Two pipes fill a tank in $10$ and $15$ minutes. The first is closed after $4$ minutes and the second…

Two pipes fill a tank in $10$ and $15$ minutes. The first is closed after $4$ minutes and the second continues alone. The total time to fill the tank is:
  1. $8$ min
  2. $9$ min
  3. $13$ min
  4. $15$ min

Solution

In $4$ min both fill $4\left(\dfrac{1}{10} + \dfrac{1}{15}\right) = 4 \cdot \dfrac{5}{30} = \dfrac{2}{3}$. Remaining $= \dfrac{1}{3}$. Second pipe (rate $\dfrac{1}{15}$) takes $\dfrac{1/3}{1/15} = 5$ min. Total $= 4 + 5 = 9$ min.

Asked in: IMO

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