Tap A can fill a tank in 8 minutes while tap B empties it in 9 minutes. How long will the tank be full if the two taps are opened at the same time?
1/8 - 1/9 = 1/72
That's how much of the tank gets filled in one minute with both taps open, right?
But the question is poorly worded...
1/24
To solve this problem, we need to find the combined rate at which the two taps fill or empty the tank.
Let's assign a positive rate for filling and a negative rate for emptying.
The filling rate of tap A is 1 tank per 8 minutes, or 1/8 tanks per minute (positive rate).
The emptying rate of tap B is 1 tank per 9 minutes, or -1/9 tanks per minute (negative rate).
To find the combined rate, we add the rates of tap A and tap B:
Combined rate = Rate of tap A + Rate of tap B
Combined rate = 1/8 - 1/9
We need to find the reciprocal of the combined rate to get the time it takes to fill the tank when both taps are open.
Time = 1 / (Combined rate)
Time = 1 / (1/8 - 1/9)
To simplify this expression, we need to find a common denominator for 8 and 9. The common denominator is 72.
Time = 1 / ((9 - 8)/72)
Time = 1 / (1/72)
Time = 72 minutes
Therefore, if both taps A and B are opened at the same time, it will take 72 minutes to fill the tank.
To find out how long it will take the tank to be full when both taps are opened at the same time, we need to consider their individual rates of filling and emptying.
Let's assign a variable to represent the time it takes for the tank to be full, which we'll call T. We can also calculate the rates of filling and emptying for each tap.
Tap A fills the tank in 8 minutes, which means it can fill 1/8th of the tank in 1 minute. Therefore, its filling rate is 1/8th of the tank per minute.
Tap B empties the tank in 9 minutes, which is equivalent to emptying 1/9th of the tank per minute. Hence, its emptying rate is 1/9th of the tank per minute.
When both taps are opened at the same time, their filling and emptying rates add up. So, the combined rate of filling the tank is the sum of the individual rates, which is (1/8 + -1/9) of tank per minute.
To calculate the combined rate, we need to find a common denominator between 8 and 9, which is 72. Converting the rates to have a denominator of 72, we get (9/72 - 8/72) of tank per minute, which simplifies to 1/72 of the tank per minute.
Therefore, when both taps are opened at the same time, the tank will fill at a rate of 1/72 of the tank per minute. To find how long it will take for the tank to be full, we need to calculate T, the time in minutes, using the formula:
1/72 * T = 1 (since the whole tank needs to be filled)
To solve for T, we multiply both sides of the equation by 72:
(1/72) * 72 * T = 1 * 72
T = 72
Hence, it will take 72 minutes for the tank to be full when both taps A and B are opened at the same time.