This article is from the Puzzles FAQ, by Chris Cole email@example.com and Matthew Daly firstname.lastname@example.org 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?
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
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