# One pump can empty a pool in 4 days, whereas a second pump can empty the pool in 12 days. How long will it take the two pumps, working together, to empty the pool?

## 1/4 + 1/12 = 1/x

looks like it will take 3 days

## To find out how long it will take the two pumps, working together, to empty the pool, we need to calculate their combined rate of pumping.

Let's say the first pump's rate of pumping is represented by "x," which means it can empty 1/x of the pool in a single day. Similarly, the second pump's rate of pumping is 1/12, as it takes 12 days for it to empty the pool.

To determine their combined rate of pumping, we add the rates together:

1/x + 1/12 = 1/t,

Where "t" represents the time it would take for both pumps, working together, to empty the pool.

To solve this equation, we need to find a common denominator:

12/x + 1/12 = 1/t.

Multiplying every term by 12x, we get:

12 + x = 12x/t.

Now, let's simplify the equation:

12t + tx = 12x.

Rearranging the equation, we have:

tx - 12x = -12t.

Factoring out "x" from the left side of the equation:

x(t - 12) = -12t.

Next, divide both sides by (t - 12):

x = -12t / (t - 12).

Since both pumps are working to empty the pool, the rate of pumping cannot be negative. Therefore, -12t must be positive:

-12t > 0.

Dividing both sides by -12, we obtain:

t < 0.

The time it takes for the two pumps, working together, to empty the pool, cannot be negative. There appears to be an error in the problem statement or the calculations. Could you please provide additional information or check if there was a mistake in the given data?

## To find out how long it will take the two pumps to empty the pool together, we need to determine their combined rate of work.

Let's assume that the pool is the same size, and the work done by each pump is proportional to the time it takes for them to empty the pool individually.

The first pump empties the pool in 4 days, so its work rate is 1/4 of the pool per day. Similarly, the second pump empties the pool in 12 days, so its work rate is 1/12 of the pool per day.

To find their combined work rate, we add up their individual work rates: 1/4 + 1/12 = 1/3.

Therefore, the two pumps working together can empty 1/3 of the pool per day.

To empty the entire pool, it will take the two pumps 1/(1/3) = 3 days.

So, the two pumps, working together, will take 3 days to empty the pool.