Mary (M) is twice as old as Ann (A) was when M was half as old as A will be when A is $3$ times as old as M was when M was $3$ times as old as A was.

Question

The combined ages of Mary and Ann is $44$ years, and Mary is twice as old as Ann was when Mary was half as old as Ann will be when Ann is three times as old as Mary was when Mary was three times as old as Ann was.

How old is Mary?

Translating into equations seems to be the way to go with this but the difficulty is where to start after the second "and".

"The combined ages of Mary and Ann is 44" clearly translates to:

$$M + A = 44$$

Any suggestions?

Answer 1

The combined ages of Mary and Ann is 44 years and Mary is twice as old as Ann was when Mary was half as old as Ann will be when Ann is three times as old as Mary was when Mary was three times as old as Ann was.

How old is Mary?

$M + A = 44.$

$k$ years ago, Mary's age was $M - K$ and Ann's age was $A - k$.

You know that $M - k = 3(A - k)$.

So, now you have $2$ linear equations, in the $3$ unknowns, $M,A,k$.

When Ann is $3(M-k)$, and Mary is (1/2) that age, Mary is $(3/2)(M-k)$.

At that point, Ann was $(3/2)(M-k) - (M-A).$

Mary is now twice that age.

Therefore,

• $M = 3(M-k) - 2(M-A) \implies 2A = 3k.$

The other two equations are

• $M + A = 44.$

• $M - k = 3(A - k) \implies [M - (2/3)A] = 3A - 2A \implies$
  $M = (5/3)A$.

So, $M = 27.5, A = 16.5$. For what it's worth, I tracked through the story, based on these results. Unless I made a mistake, it works.