Working simultaneously at their respective constant rates, M

Let's say the rate of Machine A is r_A nails per hour, and the rate of Machine B is r_B nails per hour. Then we know that:

Together, Machines A and B produce 800 nails in x hours, so their combined rate is r_A + r_B, and we have:

800 = (r_A + r_B) x

Alone, Machine A produces 800 nails in y hours, so we have:

800 = r_A y

We want to find the time it takes for Machine B to produce 800 nails alone, which we'll call t_B. We can start by using the formula:

rate = amount / time

To find the rate of Machine A:

r_A = 800 / y

And the combined rate of Machines A and B:

r_A + r_B = 800 / x

We can now solve for r_B by subtracting r_A from both sides:

r_B = 800 / x - 800 / y

Finally, we can use the formula for rate to find the time it takes Machine B to produce 800 nails alone:

r_B = 800 / t_B

Substituting the expression we found for r_B:

800 / t_B = 800 / x - 800 / y

Solving for t_B:

t_B = xy / (y - x)

Therefore, the answer is (E) xy / (y - x).