Work-rate problems drills

Combine individual rates to solve work/completion problems.

Free, timed SHSAT practice sets on work-rate problems— every question includes a full worked explanation, and there's no account required.

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How to solve work rate problems

Think in jobs per hour. If a person finishes a job in 4 hours, they do \(\tfrac{1}{4}\) of the job each hour.

  1. Write each worker's rate as \(\tfrac{1}{\text{time}}\).
  2. Add the rates to get the combined rate.
  3. Flip the combined rate to get the time together.

Example: one person takes 4 hours and another takes 12 hours. Combined rate \(= \tfrac{1}{4} + \tfrac{1}{12} = \tfrac{3}{12} + \tfrac{1}{12} = \tfrac{1}{3}\) job per hour, so together they take 3 hours.

Shortcut for two workers: time together \(= \dfrac{a \times b}{a + b}\). For 6 and 8 hours: \(\tfrac{48}{14} = 3\tfrac{3}{7}\) hours.

Traps:

Worked examples

Example 1.
Mason can clean an office building alone in 6 hours. Ava can do the same job alone in 4 hours. Working together at their individual rates, how many hours will it take them to finish the job?
Show answer and explanation
Answer:
B

Combined rate \(= \frac{1}{6} + \frac{1}{4}\). Time together \(= \dfrac{6 \times 4}{6 + 4} = 2\frac{2}{5}\) hours. (Adding the two times (\(6 + 4\)) gives 10 hours. Averaging the two times (\((6 + 4) \div 2\)) gives 5 hours. Stopping at the combined rate (the part of the job done in 1 hour) instead of taking its reciprocal gives \(\frac{5}{12}\) hours.)

Example 2.
Pipe A can fill a water tank in 2 hours. Pipe B can fill the same water tank in 6 hours. If both pipes are opened at the same time, how long will it take to fill the water tank?
Show answer and explanation
Answer:
D

Combined rate \(= \frac{1}{2} + \frac{1}{6}\). Time together \(= \dfrac{2 \times 6}{2 + 6} = 1\frac{1}{2}\) hours. (Adding the two times (\(2 + 6\)) gives 8 hours. Averaging the two times (\((2 + 6) \div 2\)) gives 4 hours. Stopping at the combined rate (the part of the job done in 1 hour) instead of taking its reciprocal gives \(\frac{2}{3}\) hours.)