The question is from Time and work: Three friends, combines with the idea of Algebra. The combination of the slowest two and the quickest two and the respective time to complete the task is given and we need to find out the time required by all three. x's and y's introduced into any topic can complicate anything, including pipes and cisterns. CAT Questions from Time and Work.

Question 6: Consider three friends A, B and C who work at differing speeds. When the slowest two work together they take n days to finish a task. When the quickest two work together they take m days to finish a task. One of them, if he worked alone would take thrice as much time as it would take when all three work together. How much time would it take if all three worked together?

- \\frac{3mn}{2(m + n)}\\)
- \\frac{2mn}{m + n}\\)
- \\frac{4mn}{3(m + n)}\\)
- \\frac{5mn}{3(m + n)}\\)

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Let A < B < C in terms of efficiency.

B and C together take n days.

A and B together take m days.

One of them, if he worked alone would take thrice as much time as it would take when all three work together. This is a crucial statement. Now, if there are three people who are all equally efficient, for each of them it would take thrice as much time as for all three together.

Now, this tells us that the person who takes thrice as much time cannot be the quickest one. If the quickest one is only one-third as efficient as the entire team, the other two cannot add up to two-thirds. By a similar logic, the slowest one cannot be the person who is one-third as efficient.

In other words, the person one-third as efficient = B

Let A, B and C together take x days. B alone would take 3x days

B and C together take n days. Or B + C in 1 day do

\\frac{1}{n}\\) of the task ........Eqn (i)

A and B together take m days. Or, A + B in 1 day do

\\frac{1}{m}\\) of the task ........Eqn (ii)

B takes 3x days to do the task. Or, B, in one day, does

\\frac{1}{3x}\\) of the task ........Eqn (iii)

Now, if we do (i) + (i) – (iii) we get

A + B + C do \\frac{1}{n}\\) + \\frac{1}{m}\\) - \\frac{1}{3x}\\) in a day. This should be equal to \\frac{1}{x}\\) as all three of them complete the task in x days.

\\frac{1}{n}\\) + \\frac{1}{m}\\) - \\frac{1}{3x}\\) = \\frac{1}{x}\\)

\\frac{1}{n}\\) + \\frac{1}{m}\\) = \\frac{4}{3x}\\)

\\frac{m + n}{mn}\\) = \\frac{4}{3x}\\)

\\frac{4mn}{3(m + n)}\\)

The question is **"How much time would it take if all three worked together?"**

Choice C is the correct answer.

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