Olympiad topic · P3–P6
Work problems
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.
Try one
★★ MediumAn estate needs a long corridor painted.
Mr Chan alone would finish it in 10 days;
Mr Lee alone would take 15 days.
Each works at a steady pace every day.
If they work on it together from the first day, how many days will it take?
- Hint 1: Pick a number that both 10 and 15 divide into, and split the whole job into that many equal parts.
- Hint 2: With the job as 30 parts, how many parts does each man do a day?
- 1.Call the whole job 30 parts (the lowest common multiple of 10 and 15).
- 2.Mr Chan does the 30 parts in 10 days: 30 ÷ 10 = 3 a day. Mr Lee takes 15 days: 30 ÷ 15 = 2 a day.
- 3.Together they do 3 + 2 = 5 parts a day.
- 4.How many lots of 5 make 30? 30 ÷ 5 = 6 days.
- ⚠️Common mistake: answering 10 + 15 = 25 days. Two working together must be quicker than either alone.
Answer: 6 days
Four levels, step by step
- ★EasyFree2 question types · unlimited questionsRates simply added.
- ★★MediumFree3 question types · unlimited questionsTwo together, a pipe filling while one empties.
- ★★★HardFree3 question types · unlimited questionsSomeone leaves part way, or turns taken.
- ★★★★Expert (competition level)VIP3 question types · unlimited questionsGrass that keeps growing, a queue that keeps joining, three teams helping each other.
Work problems: what it is and how to solve it
Several people (or machines, pipes) sharing a job: how long it takes.
For example: Mr Chan alone takes 42 days, Mr Lee 56. Together? Call the job 168 parts (a common multiple of 42 and 56): Chan does 4 a day, Lee 3, so 168 ÷ (4 + 3) = 24 days.
The key idea: Rate × time = work done. Calling the whole job a common multiple of everyone's times keeps every number whole; calling it 1, with each doing a fraction a day, gives the same answer.
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