Question

A tank is filled by three pipes with uniform flow. The first two pipes operating simultaneously fill the tank in the same time during which the tank is filled by the third pipe alone. The second pipe fills the tank 5 hours faster than the first pipe and 4 hours slower than the third pipe. The time required by the first pipe is.
$
  {\text{A}}{\text{. 6 hours}} \\
  {\text{B}}{\text{. 10 hours}} \\
  {\text{C}}{\text{. 15 hours}} \\
  {\text{D}}{\text{. 30 hours}} \\
$

Answer 1

Hint – To find the time required by the first pipe we consider it to be a variable x, we then write the time taken by second and third pipes in terms of time taken by the first pipe and solve that relation using the given data for x.

Complete step by step answer:
Let the time taken by the first pipe be x.

Given: The second pipe fills the tank 5 hours faster than the first pipe.
             The second pipe fills the tank 4 hours slower than the third pipe.

Then, second and third pipes will take (x-5) and (x-9) hours respectively to fill the tank.

Given: The first two pipes operating simultaneously fill the tank in the same time during which the tank is filled by the third pipe alone.

Let’s say the first pipe takes k hours to fill the tank, i.e. in one hour the amount of tank filled is$\dfrac{1}{{\text{k}}}$. Similarly we get $\dfrac{1}{{{\text{x - 5}}}}{\text{ and }}\dfrac{1}{{{\text{x - 9}}}}$ part of tank filled in one hour. Hence the relation is
$
  \dfrac{1}{{\text{x}}} + \dfrac{1}{{{\text{x - 5}}}} = \dfrac{1}{{{\text{x - 9}}}} \\
   \Rightarrow \dfrac{{{\text{x + }}\left( {{\text{x - 5}}} \right)}}{{{\text{x}}\left( {{\text{x - 5}}} \right)}} = \dfrac{1}{{{\text{x - 9}}}} \\
   \Rightarrow \left( {{\text{2x - 5}}} \right)\left( {{\text{x - 9}}} \right) = {\text{x}}\left( {{\text{x - 5}}} \right) \\
   \Rightarrow 2{{\text{x}}^2} - 5{\text{x - 18x + 45 = }}{{\text{x}}^2} - {\text{5x}} \\
   \Rightarrow {{\text{x}}^2} - 18{\text{x + 45 = 0}} \\
   \Rightarrow \left( {{\text{x - 15}}} \right)\left( {{\text{x - 3}}} \right) = 0 \\
   \Rightarrow {\text{x = 15}}{\text{.}} \\
$
We neglect x=3 because it does not hold true for all the given data in the question.
Hence the time required by the first pipe is 15 hours.
Hence Option C is the correct answer.

Note – In order to solve this type of questions the key is to write the sentences given the question in the form of equations and solve accordingly. Writing all the quantities in terms of one quantity simplifies the equation. While there are more than one values for the variable x after solving, we compare them with the data given in the question to verify.