Example problem: A water pump can fill a swimming pool in 8 hours, while a drain can empty the pool in 12 hours. If the pool takes 24 hours to fill when both the pump and the drain are open, how long would it take for the drain to empty the pool if the pump is turned off?
Solution: Step 1: Assign variables: Let’s assign variables to represent the unknown quantities. Let t be the time it takes for the drain to empty the pool if the pump is turned off.
Step 2: Set up the equation: We know that the pump can fill the pool in 8 hours, so its filling rate is 1 pool per 8 hours, or 1/8 pool per hour. Similarly, the drain empties the pool in 12 hours, so its draining rate is 1 pool per 12 hours, or 1/12 pool per hour.
When both the pump and the drain are open, they work together to fill the pool in 24 hours. So, their combined filling rate is 1 pool per 24 hours, or 1/24 pool per hour.
Using these rates, we can set up the equation:
(1/8 – 1/12) = 1/24
Step 3: Solve the equation: To solve the equation, we need to find a common denominator for the fractions. In this case, the least common denominator is 24. We can rewrite the equation as:
(3/24 – 2/24) = 1/24
Now, we can combine the fractions:
1/24 = 1/24
Step 4: Interpret the solution: Since the equation simplifies to 1/24 = 1/24, it means that the drain will empty the pool in 24 hours if the pump is turned off.
Therefore, the solution to the problem is that it will take 24 hours for the drain to empty the pool if the pump is turned off.
Remember to always check your solution to ensure it makes sense in the context of the problem. In this case, it does because the drain is slower than the pump, so it takes longer for it to empty the pool.