Question

Two grain-collecting machines can collect all the grain from a field 9 days faster than if the first one was doing it alone and 4 days faster than if the second one was working alone. How long does it take each grain-collecting machine to collect all the grain by itself?

Last problem on my homework, all help is appreciated, even an equation helps.

Answer 1

Let's define variables:
f = number of days for the first machine
s = number of days for the second machine
"Two grain-collecting machines can collect all the grain from a field 9 days faster than if the first one was doing it alone":
f-1 / ((1 / f) + (1 / s)) = 9
"and 4 days faster than if the second one was working alone":
s-1 / ((1 / f) + (1 / s)) = 4
We have two equations with two unknowns.
Rewriting the equations, you can reach the following expression:
s ^ 2 - 8s - 20 = 0
Factor:
(s-10) (s + 2) = 0
We ignore the negative root.
s = 10
Note: Verify that s = 10 satisfies both equations.
Answer:
It will take 10 days for the second machine to collect all the grain by itself
It will take 10 + 5 = 15 days for the first machine to collect all the grain by itself.