ASVAB Study Guide
Word Problems · Lesson 2 of 8

Ratios, Rates, and Proportions

Key rules and formulas, worked examples and 12 practice problems with step-by-step solutions.

1.Ratios

A comparison of two quantities by division.

$$a:b$$ or a/b

Shows the relative size of one value to another.

Example

A classroom has 12 boys and 18 girls. What is the simplified ratio of boys to girls?

Show solution

Step 1: Write the ratio as 12:18

Step 2: Find the greatest common divisor, which is 6

Step 3: Divide both by 6 to get $$2:3$$

Practice problems

  1. Problem 1

    A basket contains 15 red apples and 25 green apples. What is the ratio of red apples to total apples in simplest form?

    Show solution

    Answer: $$3:8$$

    Step 1: Total apples = $$15 + 25 = 40$$

    Step 2: Write the ratio as 15:40

    Step 3: Divide both sides by 5 to get $$3:8$$

  2. Problem 2

    A recipe requires 4 cups of flour and 6 tablespoons of sugar. What is the simplified ratio of flour to sugar?

    Show solution

    Answer: $$2:3$$

    Step 1: Write the ratio as $$4:6$$

    Step 2: Divide both numbers by their common factor 2

    Step 3: Simplify to get $$2:3$$

  3. Problem 3

    In a parking lot, there are 20 SUVs and 30 sedans. What is the ratio of sedans to SUVs?

    Show solution

    Answer: $$3:2$$

    Step 1: Write the ratio of sedans to SUVs as 30:20

    Step 2: Divide both numbers by 10

    Step 3: Simplify to get $$3:2$$

Remember

Ratios must be simplified to their lowest terms. Keep the original order of the quantities unchanged.

2.Unit Rates

A rate with a denominator of 1.

Rate = Amount / Unit

Used to find the price or speed for a single unit.

Example

A car travels 120 miles in 2 hours. What is the unit rate in miles per hour?

Show solution

Step 1: Write the rate as 120 miles / 2 hours

Step 2: Divide 120 by 2

Step 3: Unit rate is 60 mph

Practice problems

  1. Problem 4

    A store sells a pack of 4 soda cans for $6.00. What is the unit price per can?

    Show solution

    Answer: $1.50 per can

    Step 1: Write the rate as $$\frac{6.00}{4}$$ cans

    Step 2: Divide 6.00 by 4

    Step 3: Unit price is $1.50 per can

  2. Problem 5

    An industrial printer prints 350 pages in 5 minutes. What is the printing rate per minute?

    Show solution

    Answer: 70 pages per minute

    Step 1: Write the rate as 350 pages / 5 minutes

    Step 2: Divide 350 by 5

    Step 3: Unit rate is 70 pages per minute

  3. Problem 6

    A runner covers 12 kilometers in 1.5 hours. What is the runner's average speed in kilometers per hour?

    Show solution

    Answer: $$8\text{ km/h}$$

    Step 1: Write the rate as 12 km / 1.5 hours

    Step 2: Divide 12 by 1.5

    Step 3: Unit rate is 8 km/h

Remember

Always divide the first quantity by the second quantity to find the rate per a single unit.

3.Proportions

An equation stating that two ratios are equal.

$$\frac{a}{b} = \frac{c}{d}$$

If one ratio is known, you can find a missing value in the second by cross-multiplying $$(a \times d = b \times c)$$

Example

If 3 notebooks cost $5, how much will 9 notebooks cost?

Show solution

Step 1: Set up the proportion: $$\frac{3}{5} = \frac{9}{x}$$

Step 2: Cross-multiply: $$3 \times x = 5 \times 9$$-> $$3x = 45$$

Step 3: Divide by 3: $$x = 15$$

Practice problems

  1. Problem 7

    A map has a scale where 2 inches represents 50 miles. How many miles do 5 inches represent?

    Show solution

    Answer: 125 miles

    Step 1: Set up the proportion: $$\frac{2}{50} = \frac{5}{x}$$

    Step 2: Cross-multiply: $$2 \times x = 50 \times 5$$-> $$2x = 250$$

    Step 3: Divide by 2: $$x = 125$$ miles

  2. Problem 8

    If a factory machine produces 40 parts in 80 minutes, how many parts can it produce in 120 minutes?

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    Answer: 60 parts

    Step 1: Set up the proportion: $$\frac{40}{80} = \frac{x}{120}$$

    Step 2: Cross-multiply: $$80 \times x = 40 \times 120$$-> $$80x = 4800$$

    Step 3: Divide by 80: $$x = 60$$ parts

  3. Problem 9

    A baker uses 6 cups of flour to make 24 cookies. How many cups of flour are needed to make 36 cookies?

    Show solution

    Answer: 9 cups

    Step 1: Set up the proportion: $$\frac{6}{24} = \frac{x}{36}$$

    Step 2: Cross-multiply: $$24 \times x = 6 \times 36$$-> $$24x = 216$$

    Step 3: Divide by 24: $$x = 9$$ cups

Remember

Use cross-multiplication to solve for the missing variable. Cross-multiply and then divide.

4.Direct Variation

A relationship where one variable increases as the other increases.

$$y=k \times x$$

k is the constant of proportionality $$(k = \frac{y}{x})$$

Example

If y varies directly as x, and $$y = 10$$ when $$x = 2$$, find y when $$x = 5$$

Show solution

Step 1: Find $$k = \frac{y}{x} = \frac{10}{2} = 5$$

Step 2: Write the equation: $$y = 5 \times x$$

Step 3: Substitute $$x = 5$$: $$y = 5 \times 5 = 25$$

Practice problems

  1. Problem 10

    The total cost of tickets varies directly with the number of tickets bought. If 3 tickets cost $45, what is the cost of 7 tickets?

    Show solution

    Answer: $105

    Step 1: Find $$k = \frac{45}{3} = 15$$

    Step 2: Write the variation equation: $$y = 15 \times x$$

    Step 3: Substitute $$x = 7$$: $$y = 15 \times 7 = 105$$

  2. Problem 11

    The distance a spring stretches varies directly with the weight attached. If a 4-pound weight stretches it 12 inches, how far will a 6-pound weight stretch it?

    Show solution

    Answer: 18 inches

    Step 1: Find $$k = \frac{12}{4} = 3$$

    Step 2: Write the equation: $$y = 3 \times x$$

    Step 3: Substitute $$x = 6$$: $$y = 3 \times 6 = 18$$ inches

  3. Problem 12

    If y varies directly with x, and $$y = 18$$ when $$x = 6$$, find x when $$y = 36$$

    Show solution

    Answer: $$x = 12$$

    Step 1: Find $$k = \frac{18}{6} = 3$$

    Step 2: Write the equation: $$y = 3 \times x$$

    Step 3: Substitute $$y = 36$$: $$36 = 3 \times x$$-> $$x = 12$$

Remember

Find the constant k first by dividing y by x, then use $$y = k \times x$$ to find the unknown value.