One pump fills a tank two times as fast as another pump. If the pumps work together they fill the tank in 18 minutes. How long does it take each pump working alone to fill the tank?
If the fast pump fills the tank in x minutes, we have
1/x + 1/2x = 1/18
x = 27
Let's assume that the slower pump takes x minutes to fill the tank on its own. Since the faster pump fills the tank two times as fast, it will take x/2 minutes to fill the tank on its own.
To find the combined rate of the two pumps, we need to consider that rates add up when working together. The rate of the slower pump is 1/x (1 tank per x minutes) and the rate of the faster pump is 2/x (2 tanks per x minutes). Working together, their combined rate is 1/18 (1 tank per 18 minutes).
Therefore, we can write the equation:
1/x + 2/x = 1/18
To solve for x, we need to get rid of the denominators:
(1 + 2)/x = 1/18
Simplifying further:
3/x = 1/18
Cross-multiplying:
18 * 3 = x
x = 54
So the slower pump takes 54 minutes to fill the tank on its own, and the faster pump takes half of that time:
54/2 = 27
Therefore, the slower pump takes 54 minutes and the faster pump takes 27 minutes to fill the tank on their own.
To solve this problem, we can set up an equation based on the rates of the two pumps.
Let's assume that the slower pump takes x minutes to fill the tank alone. Since the faster pump fills the tank two times faster, it takes x/2 minutes to fill the tank alone.
If both pumps work together for 18 minutes, the combined amount of work they do is equal to filling the entire tank. Therefore, the equation can be established as follows:
1/x + 1/(x/2) = 1/18
To solve the equation, we'll first simplify it by finding the least common denominator (LCD), which is 2x:
(2 + 1)/(2x) = 1/18
3/(2x) = 1/18
Next, we'll cross multiply:
3 * 18 = 2x
54 = 2x
Finally, divide both sides of the equation by 2:
x = 54/2
x = 27
Therefore, the slower pump takes 27 minutes to fill the tank alone, and the faster pump takes half the time, which is 27/2 = 13.5 minutes.