Apple Harvest

We have to find LCM of the numbers as it would be the first or lowest amount they can bring home which will be same. To find LCM, we find the prime factorisation of the numbers and then take all the unique primes for their factors and if a prime is repeated then take its highest exponent as the LCM has to be completely divisible.

• 15 = 5 x 3

• 50 = 5 x 5 x 2

• 85 = 17 x 5

Therefore, LCM = 17 x 5 x 5 x 2 x 3 = 2550.

Therefore, the first son with 15 apples can sell one apple for 170 rupees ( or whatever currency )

The second son with 50 apples has to sell one apple for 51 rupees.

The third son with 85 apples has to sell one apple for 30 rupees.

Because the Least Common Multiple of $15$ , $50$ , and $85$ is $2550$ :

• Paul sells his $15$ apples at a price of $\frac{2550}{15}$ , or $\text{\$}170$ per apple.
• Nick sells his $50$ apples at a price of $\frac{2550}{50}$ , or $\text{\$}51$ per apple.
• Ben sells his $85$ apples at a price of $\frac{2550}{85}$ , or $\text{\$}30$ per apple.

All kids will come home with $\text{\$}2550$ each.

Note: if the prices could be decimal values, you could make it even more efficient. (I think each kid will come home with $\text{\$}25.50$ , but I'm not sure.) However, we just need to see if it's possible, so this is just one solution.

Yes it is possible.

The first buyer purchases 12 dozen apples at $1 per dozen.

Paul sells him one dozen and has three apples left; Nick sells him four-dozen and has two apples left; Ben sells him seven-dozen and has one apple left.

Then a second buyer comes along and buys all their remaining apples for $3 apiece. The three brothers head home with $10 each.

Paul gets 1$ + 3*3$ = 10$

Nick gets 4$ + 2*3$ = 10$

Ben gets 7$ + 1*3$ = 10$

Mrs. Jones was preparing them for their test on Logic.

LOL, Just Kidding, but you can think of it as a backstory to this problem