2. Working together: add daily work, not days
Today’s goal: find the time when two people work together. Add daily work first, then find time.
On the same job, A does 3 units/day and B does 2 units/day. How much together per day?
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Correct answer: D
Daily contributions add: 3 + 2 = 5 units/day.
Option A: Both contribute work; neither undoes it.
Option B: Add daily rates; do not multiply them.
Option C: The average rate is not the combined rate.
Hint: Count both contributions in the same day.
Watch one example
A takes 6 days alone; B takes 3. Choose the whole job as 6 units.
| Worker | Time alone | Daily work |
|---|---|---|
| A | 6 days | 1 unit |
| B | 3 days | 2 units |
| Together | To find | 3 units |
From daily contributions to total time
- Use the same total for both: 6 units.
- A: 6 ÷ 6 = 1 unit/day; B: 6 ÷ 3 = 2 units/day.
- Together: 1 + 2 = 3 units/day.
- Time = total work ÷ daily work = 6 ÷ 3 = 2 days.
Why 6? It divides exactly by both times. Choose 12 or 18 and the final time still comes to 2 days. The smallest such positive whole number is the LCM, or least common multiple.
Finish with help
A takes 12 days alone; B takes 6. Choose 12 total units: A does 1/day, B does 2/day. Time together?
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Correct answer: B
Together: 3 units/day. Time = 12 ÷ 3 = 4 days.
Option A: Do not add completion times.
Option C: Do not average completion times.
Option D: 3 is daily work, not days needed.
Hint: Total work ÷ combined daily work.
Try new numbers
A finishes a job in 10 days; B in 15 days. How long together?
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Correct answer: C
Use 30 units. A = 3/day, B = 2/day. Together = 5/day; time = 30 ÷ 5 = 6 days.
Option A: 25 adds the individual times.
Option B: Together should be faster than either alone.
Option D: 5 is units per day, not time.
Hint: Choose a total divisible by 10 and 15: 30.
A takes 8 days alone and B takes 12. What is their time together?
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Correct answer: A
Use 24 units: rates 3/day and 2/day. Time = 24 ÷ 5 = 4.8 days.
Option B: Adding times does not give the combined time.
Option C: Averaging times is not the method.
Option D: 5 is the combined daily rate for 24 units.
Hint: A fraction of a working day is allowed.
Check: with two positive rates and no interference, together takes less time than either alone. 4.8 days means 4 + 4/5 working days. To convert the fraction to hours, you must know the hours in one working day.
Recall: same total → daily work → add → divide.
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