you cross multiply, you will get (12)(x) = (9)(16). Which of the answers given will result in the same equation? D is the correct answer since that proportion is equivalent to your original correctly constructed one. The cross products (12x and 144) are the same for both proportions.

PRACTICE

1. A jar of peaches weighs 13 ounces. There are 16 jars in a crate. How many ounces does a crate weigh?

2. Tonya drove 343 miles the first day of her trip. How many days will it take her to drive the remaining 1,200 miles if she continues at the same rate?

3. A 6-pound ham will serve 14 people. How many people will a 9-pound ham serve? Set up a proportion and solve.

4. A fine fabric was selling for $ 10.50 a yard. Joni bought 2 1/3 yards. How much did she spend?

5. A doctor's orders are to give his patient 250 milligrams of a pain-relieving medication. If there are 50 milligrams in a capsule, then how many capsules should the patient take?

6. You have 18 days to complete your math prescription in order to retake the Test of Adult Basic Education (TABE). You've noticed that it takes you 2 days to finish 7 modules, but there are still 24 modules left to do. How many days will it take to finish? Will you finish on schedule?

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7. An engineer is proof-reading his report and finds that he has mistakenly used micrometers as a unit of measure. His experiment showed a measurement of 2 micrometers, but the government requires the measurement to be in millimeters. How should the engineer record the measurement (if there are 1000 micrometers in 1 millimeter)?

8. The next-door neighbors are spending 10 hours a week maintaining their ? acre property. You are thinking about landscaping your own property in exactly the same way, but it is a 1/3 acre lot. How many hours per week will you have to spend maintaining your property?

9. A piece of stone on Planet Y weighs seven pounds, but on Earth it weighs only 4 pounds. What would a 130-pound Earthling weigh on Planet Y?

10. Eighty-eight percent of all people who try to do this ratio and proportion worksheet find they need some help with the problems. There are now 49 people who have asked for help while doing this worksheet. How many people have worked on this worksheet so far?

11. Write a proportion based on the following information: "A new energy efficient

car travels 720 miles on 12 gallons of gas. How many gallons would be used on

a 1000 mile trip?"

A. 720 = x 12 1000

B. 12 = 1000 720 x

C. 720 = 1000 12 x

D. 720 = x 1000 12

12. Write a proportion based on the following information: "A copier uses 30 grams of toner to produce 1000 copies. How many grams are required to produce 2500 copies?"

A. 30 = 2500 1000 x

B. 30 = 2500 x 1000

C. 1000 = x 30 2500

D. 1000 = 2500 30 x

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Answer Key Proportions: Word Problems ? Explanation & Practice

1. Unknown: How many ounces in a crate? Call it X. First ratio: One jar is 13 ounces. Second ratio: 16 jars is X ounces. Proportion: 1 jar = 16 jars 13 ounces X ounces 1X= 1316 X = 208 ounces

2. Unknown: How many days to drive 1200 miles? Call it X.

First ratio: 343 miles in one day.

Second ratio: 1200 miles in X days.

Proportion: 343 miles = 1200 miles

1 day

X days

343X = 1200 343 343 X 3.5 days

3. 6 pounds = 9 pounds 14 people x people 6x= 9 ?14 6x = 126 x = 21 people

5. 250 mg = 50 mg x caps 1 cap 250 1 =50x

250 = 50x 50 50 x = 5 caps

4.

$10.50 = x dollars

1 yard

1 2 yards

3

(10.50) (2 1) = 1x 3

21 7 =1x 23

x = $24.50

6. 2 days = x days 7 mods 24 mods 2 24 = 7x

48 = 7x 77 x = 6 6 days; yes

7

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