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Two water taps together can fill a tank in $3 \frac{1}{13}$ hours. The tap of longer diameter takes $3$ hours less then smaller one to fill the bank separately. Find the time(hrs) in which tap of smaller diameter can separately fill the tank.

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5

3

8

None of these

answer is C.

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Detailed Solution

It is given that two water taps together can fill a tank in

3 \frac{1}{13}

hours and the tap of longer diameter takes 3 hours more.
Let time taken by tap of longer diameter be

x

hours.
Then time taken by tap of smaller diameter will be

x + 3

hours
Now, according to question,

\frac{1}{x} + \frac{1}{x + 3} = \frac{13}{40}

\Rightarrow \frac{2 x + 3}{x (x + 3)} = \frac{13}{40}

\Rightarrow 80 x + 120 = 13 x^{2} + 39 x

\Rightarrow 13 x^{2} - 41 x - 120 = 0

\Rightarrow 13 x^{2} + 24 x - 65 x - 120 = 0

\Rightarrow x (13 x + 24) - 5 (13 x + 24) = 0

\Rightarrow (13 x + 24) (x - 5) = 0

\Rightarrow x = \frac{- 24}{13}

x = 5

Since time cannot be negative, therefore

x = 5

.
Hence, time taken by tap of larger diameter is

5

hours and time taken by tap of smaller diameter is

8

hours.
Hence, the correct option is 3.

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