Question

A, B and C can do a work together in a certain number of days. If A leaves after half the work is done, then the work takes 4 more days for completion, but if B leaves after half the work is done, the work takes 5 more days for completion. If A takes 10 more days than B to do the work alone, then in how many days can C alone do the work?

• A. 52.5
• B. 65
• C. 80
• D. 75

Answer 1

The correct option is A

52.5

Let A alone can do the work in a days, B alone in b days and C alone in c days.
Let A, B and C together completes the work in x days.

So,$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{x}...........\left(1\right)$

Then half of the work they will complete together in $\frac{x}{2}$ days.
When A leaves, Rest half of the work is done by B and C together,

$(\frac{2}{b}+\frac{2}{c})=\frac{1}{\left\{\right(\frac{x}{2})+4\}}$

Using (1), $2\times (\frac{1}{x}-\frac{1}{a})=\frac{1}{\left\{\right(\frac{x}{2})+4\}}$

Solving we get, a $=\frac{[x\times \{\left(\frac{x}{2}\right)+4\left\}\right]}{4}................\left(2\right)$

On similar lines b $=\frac{[x\times \{\left(\frac{x}{2}\right)+5\left\}\right]}{5}................\left(3\right)$

Given that a=b+10 ..........................(4)

Thus we have $\frac{[x\times \{\left(\frac{x}{2}\right)+4\left\}\right]}{4}=\frac{[x\times \{\left(\frac{x}{2}\right)+5\left\}\right]}{5}+10$

Solving we get x = 20.
From (2) and (3) we get
a = 70 and b = 60
Put these values in (1) we get c = 52.5 days.
Hence, choice (a) is the right option