# Solution

(related to Problem: How Old Was Mary?)

The age of Mary to that of Ann must be as $5$ to $3.$ And as the sum of their ages was $44,$ Mary was $27\frac 12$ and Ann $16\frac 12.$ One is exactly $11$ years older than the other. I will now insert in brackets in the original statement the various ages specified: "Mary is $(27\frac 12)$ twice as old as Ann was $(13\frac 34)$ when Mary was half as old $(24\frac 34)$ as Ann will be $(49\frac 12)$ when Ann is three times as old $(49\frac 12)$ as Mary was $(16\frac 12)$ when Mary was $(16\frac 12)$ three times as old as Ann $(5\frac 12)$."

Now, check this backwards.

When Mary was three times as old as Ann, Mary was $16\frac 12$ and Ann $5\frac 12$ ($11$ years younger). Then we get $49\frac 12$ for the age Ann will be when she is three times as old as Mary was then. When Mary was half this she was $24\frac 34.$ And at that time Ann must have been $13\frac 34$ $(11$ years younger$).$ Therefore Mary is now twice as old — $27\frac 12,$ and Ann $11$ years younger — $16\frac 12.$

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### References

#### Project Gutenberg

1. Dudeney, H. E.: "Amusements in Mathematics", The Authors' Club, 1917

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