Question : Pipes A, B and C together can fill a cistern in 12 hours. All three pipes are opened together for 4 hours and then C is closed. A and B together take 10 hours to fill the remaining part of the cistern. C alone will fill two-thirds of the cistern in:
Option 1: 60 hours
Option 2: 40 hours
Option 3: 48 hours
Option 4: 50 hours
Correct Answer: 40 hours
Solution : Pipes A, B, and C together can fill the cistern in 12 hours. So, their combined rate is $\frac{1}{12}$ unit per hour. All three pipes are opened together for 4 hours, so they fill 4 × $\frac{1}{12}$ = $\frac{1}{3}$ of the cistern. This leaves 1 – $\frac{1}{3}$ = $\frac{2}{3}$ of the cistern to be filled. Pipes A and B together take 10 hours to fill this remaining part. So, their combined rate is $\frac{2}{3 × 10}$ = $\frac{1}{15}$ unit per hour. Since A, B, and C together have a rate of $\frac{1}{12}$ unit per hour, and A and B together have a rate of $\frac{1}{15}$ unit per hour, the rate of pipe C alone is $\frac{1}{12}-\frac{1}{15}$ = $\frac{1}{60}$ unit per hour. Therefore, pipe C alone will fill two-thirds of the cistern in $\frac{2}{3}$ × 60 = 40 hours. Hence, the correct answer is 40 hours.
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