Time & Work • Topic 2 of 4

Combined Work

To find how long several people take together, add their rates. With two workers the shortcut is ab/(a+b) days. The LCM method scales beautifully to three or more: total = LCM(days); add units/day; divide. If a third person's time is unknown but the trio's combined time is given, subtract the known rates from the combined rate.

✅ Solved examples

1. A in 12, B in 6 days. Together?
ab/(a+b) = 72/18 = 4 days.
2. A in 10, B in 15, C in 30 days. Together?
LCM 30: rates 3+2+1 = 6 units/day; 30/6 = 5 days.
3. A and B together in 8 days, A alone in 12. B alone?
Rate B = 1/8 - 1/12 = 1/24 -> 24 days.
4. A in 20, B in 30; they work together 5 days then A leaves. Days for B to finish?
LCM 60: rates 3+2=5; in 5 days 25 units done; 35 left at B rate 2/day = 17.5 days.

✏️ Practice — try these, take hints as needed

1. A in 15, B in 10. Together?
ab/(a+b).
150/25.
—
6 days
2. A in 6, B in 12, C in 12. Together (LCM 12)?
Rates 2+1+1 = 4.
12/4.
—
3 days
3. A+B in 6, B in 9. A alone?
1/6 - 1/9.
1/18.
—
18 days
4. A in 24, B in 8. Together?
ab/(a+b).
192/32.
—
6 days
5. A,B,C together in 4; A in 12, B in 18. C alone?
1/4 - 1/12 - 1/18.
LCM 36: 9-3-2=4 -> rate 4/36.
36/4.
9 days

📝 Topic test — 8 questions

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