ASVAB Study Guide
Word Problems · Lesson 4 of 8

Combined Rates and Work

Key rules and formulas, worked examples and 3 practice problems with step-by-step solutions.

4.Percent Change

Calculating the total time required for multiple participants to complete a single task together.

$$\frac{1}{T} = 1/R_1 + 1/R_2$$

When two people or machines work together, add their individual rates (work per unit of time) to find the combined rate.

Example

John can paint a house in 6 hours, and Dave can paint it in 3 hours. How many hours will it take them to paint the house together?

Show solution

Step 1: Find individual rates: John = $$\frac{1}{6}$$, Dave $$= \frac{1}{3}$$

Step 2: Add rates: $$\frac{1}{6} + \frac{1}{3} = \frac{1}{6} + \frac{2}{6} = \frac{3}{6} = \frac{1}{2}$$

Step 3: Invert the combined rate to find total time: $$T = 2$$ hours

Practice problems

  1. Problem 1

    Pipe A fills a tank in 4 hours, and Pipe B fills the same tank in 12 hours. If both pipes are turned on at the same time, how many hours will it take to fill the tank?

    Show solution

    Answer: 3 hours

    Step 1: Find individual rates: Pipe A = $$\frac{1}{4}$$, Pipe B $$= \frac{1}{12}$$

    Step 2: Add rates: $$\frac{1}{4} + \frac{1}{12} = \frac{3}{12} + \frac{1}{12} = \frac{4}{12} = \frac{1}{3}$$

    Step 3: Invert the combined rate to find total time: $$T = 3$$ hours

  2. Problem 2

    Machine X prints a batch of flyers in 10 minutes, and Machine Y prints the same batch in 15 minutes. How many minutes does it take them working together?

    Show solution

    Answer: 6 minutes

    Step 1: Find individual rates: Machine X = $$\frac{1}{10}$$, Machine Y $$= \frac{1}{15}$$

    Step 2: Add rates: $$\frac{1}{10} + \frac{1}{15} = \frac{3}{30} + \frac{2}{30} = \frac{5}{30} = \frac{1}{6}$$

    Step 3: Invert the combined rate to find total time: $$T = 6$$ minutes

  3. Problem 3

    Alice can clean an office building in 8 hours. Working together with Ben, they can clean it in 4 hours. How many hours would it take Ben to clean it alone?

    Show solution

    Answer: 8 hours

    Step 1: Write the equation: $$\frac{1}{8} + \frac{1}{\text{Ben}} = \frac{1}{4}$$

    Step 2: Subtract Alice's rate: $$\frac{1}{\text{Ben}} = \frac{1}{4} - \frac{1}{8} = \frac{2}{8} - \frac{1}{8} = \frac{1}{8}$$

    Step 3: Invert Ben's rate to find his total time: $$T = 8$$ hours

Remember

Always find the individual rate of work per single hour or minute before combining them.