# 2004 Indonesia MO Problems/Problem 2

## Problem

A trough, if filled with cold water tap, will be full in 14 minutes. To empty the full trough with opening the hole on the base of the trough, the water will be all out in 21 minutes. If the cold water tap and the hot water tap are opened simultaneously with the opening of the hole, the trough will be full in 12.6 minutes. Then, how long does it take to full the trough when only the hot water tap is opened and the hole is closed?

## Solution

Let the volume of the trough be $V$ liters. Also, let the rate of the cold water tap be $C$ liters per minute, the rate of the hot water tap be $H$ liters per minute, and the rate of the water leaving out of the hole be $E$ liters. With these variables and the rate formula, we can write three equations.

$$V = 21E$$ $$V = 14C$$ $$V = 12.6(H+C-E)$$

From the first two equations, $E = \tfrac{V}{21}$ and $C = \tfrac{V}{14}$. Substitute these into the third equation and solve for $V$.

$$V = \frac{63}{5} (H + \frac{V}{14} - \frac{V}{21})$$ $$V = \frac{63}{5}H + \frac{9}{10}V - \frac{3}{5}V$$ $$10V = 126H + 9V - 6V$$ $$7V = 126H$$ $$V = 18H$$

It takes $\boxed{18}$ minutes for the trough to fill up when the hot water tap is opened and the hole is closed.