Rates, Work & Speed • Topic 4 of 4

Combined Work

The golden rule of work problems: add rates, not times. If Ann paints a room in 4 hours, her rate is 1/4 of the room per hour; if Ben takes 6 hours, his rate is 1/6. Together they paint 1/4 + 1/6 = 5/12 of the room each hour, so the whole room takes 12/5 = 2.4 hours. For exactly two workers the shortcut is T = ab/(a + b) — the "product over sum" of the individual times. This idea covers pipes filling a tank (add fill rates), a drain working against a tap (subtract the drain’s rate), and staggered starts (one works alone for a while, then the other joins). The common wrong move is to average the times, which is meaningless — a fast helper should make the job take less time than either person alone, and only the rate method guarantees that. Keep the rates as clean fractions; the numbers are chosen so they combine neatly without the calculator.

✅ Solved examples

1. Pipe A fills a tank in 4 hours, pipe B in 6 hours. How long together?
Combined rate = 1/4 + 1/6 = 5/12 tank/hour. Time = 12/5 = 2.4 hours (2 h 24 min).
2. A can do a job in 10 days, B in 15 days. Working together, how many days?
T = ab/(a+b) = (10×15)/(10+15) = 150/25 = 6 days.
3. A tank is filled by a tap in 3 hours but drained by a leak in 6 hours. With both open, how long to fill?
Net rate = 1/3 − 1/6 = 1/6 tank/hour, so it fills in 6 hours.
4. Quantitative Comparison. A finishes a task in 8 hours, B in 8 hours. Quantity A: time for A and B working together. Quantity B: 4 hours. Which is greater?
Combined rate = 1/8 + 1/8 = 1/4, so together they take exactly 4 hours. The quantities are equal.

✏️ Practice — try these, take hints as needed

1. X paints a fence in 12 hours, Y in 6 hours. How long together?
Add rates: 1/12 + 1/6.
= 1/12 + 2/12 = 3/12 = 1/4.
Time = reciprocal.
4 hours
2. Two pipes fill a pool in 20 and 30 minutes respectively. How long together?
T = ab/(a+b).
(20×30)/(20+30).
600/50.
12 minutes
3. A does a job in 9 days, B in 18 days. Together, how many days?
1/9 + 1/18.
= 2/18 + 1/18 = 3/18 = 1/6.
Reciprocal.
6 days
4. A tap fills a tank in 5 hours; a drain empties it in 10 hours. Both open — fill time?
Net rate = 1/5 − 1/10.
= 2/10 − 1/10 = 1/10.
Reciprocal.
10 hours
5. A and B together finish in 4 hours; A alone takes 6 hours. How long for B alone?
1/A + 1/B = 1/4.
1/6 + 1/B = 1/4.
1/B = 1/4 − 1/6 = 1/12.
12 hours

📝 Topic test — 8 questions

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