### Brain Teasers

# Orchard

Last year, the average number of apples on each tree equalled the number of trees in the orchard. However this year there were 640 less apples than last year's harvest. If the average number of apples per tree this year was 72, how many trees are there in my orchard?

Please Note: No extra trees have been planted or any removed! There are no tricks to this question.

Please Note: No extra trees have been planted or any removed! There are no tricks to this question.

### Answer

There are 80 trees in my lovely little orchard.Let T = the number of trees and N = the total number of apples. Set up two equations, one for each year:

Last year: N/T = T or N = T^2

This year: (N-640)/T = 72

Substitute N = T^2 into the second equation to get: (T^2 - 640)/T = 72.

Multiplying both sides by T and rearranging gives you the polynomial: T^2 - 72*T - 640 = 0. Solving this gives roots of -8 and 80. Obviously, there can't be -8 trees in the orchard, so there must be 80.

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