how many lettuces out of every 20 does he grow unsuccessfully?

Question

When growing lettuces in a greenhouse, a gardener unsuccessfully grows  a number of them. He plants 15000 in one patch. If the gardener makes R1.72 profit on  each lettuce, and makes an overall profit of R15 380, how many lettuces out of every 20 does he grow unsuccessfully?

Answer 1

Out of 15000 lettuces n lettuces are unsuccessful and can’t be sold. Let c=cost of producing one lettuce (successful or unsuccessful). Total cost of producing all lettuces=15000c. No fixed costs are mentioned. If s=sale price of 1 lettuce, income is (15000-n)s.

Total profit = (15000-n)s-15000c=15000s-ns-15000c=15000(s-c)-ns.

s-c=R1.72 for a successful lettuce, so total profit=25800-ns=15380. Therefore ns=R10420. This is the total of lost sales because of unsuccessful lettuces. We don’t know the sale price of a lettuce. If s=R5, then n=2084, if s=R4, n=2605, if s=R10, n=1042, if s=R20, n=521. If we assume n=521, then 521/15000 are unsuccessful, and this means that in 20 lettuces, less than 1 would be unsuccessful.

Now assume n=1042, then about 1.39 in 20 are unsuccessful; n=2605, 3.47 unsuccessful in 20; n=2084, 2.78 unsuccessful in 20.