Fix the mistake: Work Rate Problems

Work-Rate Problems ① Thinking in Terms of One Day’s Work

Work-rate problems are solved by expressing as a fraction how much work can be done in one day (or one hour) when the total amount of work is taken as 1.

How to Think About Work-Rate Problems

Set the total amount of work as “1.” Then,

  • One day’s work for a person who finishes in ○ days = 1/○
  • Number of days required = 1 ÷ (one day’s work)

can be written.

Example Problem
A person working alone can finish a certain job in 12 days, and B working alone can finish it in 18 days. How many days will it take if they work together?

Solution:
① Set the total amount of work as 1.
② A’s one-day work: 1/12
③ B’s one-day work: 1/18
④ Their combined one-day work: 1/12 + 1/18 = 3/36 + 2/36 = 5/36
⑤ Number of days required: 1 ÷ (5/36) = 36/5 = 7.2 days (7 days and 1/5 of a day)

A Useful Technique: Set the Total Amount as a Number

When fraction calculations are troublesome, it is easier to use a specific number for the total amount of work, such as “the least common multiple of 12 and 18 = 36.”

  • Set the total amount as 36.
  • A’s one-day work = 36 ÷ 12 = 3
  • B’s one-day work = 36 ÷ 18 = 2
  • Together, they can do 5 units of work in one day.
  • Number of days required = 36 ÷ 5 = 7.2 days
Key Points
  • In work-rate problems, it is customary to set the total amount as “1” or as the least common multiple.
  • Remember clearly: “one day’s work = 1 ÷ ○ days.”
  • When adding fractions, do not forget to find a common denominator.

Practice Problems

  1. A person working alone can finish a certain job in 10 days, and Y working alone can finish it in 15 days. How many days will it take if they work together?
  2. It takes pump A alone 8 hours and pump B alone 12 hours to fill a pond completely. How many hours will it take if both pumps are operated at the same time?
  3. Company P alone can complete a certain construction project in 20 days, and company Q alone can complete it in 30 days. How many days will it take if both companies work together?

Answers and Explanations

  1. Answer: 6 days
    Explanation: Set the total amount of work as 30 (the LCM of 10 and 15). X’s one-day work = 30 ÷ 10 = 3. Y’s one-day work = 30 ÷ 15 = 2. Their combined one-day work = 3 + 2 = 5. Number of days required = 30 ÷ 5 = 6 days. Check: 5 × 6 = 30✓
  2. Answer: 4 hours 48 minutes (24/5 hours)
    Explanation: Set the total amount as 24 (the LCM of 8 and 12). A’s one-hour work = 24 ÷ 8 = 3. B’s one-hour work = 24 ÷ 12 = 2. Their combined one-hour work = 3 + 2 = 5. Time required = 24 ÷ 5 = 4.8 hours = 4 hours 48 minutes.
  3. Answer: 12 days
    Explanation: Set the total amount as 60 (the LCM of 20 and 30). Company P’s one-day work = 60 ÷ 20 = 3. Company Q’s one-day work = 60 ÷ 30 = 2. Their combined one-day work = 3 + 2 = 5. Number of days required = 60 ÷ 5 = 12 days. Check: 5 × 12 = 60✓

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