Henry Ernest Dudeney/Puzzles and Curious Problems/43 - Mike's Age/Solution

by : $43$

 * Mike's Age
 * Pat O'Connor is $1 \tfrac 1 3$ times as old as when he built the pigsty.
 * Little Mike, who was $40$ months old when Pat built the sty, is now two years more than half as old as Pat's wife, Biddy, was when Pat built the sty,
 * so that when Little Mike is as old as Pat was when he built the sty,
 * their three ages combined will amount to just one hundred years.


 * How old is Little Mike?

Solution
Mike is currently $10 \tfrac {16} {21}$ years old.

Pat was $22 \tfrac 2 7$ when he built the pigsty, which was $7 \tfrac 3 7$ years ago, and is now $29 \tfrac 5 7$ years old.

Biddy was $17 \tfrac {11} {21}$ when Pat built the pigsty, and is now $24 \tfrac {20} {21}$ years old

Proof
Let $P$ years be Pat's age when he built the pigsty.

Let $M$ years be Mike's age now.

Let $B$ years be Biddy's age when Pat built the pigsty.

First we note that from:
 * Pat O'Connor is $1 \tfrac 1 3$ times as old as when he built the pigsty

$P$ built the pigsty $\dfrac P 3$ years ago.

We have:

The remaining ages are calculated hence.