Advanced Word Problems • Topic 2 of 5

Work Problems

Work problems are about rates: if a worker finishes a job in x hours, the rate is 1/x of the job per hour. To combine workers, add their rates: together they do 1/x + 1/y per hour, so the whole job takes (x·y)/(x + y) hours. For staffing questions, the number of worker-days (or worker-hours) is constant: M workers × D days = the same total, so N workers take (M·D)/N days — more workers, fewer days (inverse proportion). Identify whether rates add (working together) or scale inversely (more workers), set up the constant, and solve.

Combining work rates: 1/x + 1/y per hourCombining work ratesA: 1 jobin x h+B: 1 jobin y h=rate 1/x + 1/yTogether they finish in (x × y)/(x + y) hours

✅ Solved examples

1. A does a job in 6 h, B in 12 h. Together?
(6·12)/(6 + 12) = 72/18 = 4 hours.
2. 4 workers build a wall in 6 days. How long for 8 workers?
Worker-days = 24; 24/8 = 3 days.
3. A in 10 h, B in 15 h. Together?
(10·15)/25 = 150/25 = 6 hours.
4. 5 machines fill an order in 8 hours. How long for 10 machines?
5·8 = 40 machine-hours; 40/10 = 4 hours.

✏️ Practice — try these, take hints as needed

1. A does a job in 4 h, B in 12 h. Together?
(xy)/(x + y).
(4·12)/16.
48/16.
3 hours.
2. 6 workers finish in 10 days. How long for 12 workers?
Worker-days constant = 60.
60/12.
—
5 days.
3. A in 20 h, B in 30 h. Together?
(20·30)/50.
600/50.
—
12 hours.
4. 3 painters take 8 hours. How long for 4 painters?
24 painter-hours.
24/4.
—
6 hours.
5. A in 8 h, B in 8 h. Together?
(8·8)/16.
—
—
4 hours.

📝 Topic test — 8 questions

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