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CHAPTER 16: Percents, Ratio, and Proportion 181
RATIO AND PROPORTION
Ratio and proportion questions have long been a popular type of arithmetic problem given on Civil Service Exams. This section will help you understand the rules governing ratio and proportion problems.
Solving Ratio Problems
A ratio expresses the relationship between two (or more) quantities in terms of numbers. The mark used to indicate ratio is the colon (:) and is read “to.” For example, the ratio 2:3 is read “2 to 3.”
A ratio also represents division. Therefore, any ratio of two terms can be written as a fraction, and any fraction can be written as a ratio. For example, 3:4 = 3
4 . Follow these steps to solve problems in which the ratio is given:
1
Add the terms in the ratio.2
Divide the total amount that is to be put into a ratio by this sum.3
Multiply each term in the ratio by this quotient.For example, the sum of $360 is to be divided among three people according to the ratio 3:4:5. How much does each one receive?
1
Add the terms in the ratio: 3 + 4 + 5 = 12.2
Divide the total amount to be put into the ratio by this sum: $360 ÷ 12 = $30.3
Multiply each term in the ratio by this quotient: $30 × 3 = $90; 30 × 4 = $120;$30 × 5 = $150.
The money is divided thus: $90, $120, and $150.
To simplify any complicated ratio of two terms containing fractions, decimals, or percents, you only need to divide the first term by the second. Reduce the answer to its lowest terms, and write the fraction as a ratio. For example, simplify the ratio
5 6:7
8→56÷78=20
21= 20:21.
Solving Proportion Problems
A proportion indicates the equality of two ratios. For example, 2:4 = 5:10 is a proportion.
This is read, “2 is to 4 as 5 is to 10.” The two outside terms (2 and 10) are the extremes, and the two inside terms (4 and 5) are the means. Proportions are often written in fractional form. For example, the proportion 2:4 = 5:10 can be written as 2
4 = 5
10. In any proportion, the product of the means equals the product of the extremes. If the proportion is in fractional form, the products can be found by cross-multiplication. For example, in the proportion 2
4 = 5
10, 4 × 5 = 2 × 10.
Many problems in which three terms are given and one term is unknown can be solved using proportions. To solve such problems, follow these steps:
1
Formulate the proportion very carefully according to the facts given. (If any term is misplaced, the solution will be incorrect.) Any symbol can be written in place of the missing term.2
Determine by inspection whether the means or the extremes are known. Multiply the pair that has both terms given.3
Divide this product by the third term given to find the unknown term.Try this example problem:
1. The scale on a map shows that 2 centimeters represent 30 miles of actual length. What is the actual length of a road that is represented by 7 centimeters on the map?
In this problem, the map lengths and the actual lengths are in proportion; that is, they have equal ratios. If m stands for the unknown length, the proportion is
2 7 = 30
m. As the proportion is written, m is an extreme and is equal to the product of the means, divided by the other extreme: m = 7 × 30 ÷ 2 = 210 ÷ 2 = 105. Therefore, 7 cm on the map represent 105 miles.
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CHAPTER 16: Percents, Ratio, and Proportion 183
exer cises
EXERCISE 1
Directions: Each question has four suggested answers. Select the correct one.
4. If there are 16 boys and 12 girls in a class, the ratio of the number of girls to the number of children in the class is (A) 3 to 4
(B) 3 to 7 (C) 4 to 7 (D) 4 to 3 5. 259 is to 37 as
(A) 5 is to 1 (B) 63 is to 441 (C) 84 is to 12 (D) 130 is to 19 1. The ratio of 24 to 64 is
(A) 8:3 (B) 24:100 (C) 3:8 (D) 64:100
2. The Baltimore Ravens won 8 games and lost 3. The ratio of games won to games played is
(A) 8:11 (B) 3:11 (C) 8:3 (D) 3:8 3. The ratio of 1
4 to 3
5 is (A) 1 to 3
(B) 3 to 20 (C) 5 to 12 (D) 3 to 4
EXERCISE 2
Directions: Each question has four suggested answers. Select the correct one.
5. The actual length represented by 31 inches on a drawing having a scale of2
1
6. Aluminum bronze consists of copper and aluminum, usually in the ratio of 10:1 by weight. If an object made of this alloy weighs 77 pounds, how many pounds of aluminum does it contain?
(A) 7.7 (B) 7.0 (C) 70.0 (D) 62.3
7. It costs 31 cents a square foot to lay vinyl flooring. To lay 180 square feet of flooring, it will cost days, the amount that he will earn in 117 days is most nearly
(A) $3,050 (B) $2,575 (C) $2,285 (D) $2,080 1. Two dozen cans of dog food at the rate of
three cans for $1.45 would cost (A) $10.05
(B) $11.20 (C) $11.60 (D) $11.75
2. A snapshot measures 21
2 inches by 17
8
inches. It is to be enlarged so that the longer dimension will be 4 inches. The length of the enlarged shorter dimension will be (D) None of these
3. Men’s white handkerchiefs cost $2.29 for three. The cost per dozen handkerchiefs is (A) $27.48
(B) $13.74 (C) $9.16 (D) $6.87
4. A certain pole casts a shadow 24 feet long.
Another pole 3 feet high casts a shadow 4 feet long. How high is the first pole, given that the heights and shadows are in proportion?
(A) 18 feet (B) 19 feet (C) 20 feet (D) 21 feet
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CHAPTER 16: Percents, Ratio, and Proportion 185
exer cises
9. Assuming that on a blueprint, 1
8 inch equals 12 inches of actual length, the actual length in inches of a steel bar represented on the blueprint by a line 33
4
inches long is (A) 3
(B) 30 (C) 450 (D) 360
10. A, B, and C invested $9,000, $7,000, and
$6,000, respectively. Their profits were to be divided according to the ratio of their investments. If B uses his share of the firm’s profit of $825 to pay a personal debt of $230, how much will he have left?
(A) $30.50 (B) $32.50 (C) $34.50 (D) $36.50
ANSWER KEY AND EXPLANATIONS
Exercise 1
1. C 2. A 3. C 4. B 5. C
1. The correct answer is (C). The ratio 24 to 64 can be written 24:64, or 24
64. In fraction form, the ratio can be reduced to
3
8, or 3:8.
2. The correct answer is (A). The num-ber of games played was 3 + 8 = 11. The ratio of games won to games played is 8:11.
3. The correct answer is (C). 1
4 : 3
5 =
1
4 ÷ 35 = 5
12 = 5:12
4. The correct answer is (B). There are 16 + 12 = 28 children in the class. The ratio of number of girls to number of children is 12:28, which can be reduced to 3:7.
5. The correct answer is (C). The ratio
259
37 reduces by 37 to 7
1 . The ratio 84
12 also reduces to 7
1 . Therefore, 259
37 = 84
12 is a proportion.
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CHAPTER 16: Percents, Ratio, and Proportion 187
answers
Exercise 2
1. C 3. C 5. B 7. C 9. D
2. B 4. A 6. B 8. B 10. B
6. The correct answer is (B). Because only two parts of a proportion are known (77 is the total weight), the problem must be solved by the ratio method. The ratio of 10:1 means that if the alloy were sepa-rated into equal parts, 10 of those parts would be copper and 1 would be alumi-num, for a total of 11 parts. 77 ÷ 11 = 7 pounds per part. The alloy has 1 part aluminum: 7 × 1 = 7 pounds aluminum.
7. The correct answer is (C). The cost (c) is proportional to the number of square feet: .31
c = 1
180. c = .31 × 180 ÷ 1 = $55.80.
8. The correct answer is (B). The amount earned is proportional to the number of days worked. If a is the unknown amount, the proportion is: $352
a = 16
117. a = 352 × 117 ÷ 16 = $2,575.
9. The correct answer is (D). If n is the unknown length, the proportion is:
1 8 3
34 = 12
n . n = 12 × 334 ÷ 18 = 360.
10. The correct answer is (B). The ratio of investment is: 9000:7000:6000 or 9:7:6.
9 + 7 + 6 = 22. Each share of the profit is
$825 ÷ 22 = $37.50. B’s share of the profit is 7 × 37.50 = $262.50. The amount B has left is $262.50 – $230.00 = $32.50.
1. The correct answer is (C). The num-ber of cans is proportional to the price.
Let p represent the unknown price:
3
24 = 1 45.p . p = 1.45 × 24 ÷ 3 = 34.80 ÷ 3
= $11.60.
2. The correct answer is (B). Let s rep-resent the unknown shorter dimension:
2 cost per dozen (12), the proportion is:
3
12 = 2 29.p . p = 12 × 2.29 ÷ 3 = $9.16.
4. The correct answer is (A). If f is the height of the first pole, the proportion is:
f 24= 3
4. f = 24 × 3 ÷ 4 = 18 ft.
5. The correct answer is (B). If y is the unknown length, the proportion is:
312
1 8
= 1y. y = 312 × 1 ÷ 18 = 28 ft.
chapter 17
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Graphs and Tables
OVERVIEW
• Graphs
• Tabular completions
GRAPHS
A graph is a picture that illustrates comparisons and trends in statistical information. This section will prepare you to see the “complete picture” in a graph and supply the correct answers based on the data. The following are the most commonly used graphs: