I need help with the working in a very hard math problem please

heres the question
its not homework or anything i just really want to know the WORKING behind it i dont care about the anwser


A high-speed pump and a low-speed pump can be used together at the same time to fill a swimming pool in 8 hours.
However, if only the low-speed pump is used, it takes 3 hours longer than if only the high-speed pump is used.
How long does each pump take to fill the swimming pool if used alone?

Let's say that the high-speed pump (on its own) can fill the pool in "x" hours.
    • In 1 hour, it can fill 1/x of the pool.

The low-speed pump (on its own) can fill the pool in "x + 3" hours.
    • In 1 hour, it can fill 1/(x + 3) of the pool.

The two pumps, working together, can fill 1/x + 1/(x + 3) of the pool in 1 hour.

We're told that the pool can be filled by the two pumps in 8 hours, so …

1/x + 1/(x + 3) = 1/8

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Can you take it from here?

You find many problems of a similar nature here on Y!A (and presumably in the high school syllabus) … the working method is pretty much the same each time, with slight variations.

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Okay. It's basic algebra from here on. Multiply both sides by 8x(x + 3) to clear the fractions.

8x(x + 3)/x + 8x(x + 3)/(x + 3) = 8x(x + 3)/8
8(x + 3) + 8x = x² + 3x
8x + 24 + 8x = x² + 3x
x² - 13x - 24 = 0

You could use the quadratic equation … or complete the square, as follows:
x² - 13x = 24
x² - 13x + (13/2)² = 24 + (13/2)²
(x - 13/2)² = 24 + 169/4 = 265/4

Take the square roots of both sides:
x - 13/2 = ±(√265)/2
x = 13/2 ± (√265)/2

x = -1.64, 14.64 (to two decimal places) … ignore the negative value.

SOLUTION:
• the high-speed pump on its own can fill the pool in 14.64 hours.
                        (that's about 14 hours, 38½ minutes)
• the high-speed pump on its own can fill the pool in three hours more: 17.64 hours.