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Math Practice

Work Rate Practice Problems

Twelve hand-built work-rate problems with full worked solutions. Each uses the fact that a worker finishing a job in a hours completes 1/a of it per hour, and combined rates add up to one whole job.

  1. 1.

    Ana paints a fence in 5 hours; Ben paints it in 20 hours. How long together?

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    $$ \frac{t}{5} + \frac{t}{20} = 1 \quad\Longrightarrow\quad 4t + t = 20 \quad\Longrightarrow\quad 5t = 20 \quad\Longrightarrow\quad t = 4 \text{ hours} $$

  2. 2.

    One hose fills a pool in 6 hours; another in 12 hours. Both open. How long to fill?

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    $$ \frac{t}{6} + \frac{t}{12} = 1 \quad\Longrightarrow\quad 2t + t = 12 \quad\Longrightarrow\quad t = 4 \text{ hours} $$

  3. 3.

    Three workers finish alone in 4, 6, and 12 hours. How long together?

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    $$ \frac{t}{4} + \frac{t}{6} + \frac{t}{12} = 1 $$

    Multiply by 12: \(3t + 2t + t = 12\), so \(6t = 12\), \(t = 2\) hours.

  4. 4.

    Pipe A fills in 4 hours, pipe B in 6 hours, a drain empties in 12 hours. All open. How long to fill?

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    $$ \frac{t}{4} + \frac{t}{6} - \frac{t}{12} = 1 $$

    Multiply by 12: \(3t + 2t - t = 12\), so \(4t = 12\), \(t = 3\) hours.

  5. 5.

    Together, two crews clear a lot in 3 hours. Crew A alone takes 4 hours. How long does Crew B take alone?

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    $$ \frac{3}{4} + \frac{3}{b} = 1 \quad\Longrightarrow\quad \frac{3}{b} = \frac{1}{4} \quad\Longrightarrow\quad b = 12 \text{ hours} $$

  6. 6.

    Rex mows a field in 9 hours. With a helper it takes 6 hours. How long would the helper take alone?

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    $$ \frac{6}{9} + \frac{6}{h} = 1 \quad\Longrightarrow\quad \frac{6}{h} = \frac{1}{3} \quad\Longrightarrow\quad h = 18 \text{ hours} $$

  7. 7.

    A tub fills in 10 minutes with the drain closed; an open drain empties a full tub in 15 minutes. With the drain left open, how long to fill?

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    $$ \frac{t}{10} - \frac{t}{15} = 1 $$

    Multiply by 30: \(3t - 2t = 30\), so \(t = 30\) minutes.

  8. 8.

    A pool fills in 8 hours through an inlet and drains in 24 hours through an open valve. Starting empty with both open, how long to fill?

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    $$ \frac{t}{8} - \frac{t}{24} = 1 $$

    Multiply by 24: \(3t - t = 24\), so \(t = 12\) hours.

  9. 9.

    Machine A does a job in 8 hours, machine B in 12 hours. They start together, but A stops 2 hours before the job ends. How long is the whole job?

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    Let \(t\) be total time; A runs \(t - 2\).

    $$ \frac{t - 2}{8} + \frac{t}{12} = 1 $$

    Multiply by 24: \(3(t - 2) + 2t = 24\), so \(5t - 6 = 24\), \(t = 6\) hours.

  10. 10.

    Two hoses fill a tank in 3 and 6 hours alone. They run together 1 hour, then the faster one shuts off. How much longer does the slower hose need?

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    In 1 hour together: \(\tfrac{1}{3} + \tfrac{1}{6} = \tfrac{1}{2}\) done. The slower hose finishes the other half: \(\tfrac{s}{6} = \tfrac{1}{2}\), so \(s = 3\) hours.

  11. 11.

    Working together, Sara and Tom build a shed in 4 hours. Alone, Tom takes 6 hours longer than Sara. How long does each take alone?

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    Let Sara's solo time be \(x\); Tom's is \(x + 6\).

    $$ \frac{4}{x} + \frac{4}{x + 6} = 1 $$

    Multiply by \(x(x + 6)\): \(4(x + 6) + 4x = x(x + 6)\), so \(8x + 24 = x^{2} + 6x\), giving \(x^{2} - 2x - 24 = 0\), \((x - 6)(x + 4) = 0\). Reject \(x = -4\). Sara takes \(6\) hours, Tom takes \(12\).

  12. 12.

    A large printer prints a run in 20 minutes, a small one in 30 minutes. Both run 6 minutes, then the large one jams. How much longer for the small printer?

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    In 6 minutes together: \(\tfrac{6}{20} + \tfrac{6}{30} = \tfrac{3}{10} + \tfrac{1}{5} = \tfrac{1}{2}\) done. The small printer finishes: \(\tfrac{s}{30} = \tfrac{1}{2}\), so \(s = 15\) minutes.
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Work Rate Practice Problems

The Work Problems lesson explains why rates add and works through every case; this set is 12 problems to practice on, each with a full solution.

How to use this set

These problems are hand-written, not auto-generated. For each problem, write each worker’s rate as , multiply by the time worked, set the sum of the parts to , and solve. Then check your answer against the solution.

Common mistakes to watch for

Adding the times instead of the rates. “5 hours and 20 hours, so 25 together” is the classic wrong answer.

Forgetting to invert. The combined rate is a sum; the combined time is its reciprocal.

Adding a drain’s term instead of subtracting it.

Accepting a together-time larger than the fastest solo time.

Where to go next

The same chart and reciprocal idea drive the Distance, Rate, and Time Practice Problems. Clearing the denominators is the Linear Equations with Fractions Practice Problems skill. For every case, see the Work Problems lesson.

Frequently Asked Questions

What is the main idea behind a work-rate problem?+

Rates of work add. A worker who does a job in hours has rate ; working with a second worker of rate , the combined rate is , and the time together is the reciprocal of that.

Why is the whole job the number 1?+

One finished job is 100% of the work, which is . Each worker's contribution is the fraction of that job they complete, and the fractions sum to when the job is done.

What is the two-worker formula?+

If they take and hours alone, together they take where , which gives .

How do I handle a worker who leaves early?+

Give that worker a shorter time expression, like instead of . Each term is still (that worker's time) over (that worker's solo time).

How does a drain fit in?+

A drain does negative work, so its term is subtracted: . This only has a positive solution when the fill is faster than the drain.

What is a quick sanity check?+

For a 'working together' problem, the combined time must be shorter than the faster worker's solo time. A larger answer means a reciprocal or sign error.

More practice problems

Distance, Rate, and Time Practice Problems Algebra Word Problems Practice Problems Mixture Problems Practice Problems