A sink is one quarter full when both the faucet and the drain are opened. The faucet alone can fill the sink in 5 minutes, whereas it takes 8 minutes to empty it with the drain. How long will it take to fill the remaining three quarters of the sink?
1/5 - 1/8 = (3/4)/x
Let's assume that the total capacity of the sink is represented by "x".
We know that when both the faucet and the drain are opened, the sink is one quarter full. Therefore, the amount of water emptied by the drain in 1 minute is (1/4)x.
We also know that the faucet can fill the sink in 5 minutes. Therefore, the amount of water filled by the faucet in 1 minute is (1/5)x.
Since the drain empties the sink in 8 minutes, the amount of water emptied by the drain in 1 minute is (1/8)x.
To find out how long it will take to fill the remaining three quarters of the sink, we need to calculate the difference between the amount of water filled by the faucet and the amount of water emptied by the drain in 1 minute.
This can be calculated as follows:
(1/5)x - (1/8)x = (8x - 5x)/40 = (3/40)x
Since we want to find the time needed to fill the remaining three quarters of the sink, which is three times the amount of water that is already in the sink, we multiply the amount calculated above by 3.
Therefore, it will take (3/40)x * 3 = (9/40)x minutes to fill the remaining three quarters of the sink.
To solve this problem, let's break it down step by step.
1. Determine the rate at which water is entering the sink when both the faucet and the drain are opened. Since both are open, the faucet fills the sink at a rate of 1/5 of the sink's capacity per minute, and the drain empties it at a rate of 1/8 of the sink's capacity per minute. Therefore, the net rate at which the sink is being filled is (1/5) - (1/8) = (8/40) - (5/40) = 3/40 of the sink's capacity per minute.
2. As given, the sink is one quarter full when both the faucet and the drain are opened. In other words, the sink is already filled by 1/4 of its capacity. Therefore, the remaining empty portion of the sink that needs to be filled is 1 - 1/4 = 3/4 of its capacity.
3. Now, we need to find out how long it will take to fill this remaining three quarters of the sink. Let's denote the time required as "t" minutes.
4. Since the net filling rate is 3/40 of the sink's capacity per minute, we can set up the equation (3/40) * t = 3/4 to represent the filling process.
5. Solving this equation, we can cross-multiply: 3 * t = (3/4) * 40. Simplifying, we have 3 * t = 30, which gives us t = 10.
Therefore, it will take 10 minutes to fill the remaining three quarters of the sink when both the faucet and the drain are opened.