Math worksheet with work word problems for calculating combined work rates.

Let's solve each of these work word problems step by step. These types of problems involve rates of work, and the key idea is:

> If someone can complete a job in $ x $ hours, their work rate is $ \frac{1}{x} $ jobs per hour.

When people work together, their rates add up.

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Problem 1)

Mary can paint a house in 12 hours. If Nancy helps, it takes them 2 hours. Without help, how long would it take Nancy to complete the same job?

- Mary’s rate: $ \frac{1}{12} $ houses/hour
- Let Nancy’s rate be $ \frac{1}{n} $ (we want to find $ n $)
- Together: $ \frac{1}{12} + \frac{1}{n} = \frac{1}{2} $ (since they finish in 2 hours)

Solve:
$$
\frac{1}{12} + \frac{1}{n} = \frac{1}{2}
$$
$$
\frac{1}{n} = \frac{1}{2} - \frac{1}{12} = \frac{6 - 1}{12} = \frac{5}{12}
$$
$$
n = \frac{12}{5} = 2.4 \text{ hours} = 2 \text{ hours } 24 \text{ minutes}
$$

✔ Answer: $ \boxed{2.4} $ hours

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Problem 2)

Sara takes 6 hours to spread mulch. Joan takes 10 hours to finish this job. How long would it take if they worked together?

- Sara’s rate: $ \frac{1}{6} $
- Joan’s rate: $ \frac{1}{10} $
- Combined rate: $ \frac{1}{6} + \frac{1}{10} = \frac{5 + 3}{30} = \frac{8}{30} = \frac{4}{15} $

Time together: $ \frac{1}{\frac{4}{15}} = \frac{15}{4} = 3.75 $ hours

$ 0.75 \times 60 = 45 $ minutes → $ 3 $ hours $ 45 $ minutes

✔ Answer: $ \boxed{3.75} $ hours

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Problem 3)

Keith takes 8 hours, Dan takes 5 hours. How long together?

- Keith: $ \frac{1}{8} $
- Dan: $ \frac{1}{5} $
- Together: $ \frac{1}{8} + \frac{1}{5} = \frac{5 + 8}{40} = \frac{13}{40} $
- Time: $ \frac{1}{\frac{13}{40}} = \frac{40}{13} \approx 3.077 $ hours

Convert: $ 0.077 \times 60 \approx 4.6 $ minutes → ~3 hours 4.6 min

✔ Answer: $ \boxed{\frac{40}{13}} $ or $ \boxed{3.077} $ hours

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Problem 4)

Tim takes 9 hours, Dan takes 8 hours. Together?

- Tim: $ \frac{1}{9} $
- Dan: $ \frac{1}{8} $
- Together: $ \frac{1}{9} + \frac{1}{8} = \frac{8 + 9}{72} = \frac{17}{72} $
- Time: $ \frac{72}{17} \approx 4.235 $ hours

✔ Answer: $ \boxed{\frac{72}{17}} $ or $ \boxed{4.235} $ hours

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Problem 5)

Sara cuts shrubs in 12 hours. With Tom, it takes 10 hours. How long for Tom alone?

- Sara: $ \frac{1}{12} $
- Let Tom’s time = $ t $ → rate $ \frac{1}{t} $
- Together: $ \frac{1}{12} + \frac{1}{t} = \frac{1}{10} $

Solve:
$$
\frac{1}{t} = \frac{1}{10} - \frac{1}{12} = \frac{6 - 5}{60} = \frac{1}{60}
\Rightarrow t = 60
$$

✔ Answer: $ \boxed{60} $ hours

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Problem 6)

Sally and Benny together: 7 hours. Benny alone: 11 hours. How long for Sally?

- Benny: $ \frac{1}{11} $
- Let Sally’s time = $ s $ → rate $ \frac{1}{s} $
- Together: $ \frac{1}{s} + \frac{1}{11} = \frac{1}{7} $

Solve:
$$
\frac{1}{s} = \frac{1}{7} - \frac{1}{11} = \frac{11 - 7}{77} = \frac{4}{77}
\Rightarrow s = \frac{77}{4} = 19.25 \text{ hours}
$$

✔ Answer: $ \boxed{19.25} $ hours

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Problem 7)

Nancy and Mike mop store in 7 hours. Mike alone: 8 hours. How long for Nancy?

- Mike: $ \frac{1}{8} $
- Let Nancy’s time = $ n $
- Together: $ \frac{1}{n} + \frac{1}{8} = \frac{1}{7} $

Solve:
$$
\frac{1}{n} = \frac{1}{7} - \frac{1}{8} = \frac{8 - 7}{56} = \frac{1}{56}
\Rightarrow n = 56
$$

✔ Answer: $ \boxed{56} $ hours

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Problem 8)

Nancy: 11 hours, Tom: 8 hours. Together?

- Nancy: $ \frac{1}{11} $
- Tom: $ \frac{1}{8} $
- Together: $ \frac{1}{11} + \frac{1}{8} = \frac{8 + 11}{88} = \frac{19}{88} $
- Time: $ \frac{88}{19} \approx 4.632 $ hours

✔ Answer: $ \boxed{\frac{88}{19}} $ or $ \boxed{4.632} $ hours

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Problem 9)

Mike: 6 hours, Melanie: 3 hours. Together?

- Mike: $ \frac{1}{6} $
- Melanie: $ \frac{1}{3} $
- Together: $ \frac{1}{6} + \frac{1}{3} = \frac{1 + 2}{6} = \frac{3}{6} = \frac{1}{2} $
- Time: $ \frac{1}{\frac{1}{2}} = 2 $ hours

✔ Answer: $ \boxed{2} $ hours

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Problem 10)

Joan: 4 hours, Fred: 6 hours. Together?

- Joan: $ \frac{1}{4} $
- Fred: $ \frac{1}{6} $
- Together: $ \frac{1}{4} + \frac{1}{6} = \frac{3 + 2}{12} = \frac{5}{12} $
- Time: $ \frac{12}{5} = 2.4 $ hours

$ 0.4 \times 60 = 24 $ minutes → 2 hours 24 minutes

✔ Answer: $ \boxed{2.4} $ hours

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✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | 2.4 hours |
| 2 | 3.75 hours |
| 3 | $ \frac{40}{13} \approx 3.077 $ hours |
| 4 | $ \frac{72}{17} \approx 4.235 $ hours |
| 5 | 60 hours |
| 6 | 19.25 hours |
| 7 | 56 hours |
| 8 | $ \frac{88}{19} \approx 4.632 $ hours |
| 9 | 2 hours |
| 10 | 2.4 hours |

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