We will learn how to solve problems in which two people cooperate, including cases where “one person starts first and the other joins later” or “one person leaves partway through.”
The “Joining Partway Through” Pattern
Example Problem
A person working alone can finish a certain job in 12 days, and B working alone can finish it in 8 days. A works alone for the first 4 days, and then A and B work together. How many days in total does it take to finish the job?
Solution: Set the total amount as 1.
① A’s one-day work = 1/12, B’s one-day work = 1/8
② Amount A completes in 4 days: 1/12 × 4 = 4/12 = 1/3
③ Remaining amount of work: 1 − 1/3 = 2/3
④ Combined one-day work: 1/12 + 1/8 = 2/24 + 3/24 = 5/24
⑤ Number of days for the remainder: (2/3) ÷ (5/24) = (2/3) × (24/5) = 16/5 = 3.2 days
⑥ Total: 4 + 3.2 = 7.2 days (7 days and 1/5 of a day)
A person working alone can finish a certain job in 12 days, and B working alone can finish it in 8 days. A works alone for the first 4 days, and then A and B work together. How many days in total does it take to finish the job?
Solution: Set the total amount as 1.
① A’s one-day work = 1/12, B’s one-day work = 1/8
② Amount A completes in 4 days: 1/12 × 4 = 4/12 = 1/3
③ Remaining amount of work: 1 − 1/3 = 2/3
④ Combined one-day work: 1/12 + 1/8 = 2/24 + 3/24 = 5/24
⑤ Number of days for the remainder: (2/3) ÷ (5/24) = (2/3) × (24/5) = 16/5 = 3.2 days
⑥ Total: 4 + 3.2 = 7.2 days (7 days and 1/5 of a day)
Using the Least Common Multiple as the Total Makes It Easier!
If you set 24, the least common multiple of 12 and 8, as the “total amount,” the calculation is easier because no fractions appear.
- A’s one-day work = 24 ÷ 12 = 2, B’s one-day work = 24 ÷ 8 = 3
- A completed 2 × 4 = 8 units in 4 days.
- Remaining amount = 24 − 8 = 16
- Together, they complete 2 + 3 = 5 units per day.
- Days for the remainder = 16 ÷ 5 = 3.2 days
- Total = 4 + 3.2 = 7.2 days
Key Points
- When someone joins partway through, first subtract the amount completed by one person, then calculate the remainder.
- Setting the total amount as the least common multiple removes fractions and makes the calculation much easier.
- The answer is not always an integer. It is fine if the number of days is a decimal.
Practice Problems
- A certain job takes A 12 days when working alone. A and B start at the same time. A works for 6 days, and B takes a 2-day break partway through, so the job takes 10 days in total. How many days would B take working alone?
- A certain job takes P 15 days and Q 10 days when working alone. They worked together at first, but P quit partway through, and Q finished the rest alone. The job was completed in a total of 9 days from the start. How many days did P work?
- It takes pump A alone 6 hours and pump B alone 4 hours to fill a tank completely. Both pumps are operated at the same time, and A is stopped after 2 hours. How many more hours will B alone need to fill the tank?
Answers and Explanations
- Answer: B would take 20 days working alone
Explanation: Consider the condition that A takes 12 days working alone. Set the total amount as 12. A’s one-day work = 1. The amount A completes in 6 days = 6. The remainder = 12 − 6 = 6. B completes the remainder in 4 days (10 − 6 days of A and B working together, excluding the 2 days B was resting) … Because the wording of the problem is complicated, here is another solution: Set the total amount as 12, with A completing 1 unit per day. The amount A completes in 6 days = 6/12 = 1/2. The remainder = 1/2, completed by B during 8 actual working days: 10 − 2 = 8. B’s one-day work = (1/2) ÷ 8 = 1/16. Therefore, B would take 16 days working alone. - Answer: P worked for 1.5 days
Explanation: Set the total amount of work as 30. P completes 2 units per day, and Q completes 3 units per day. Q worked for 9 days and P worked for x days, so 2x + 3 × 9 = 30. 2x = 3, so x = 1.5 days. Check: 2 × 1.5 + 3 × 9 = 30. - Answer: Another 2/3 of an hour (40 minutes)
Explanation: Set the total amount as 12 (the LCM of 6 and 4). A’s one-hour work = 2, and B’s one-hour work = 3. The amount completed by both pumps in 2 hours = (2 + 3) × 2 = 10. Remaining amount = 12 − 10 = 2. Time for B alone to complete the remainder = 2 ÷ 3 = 2/3 hour (40 minutes). Check: both pumps for 2 hours + B alone for 2/3 hour = 10 + 2 = 12✓