SOLUTION: 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 alo

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Question 1019704: 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?
Answer by ikleyn(52890) About Me  (Show Source):
You can put this solution on YOUR website!
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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?
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Let f be the number of days for the first  grain-collecting machines to collect all the grain from a field, if it works alone.

Let s be the number of days for the second grain-collecting machines to collect all the grain from a field, if it works alone.

Then  the rate of the first  machine is 1%2Ff of the "field-per-day", 
while the rate of the second machine is 1%2Fs of the "field-per-day".
When both machines work simultaneously, their combined rate is 1%2Ff+%2B+1%2Fs of the "field-per-day".
Hence, two machines working together can can collect all the grain from the field in 1%2F%281%2Ff+%2B+1%2Fs%29 days.
Therefore, from the first part of the condition you have this equation:
f - 1%2F%281%2Ff+%2B+1%2Fs%29 = 9.   (1)

From the second part of the condition you have this equation:
s - 1%2F%281%2Ff+%2B+1%2Fs%29 = 4.   (2)

Thus you have the system of two non-linear equations (1) and (2). 
Do not be scared: still there is a way to solve it.

Distract equation (2) (both sides) from equation (1). You will get

f - s = 5,   or   f = s + 5.   (3)

Now substitute the expression  f = s + 5  from (3) into equation (1). You will get a single equation for the unknown s:

(s+5) - 1%2F%281%2F%28s%2B5%29+%2B+1%2Fs%29 = 9.

Simplify and solve it, step by step:

(s+5) - 1%2F%28s%2F%28s%2A%28s%2B5%29%29+%2B+%28s%2B5%29%2F%28s%2A%28s%2B5%29%29%29%29 = 9,  --->  s+5 - %28s%2A%28s%2B5%29%29%2F%282s%2B5%29 = 9,  --->  . . .  --->  s%5E2+-+8s+-+20 = 0.

To solve the last quadratic equation, factor it: s%5E2+-+8s+-+20 = (s-10)*(s+2),
and you will get the unique solution to the problem  s = 10.  (The other root is negative and, therefore, doesn't suit).

Answer. It will take 10 days for the second machine to complete the job working alone.
        It will take 15 = 10+5 days for the first  machine to complete the job working alone.