Question

It takes $ 24\;hours $ to fill a swimming pool using two pipes. If the pipe of larger diameter is used for $ 8\;hours $ and the pipe of the smaller diameter is used for $ 18\;hours $ . Only half of the pool is filled. How long would each pipe take to fill the swimming pool.

Answer 1

Hint: We have been given the time taken to fill a swimming pool using two pipes. We have to find how much time each pipe would take to fill the swimming pool by forming two equations. These equations can be formed by the given conditions. After that we can find the time taken by each of them to fill the swimming pool.

Complete step by step solution:
We have been given that two pipes of different diameters when used together takes $ 24\;hours $ to fill a swimming pool. Since we do not know how much time each pipe would take to fill the swimming pool separately, we can assume that,
The larger pipe takes $ x\;hours $ and the smaller pipe takes $ y\;hours $ to fill the swimming pool.
Then large pipe can fill $ \dfrac{1}{x} $ of the pool in $ 1\;hour $ and the small pipe can fill $ \dfrac{1}{y} $ of the pool in $ 1\;hour $ .
And large pipe can fill $ \dfrac{{24}}{x} $ of the pool in $ 24\;hours $ and the small pipe can fill $ \dfrac{{24}}{y} $ of the pool in $ 24\;hours $ .
Since they both fill the pool in $ 24\;hours $ , therefore,
 $ \dfrac{{24}}{x} + \dfrac{{24}}{y} = 1\;\;\;...\left( 1 \right) $
Also, the large pipe can fill $ \dfrac{8}{x} $ of the pool in $ 8\;hours $ and the small pipe can fill $ \dfrac{{18}}{y} $ of the pool in $ 18\;hours $ .
Since they both fill half the pool, therefore,
 $ \dfrac{8}{x} + \dfrac{{18}}{y} = \dfrac{1}{2}\;\;\;...\left( 2 \right) $
We can solve these equations by multiplying equation $ \left( 2 \right) $ by $ 3 $ and subtracting it from equation $ \left( 1 \right) $ .
 $
   \Rightarrow \left( {\dfrac{{24}}{x} + \dfrac{{24}}{y}} \right) - 3 \times \left( {\dfrac{8}{x} + \dfrac{{18}}{y}} \right) = 1 - \left( {3 \times \dfrac{1}{2}} \right) \\
   \Rightarrow \;\left( {\dfrac{{24}}{x} + \dfrac{{24}}{y}} \right) - \left( {\dfrac{{24}}{x} + \dfrac{{54}}{y}} \right) = 1 - \dfrac{3}{2} \\
   \Rightarrow - \dfrac{{30}}{y} = - \dfrac{1}{2} \Rightarrow y = 60 \\
  $
And,
 $
  \dfrac{{24}}{x} + \dfrac{{24}}{y} = 1 \\
   \Rightarrow \dfrac{{24}}{x} + \dfrac{{24}}{{60}} = 1 \Rightarrow \dfrac{{24}}{x} = \dfrac{{36}}{{60}} \\
   \Rightarrow x = 24 \times \dfrac{{60}}{{36}} = 40 \;
  $
Hence, the larger pipe would take $ 40\;hours $ and the smaller pipe would take $ 60\;hours $ to fill the swimming pool.

Note: We used the given conditions to form two equations to find two unknowns which were the time taken by each pipe to fill the pool. The forming of the equation required a simple unitary method. We have to ensure in the result that the larger pipe would take less time than the smaller pipe.