A hard question on work problems for P5, with a step-by-step solution, diagrams and the answer. Common in the Hua Xia Cup's P3 to P6 papers; harder ones have someone leaving partway, people taking turns, or cows grazing a field that keeps growing.
The question
The school library is reorganising all its books.
Ms Wong alone would take 12 days;
Mr Leung alone would take 36 days.
They start together; after some days Ms Wong leaves to take pupils on a study tour, and Mr Leung finishes the rest by himself.
From start to finish it takes 24 days.
Each works at a steady pace every day.
For how many days did they work together?
Show the hints (3)
💡Hint 1: Mr Leung worked from the first day to the last. How many days is that?
💡Hint 2: Call the whole job 36 parts: Ms Wong does 3 a day and Mr Leung 1.
💡Hint 3: In 24 days Mr Leung did 1 × 24 = 24 parts; the other 12 were all Ms Wong's.
Worked solution
Have a go first, then read the solution step by step.
1.Call the job 36 parts (the lowest common multiple of 12 and 36).
2.Ms Wong does 36 ÷ 12 = 3 a day, Mr Leung 36 ÷ 36 = 1.
3.Mr Leung worked from the first day to the last, all 24 days: 1 × 24 = 24 parts.
4.The other 36 − 24 = 12 parts were Ms Wong's: 12 ÷ 3 = 4 days.
5.Ms Wong only worked while they were together, so they worked together for 4 days.
⚠️Common mistake: counting only Mr Leung's days alone, forgetting he was also working on the days they worked together.
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