3. Someone leaves: count the work still left
Today’s goal: split the question into stages when a worker leaves. Separate work completed from work left.
Together A and B do 5 units/day. Total work is 30 units. After 2 days, how much is left?
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Correct answer: C
Done = 5 × 2 = 10. Left = 30 − 10 = 20.
Option A: 10 is completed work.
Option B: 25 subtracts only one day’s work.
Option D: Subtract work-units, not days.
Hint: Find completed work first.
Follow the work
A takes 6 days alone; B takes 12. They work together for 2 days, then B leaves. Use 12 total units. A does 2/day; B does 1/day.
| Stage | Who works? | Work |
|---|---|---|
| First 2 days | A + B, 3 units/day | 6 units completed |
| B has left | A alone, 2 units/day | 6 units remain |
Solve in two stages
- Together for 2 days: (2 + 1) × 2 = 6 units completed.
- Work left: 12 − 6 = 6 units.
- A alone: 6 ÷ 2 = 3 MORE days.
- TOTAL time: 2 days already spent + 3 more = 5 days.
Finish the last step
B leaves. 6 units remain; A alone does 2 units/day. How many MORE days are needed?
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Correct answer: A
Extra time = 6 ÷ 2 = 3 days.
Option B: Earlier days are added only when total time is asked.
Option C: 2 is the daily rate, not the remaining time.
Option D: Divide remaining work by the daily rate.
Hint: Use the rate of the worker still present.
Make your own two-stage table
A takes 8 days alone, B takes 24. They work together for 3 days, then B leaves. How many MORE days does A need?
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Correct answer: D
Total 24; rates 3 and 1. Done in 3 days = 12; left = 12. A needs 12 ÷ 3 = 4 more days.
Option A: 3 days were already spent together.
Option B: 7 is total time: 3 + 4.
Option C: Find remaining work and divide by A’s rate.
Hint: Separate the two-worker and one-worker stages.
A takes 10 days alone, B takes 15. They work together for 3 days, then B leaves. What is the TOTAL time from the start?
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Correct answer: B
Total 30; rates 3 and 2. Together complete 15 units in 3 days. A needs 15 ÷ 3 = 5 more days. Total = 3 + 5 = 8.
Option A: 5 is extra time only.
Option C: 6 assumes both work to the end.
Option D: 10 ignores B’s help during the first 3 days.
Hint: Underline TOTAL.
Catch three mistakes: using both rates after one worker leaves; dividing completed work instead of work left; giving extra days when total time was asked.
Before moving on, explain whose rate you use after B leaves, and why.
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