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Work Time : Rates Practice

This aptitude practice question is a Work Time problem solving question. Concept tested: The crux of solving this question entails framing proper equations from the information given in the word problem.

Question : Part 1

Working together, A and B can do a job in 6 days. B and C can do the same job in 10 days, while C and A can do it in 7.5 days. How long will it take if all A, B, and C work together to complete the job?

  1. 8 days
  2. 5 days
  3. 3 days
  4. 7 days

  Correct Answer      Choice (2). If all 3 worked together, the will complete the job in 5 days.


Explanatory Answer

We know that if A and B work together, they can complete the job in 6 days. Therefore, if all three of them A, B and C work together the number of days it will take to complete the job will surely be less than 6 days. Hence, we can eliminate answer choices (1) and (4) right away.

Let A be the number of days that A will take to complete the job if A worked alone, B days for B to complete the job if B worked alone and C days for C to complete the job if C worked alone.

A and B can do a job in 6 days. They complete \\frac{1}{6}^{th}) of the job in a day.
i.e., \\frac{1}{A}) + \\frac{1}{B}) = \\frac{1}{6}) ------ (1)
Similarly, B and C will complete \\frac{1}{10}^{th}) of the job in a day.
i.e., \\frac{1}{B}) + \\frac{1}{C}) = \\frac{1}{10}) ------ (2)
And C and A will complete \\frac{1}{7.5}) or \\frac{2}{15}^{th}) of the job in a day
i.e., \\frac{1}{C}) + \\frac{1}{A}) = \\frac{2}{15}) ------ (3).

Adding (1), (2) and (3) we get \\frac{1}{A}) + \\frac{1}{B}) + \\frac{1}{B}) + \\frac{1}{C}) + \\frac{1}{C}) + \\frac{1}{A}) = \\frac{1}{6}) + \\frac{1}{10}) + \\frac{2}{15})
=> \\frac{2}{A}) + \\frac{2}{B}) = \\frac{2}{C}) = \\frac{5 + 3 + 4}{30}) = \\frac{12}{30})
Or \\frac{1}{A}) + \\frac{1}{B}) + \\frac{1}{C}) = \\frac{6}{30}) = \\frac{1}{5}).
i.e., working together, A, B and C complete 1/5th of the job in a day.
Therefore, they will complete the job in 5 days.

  The correct answer is Choice (2).

Question Part 2

Working together, A and B can do a job in 6 days. B and C can do the same job in 10 days, while C and A can do it in 7.5 days. How long will A alone take to complete the job?

  1. 8 days
  2. 6 days
  3. 10 days
  4. 20 days

  Correct Answer      Choice (3). A alone will take 10 days


Explanatory Answer

From part 1, we know \\frac{1}{A}) + \\frac{1}{B}) + \\frac{1}{C}) = \\frac{1}{5}) .... (4)
From equation (2) in part 1, we know \\frac{1}{B}) + \\frac{1}{C}) = \\frac{1}{10})
Subtract (2) from (4): \\frac{1}{A}) + \\frac{1}{B}) + \\frac{1}{C}) - {\\frac{1}{B}) + \\frac{1}{C})} = \\frac{1}{5}) - \\frac{1}{10})
Or \\frac{1}{A}) = \\frac{2-1}{10}) = \\frac{1}{10})
If A completes \\frac{1}{10}) of the work in a day, A will take 10 days to complete the task if A worked alone.

  The correct answer is Choice (3).

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