March 29, 1999
Problem of the Month
The Banana Problem
A rope over the top of a fence has the same length on
each side, and weighs one-third of a pound per foot.
On
one end hangs a monkey holding a banana, and on the
other end a weight equal to the weight of the monkey.
The
banana weighs 2 ounces per inch.
The length of the rope
in feet is the same as the age of the monkey, and the
weight of the monkey in ounces is as much as the age of
the monkey's mother.
The combined ages of the monkey
and its mother are 30 years.
One-half the weight of the
monkey, plus the weight of the banana is one-fourth the
sum of the weights of the rope and the weight.
The
monkey's mother is one-half as old as the monkey will be
when it is three times as old as its mother was when she
was one-half as old as the monkey will be when it is as old
as its mother will be when she is four times as old as the
monkey was when it was twice as old as its mother was
when she was one-third as old as the monkey was when it
was as old as its mother was when she was three times as
old as the monkey was when it was one-fourth as old as it
is now.
How long is the banana?
Solution to the Problem:
The answer is 5.75 inches long.
Let A = age of the monkey now and let M = age of
his Mother.
Then A + M = 30.
From the long sentence,
M = .5(3(.5(4(2(1/3(3(.25A))))))
Solving simultaneously, 2.5 A = 30
so A = 12 (the age of the monkey) and
M = 18 (the age of the monkey's mother).
So the weight of the monkey is 18 oz.
The length of the rope is 12 feet.
The rope weighs 2 pounds.
The weight weighs 18 ounces.
The banana weighs 11.5 ounces, so
the banana is 5.75 inches long.