Henry Ernest Dudeney/Puzzles and Curious Problems/132 - Sharing the Apples/Solution

Puzzles and Curious Problems by Henry Ernest Dudeney: $132$

If $3$ boys had $169$ apples which they shared in the ratio of one-half, one-third and one-fourth, how many apples did each receive?

The first boy got $78$ apples.

The second boy got $52$ apples.

The third boy got $39$ apples.

We have that:

$\dfrac 1 2 + \dfrac 1 3 + \dfrac 1 4 = \dfrac {13} {12}$

We have that

$169 = \dfrac {13} {12} 156$

Hence:

the first boy got $\dfrac 1 2 156 = 78$ apples

the second boy got $\dfrac 1 3 156 = 52$ apples

the first boy got $\dfrac 1 4 156 = 39$ apples

We note that $78 + 52 + 39 = 169$ and all is well with the world.

$\blacksquare$