Two pumps can fill a water tank in 256 minutes when working together. Alone, the second pump takes times as long as the first to fill the tank.
How many minutes would it take the first pump to fill the tank?
? times as long?
To solve this problem, let's assign variables to the time it takes each pump to fill the tank. Let's call the time it takes for the first pump to fill the tank "x" minutes. Since the second pump takes times as long as the first pump, the time it takes for the second pump to fill the tank is "x times."
When the two pumps work together, they can fill the tank in 256 minutes. This means that in one minute, the two pumps can complete 1/256th of the tank's capacity.
Let's set up an equation based on this information:
(1/x) + (1/(x times)) = 1/256
Now, let's simplify this equation. We can do this by finding a common denominator and combining the fractions:
[(x times) + x] / (x(x times)) = 1/256
[(x + x^2) / (x^2)] = 1/256
Cross-multiplying, we get:
256(x + x^2) = x^2
256x + 256x^2 = x^2
256x^2 - x^2 + 256x = 0
255x^2 + 256x = 0
Now we can solve this quadratic equation. Factoring it, we get:
x(255x + 256) = 0
Setting each factor equal to zero, we have:
x = 0 or x = -256/255
Since time cannot be negative or zero in this context, we can ignore the solution x = 0. Therefore, x = -256/255 is not a valid time for the first pump to fill the tank.
Hence, the first pump will take 256 minutes to fill the tank.