Worksheet - Ratios, Proportions, and Percent Change

DIRECTIONS: Each question has five answer choices. Select the one best answer. Do not use a calculator.

Section A - Ratio and Proportion Fundamentals - Questions 1 to 7

1. The ratio of boys to girls in a class is 3:5. If there are 15 boys in the class, how many girls are there?

1. 9
2. 18
3. 20
4. 25
5. 30

2. If \(\frac{x}{12} = \frac{5}{8}\), what is the value of \(x\)?

1. 4.8
2. 7.5
3. 9.6
4. 15
5. 19.2

3. A recipe calls for flour and sugar in the ratio 4:1. If 20 cups of flour are used, how many cups of sugar are needed?

1. 4
2. 5
3. 8
4. 16
5. 80

4. The ratio of the length to the width of a rectangle is 7:3. If the width is 21 inches, what is the length in inches?

1. 9
2. 35
3. 42
4. 49
5. 63

5. What percent of 80 is 20?

1. 4%
2. 20%
3. 25%
4. 40%
5. 60%

6. A price increases from $50 to $60. What is the percent increase?

1. 10%
2. 15%
3. 20%
4. 25%
5. 30%

7. If 30% of a number is 18, what is the number?

1. 5.4
2. 24
3. 48
4. 54
5. 60

Section B - Multi-Step Ratio and Percent Problems - Questions 8 to 14

8. The ratio of red marbles to blue marbles to green marbles in a bag is 2:3:5. If there are 60 marbles in total, how many are blue?

1. 12
2. 15
3. 18
4. 24
5. 30

9. A store reduces the price of a jacket by 20% and then reduces the new price by an additional 10%. If the original price was $100, what is the final price?

1. $68
2. $70
3. $72
4. $75
5. $80

10. If \(\frac{a}{b} = \frac{3}{4}\) and \(\frac{b}{c} = \frac{2}{5}\), what is the ratio \(a:c\)?

1. 3:10
2. 3:5
3. 5:8
4. 6:20
5. 8:15

11. A population of bacteria increases from 400 to 700. What is the percent increase?

1. 42.9%
2. 57.1%
3. 75%
4. 175%
5. 300%

12. Two numbers are in the ratio 5:8. If their sum is 91, what is the smaller number?

1. 28
2. 35
3. 40
4. 56
5. 63

13. A mixture contains water and juice in the ratio 3:2. If there are 15 liters of the mixture, how many liters of water must be added to make the ratio 2:1?

1. 1.5
2. 2
3. 3
4. 4
5. 5

14. After a 25% discount, a television costs $600. What was the original price?

1. $450
2. $750
3. $800
4. $825
5. $900

Section C - Advanced Application - Questions 15 to 20

15. The ratio of Alice's age to Bob's age is currently 3:4. In 6 years, the ratio of their ages will be 4:5. What is Alice's current age?

1. 12
2. 15
3. 18
4. 24
5. 30

16. A merchant marks up the cost of an item by 40% and then offers a 20% discount on the marked price. If the item cost the merchant $50, what is the selling price after the discount?

1. $52
2. $54
3. $56
4. $60
5. $64

17. The value of a car depreciates by 15% each year. If the car is worth $17,000 after one year, what was its original value?

1. $19,550
2. $20,000
3. $20,500
4. $22,000
5. $23,800

18. In a school, the ratio of students to teachers is 15:1. If there are 480 people in total (students and teachers), how many teachers are there?

1. 28
2. 30
3. 32
4. 35
5. 40

19. A solution contains alcohol and water in the ratio 2:7. How many liters of alcohol must be added to 45 liters of this solution to make the ratio 1:2?

1. 5
2. 7.5
3. 10
4. 12
5. 15

20. The price of stock A increased by 20% and the price of stock B decreased by 10%. If both stocks now cost $60 per share, what was the difference between their original prices?

1. $10
2. $16.67
3. $18
4. $20
5. $25

Answer Key

Quick Reference

1-D 2-B 3-B 4-D 5-C 6-C 7-E 8-C 9-C 10-A

11-C 12-B 13-C 14-C 15-C 16-C 17-B 18-B 19-B 20-B

Detailed Explanations

Question 1 - Correct Answer: D

The ratio of boys to girls is 3:5.
There are 15 boys.
Set up the proportion: \(\frac{3}{5} = \frac{15}{g}\) where \(g\) is the number of girls.
Cross multiply: \(3g = 5 \times 15\)
\(3g = 75\)
\(g = 25\)

Choice A results from dividing 15 by 5 and multiplying by 3, reversing the ratio.

Question 2 - Correct Answer: B

\(\frac{x}{12} = \frac{5}{8}\)
Cross multiply: \(8x = 12 \times 5\)
\(8x = 60\)
\(x = \frac{60}{8} = 7.5\)

Choice C results from computing \(12 \times 0.8\) instead of cross multiplying correctly.

Question 3 - Correct Answer: B

The ratio of flour to sugar is 4:1.
20 cups of flour are used.
Set up the proportion: \(\frac{4}{1} = \frac{20}{s}\) where \(s\) is cups of sugar.
Cross multiply: \(4s = 20\)
\(s = 5\)

Choice A results from reversing the ratio and computing \(20 \div 5\).

Question 4 - Correct Answer: D

The ratio of length to width is 7:3.
Width is 21 inches.
Set up the proportion: \(\frac{7}{3} = \frac{l}{21}\) where \(l\) is the length.
Cross multiply: \(3l = 7 \times 21\)
\(3l = 147\)
\(l = 49\)

Choice A results from computing \(21 \div 7 \times 3\), reversing the ratio.

Question 5 - Correct Answer: C

Find what percent of 80 is 20.
\(\frac{20}{80} \times 100\%\)
\(\frac{1}{4} \times 100\% = 25\%\)

Choice D results from computing \(\frac{80}{20} \div 10\), reversing the fraction.

Question 6 - Correct Answer: C

Original price: $50
New price: $60
Increase: \(60 - 50 = 10\)
Percent increase: \(\frac{10}{50} \times 100\% = 20\%\)

Choice A results from computing the increase as a percent of the new price instead of the original price.

Question 7 - Correct Answer: E

30% of a number is 18.
Let the number be \(n\).
\(0.30n = 18\)
\(n = \frac{18}{0.30} = 60\)

Choice D results from computing \(18 \times 3\) instead of dividing by 0.30.

Question 8 - Correct Answer: C

The ratio is 2:3:5 for red:blue:green.
Total parts: \(2 + 3 + 5 = 10\)
Total marbles: 60
Each part represents: \(\frac{60}{10} = 6\) marbles
Blue marbles: \(3 \times 6 = 18\)

Choice C results from the correct calculation. Choice A results from computing \(2 \times 6\), using the wrong part of the ratio.

Question 9 - Correct Answer: C

Original price: $100
First reduction of 20%: \(100 \times 0.80 = 80\)
Second reduction of 10% on $80: \(80 \times 0.90 = 72\)
Final price: $72

Choice B results from subtracting 30% from the original price in one step instead of applying successive discounts.

Question 10 - Correct Answer: A

\(\frac{a}{b} = \frac{3}{4}\) means \(a = \frac{3b}{4}\)
\(\frac{b}{c} = \frac{2}{5}\) means \(c = \frac{5b}{2}\)
\(\frac{a}{c} = \frac{\frac{3b}{4}}{\frac{5b}{2}} = \frac{3b}{4} \times \frac{2}{5b} = \frac{6}{20} = \frac{3}{10}\)
The ratio \(a:c = 3:10\)

Choice D is the unreduced form of the correct answer but the question asks for the ratio, and choice A presents it in simplest form.

Question 11 - Correct Answer: C

Original population: 400
New population: 700
Increase: \(700 - 400 = 300\)
Percent increase: \(\frac{300}{400} \times 100\% = 75\%\)

Choice A results from computing \(\frac{300}{700} \times 100\%\), using the new value as the base.

Question 12 - Correct Answer: B

The ratio is 5:8.
Total parts: \(5 + 8 = 13\)
Sum of numbers: 91
Each part: \(\frac{91}{13} = 7\)
Smaller number: \(5 \times 7 = 35\)

Choice D results from computing \(8 \times 7\), finding the larger number instead.

Question 13 - Correct Answer: C

Current ratio water:juice is 3:2.
Total mixture: 15 liters
Water: \(\frac{3}{5} \times 15 = 9\) liters
Juice: \(\frac{2}{5} \times 15 = 6\) liters
Desired ratio water:juice is 2:1, so water should be twice the juice.
Juice remains 6 liters, so water needs to be 12 liters.
Water to add: \(12 - 9 = 3\) liters

Choice A results from computing the difference between the ratios \(\frac{3}{2} - \frac{2}{1}\) without proper consideration of the quantities.

Question 14 - Correct Answer: C

After 25% discount, price is $600.
The discounted price represents 75% of the original price.
Let original price be \(p\).
\(0.75p = 600\)
\(p = \frac{600}{0.75} = 800\)

Choice B results from computing \(600 \times 1.25\), adding 25% to $600 instead of recognizing that $600 is 75% of the original.

Question 15 - Correct Answer: C

Current ratio Alice:Bob is 3:4.
Let Alice's current age be \(3x\) and Bob's current age be \(4x\).
In 6 years, Alice will be \(3x + 6\) and Bob will be \(4x + 6\).
The ratio will be 4:5.
\(\frac{3x + 6}{4x + 6} = \frac{4}{5}\)
Cross multiply: \(5(3x + 6) = 4(4x + 6)\)
\(15x + 30 = 16x + 24\)
\(30 - 24 = 16x - 15x\)
\(6 = x\)
Alice's current age: \(3x = 3 \times 6 = 18\)

Choice C is correct. Choice D results from computing Bob's current age \(4 \times 6 = 24\) instead of Alice's.

Question 16 - Correct Answer: C

Cost to merchant: $50
Markup by 40%: \(50 \times 1.40 = 70\)
Marked price: $70
Discount of 20%: \(70 \times 0.80 = 56\)
Selling price: $56

Choice D results from computing a net 20% increase on the cost \(50 \times 1.20 = 60\), incorrectly combining the percentages.

Question 17 - Correct Answer: B

The car depreciates by 15% each year.
After one year, value is 85% of original.
Current value: $17,000
Let original value be \(v\).
\(0.85v = 17000\)
\(v = \frac{17000}{0.85} = 20000\)

Choice A results from computing \(17000 \times 1.15\), incorrectly adding 15% to the current value.

Question 18 - Correct Answer: B

Ratio students:teachers is 15:1.
Total parts: \(15 + 1 = 16\)
Total people: 480
Each part: \(\frac{480}{16} = 30\)
Teachers: \(1 \times 30 = 30\)

Choice E results from computing \(\frac{480}{12}\), using an incorrect total number of parts.

Question 19 - Correct Answer: B

Current ratio alcohol:water is 2:7.
Total solution: 45 liters
Alcohol: \(\frac{2}{9} \times 45 = 10\) liters
Water: \(\frac{7}{9} \times 45 = 35\) liters
Desired ratio alcohol:water is 1:2.
Water remains 35 liters.
For ratio 1:2, alcohol should be: \(\frac{35}{2} = 17.5\) liters
Alcohol to add: \(17.5 - 10 = 7.5\) liters

Choice A results from computing the difference in the ratio parts without proper scaling to the actual quantities.

Question 20 - Correct Answer: B

Stock A increased by 20% to $60.
Let original price of A be \(a\).
\(1.20a = 60\)
\(a = \frac{60}{1.20} = 50\)
Stock B decreased by 10% to $60.
Let original price of B be \(b\).
\(0.90b = 60\)
\(b = \frac{60}{0.90} = 66.67\) (or \(\frac{200}{3}\))
Difference: \(66.67 - 50 = 16.67\)

Choice A results from computing \(60 - 50\), comparing the current prices to one original price incorrectly.