This article is from the Puzzles FAQ, by Chris Cole chris@questrel.questrel.com and Matthew Daly mwdaly@pobox.com with numerous contributions by others.

1) Ten years from now Tim will be twice as old as Jane was when Mary was

nine times as old as Tim.

2) Eight years ago, Mary was half as old as Jane will be when Jane is one year

older than Tim will be at the time when Mary will be five times as old as

Tim will be two years from now.

3) When Tim was one year old, Mary was three years older than Tim will be when

Jane is three times as old as Mary was six years before the time when Jane

was half as old as Tim will be when Mary will be ten years older than Mary

was when Jane was one-third as old as Tim will be when Mary will be three

times as old as she was when Jane was born.

HOW OLD ARE THEY NOW?

logic/ages.s

The solution: Tim is 3, Jane is 8, and Mary is 15. A little grumbling

is in order here, as clue number 1 leads to the situation a year and a

half ago, when Tim was 1 1/2, Jane was 6 1/2, and Mary was 13 1/2.

This sort of problem is easy if you write down a set of equations. Let

t be the year that Tim was born, j be the year that Jane was born, m be

the year that Mary was born, and y be the current year. As indefinite

years come up, let y1, y2, ... be the indefinite years. Then the

equations are

y + 10 - t = 2 (y1 - j)

y1 - m = 9 (y1 - t)

y - 8 - m = 1/2 (y2 - j)

y2 - j = 1 + y3 - t

y3 - m = 5 (y + 2 - t)

t + 1 - m = 3 + y4 - t

y4 - j = 3 (y5 - 6 - m)

y5 - j = 1/2 (y6 - t)

y6 - m = 10 + y7 - m

y7 - j = 1/3 (y8 - t)

y8 - m = 3 (j - m)

t = y - 3

j = y - 8

m = y - 15

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