Question : Three pipes, A, B, and C, can fill a cistern in 12, 18 and 24 minutes, respectively. If all the pipes are opened together for 7 minutes, what will be the volume of the water that overflows as the percentage of the total volume of the cistern?
Option 1: $23 \frac{2}{3}\%$
Option 2: $26 \frac{5}{18}\%$
Option 3: $23 \frac{1}{3}\%$
Option 4: $26 \frac{7}{18}\%$
Correct Answer: $26 \frac{7}{18}\%$
Solution :
Let the volume of the cistern be 72 units.
Pipe A can fill the cistern in 12 minutes.
The rate of filling = $\frac{72}{12}$ = 6 units per minute
Similarly, pipe B can fill the cistern in 18 minutes.
The rate of filling = $\frac{72}{18}$ = 4 units per minute
Finally, pipe C can fill the cistern in 24 minutes.
The rate of filling = $\frac{72}{24}$ = 3 units per minute
When all the pipes are opened together, the total rate of filling = 6 + 4 + 3 = 13 units per minute
Therefore, in 7 minutes the total volume of water filled in the cistern = 13 × 7 = 91 units.
The volume of the water that overflows is the difference between the volume of water filled in the cistern and the volume of the cistern itself = 91 – 72 = 19 units
Therefore, the percentage of the total volume of water that overflows is,
$=\frac{19}{72} \times 100 = 26 \frac{7}{18}\%$
Hence, the correct answer is $26 \frac{7}{18}\%$.
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