Question

Two tractors working together cultivated 2 3 of a field in 4 days. how long would it take each tractor to cultivate the whole field by itself if the first tractor can do the job 5 days faster than the second?

Answer 1

Let the time for the 1st tractor alone be A, and the time for the 2nd tractor alone be B. Then in 1 day, the 1st tractor does 1/A, and the 2nd does 1/B. Together, both complete 2/3 of the field in 4 days:
(1/A + 1/B)(4 days) = 2/3
1/A + 1/B = 1/6
If the 1st tractor can finish in 5 days less than the 2nd:
A = B - 5
Substituting:
1/(B - 5) + 1/B = 1/6
(2B - 5)/B(B - 5) = 1/6
6(2B - 5) = B(B - 5)
12B - 30 = B^2 - 5B
B^2 - 17B - 30 = 0
(B - 15)(B - 2) = 0
B = 15 (B = 2 is extraneous since A would then be -3)
A = B - 5 = 10
Therefore, the 1st tractor would take 10 days, and the 2nd would take 15.