Problem 10.
SOME MONKEY BUSINESS: Hanging over a pulley is a rope. At one end of the rope is a sand bag and at the other end is a monkey. The weight of the monkey is the same as the weight of the sand bag. The rope weighs 4 ounces per foot.
The sum of the ages of the monkey and the monkey's mother is 4 years. The monkey's weight in pounds is the same as the mother's age in years.
The mother is twice as old as the monkey was when the mother was half as old as the monkey will be when the monkey is three times as old as the mother was when she was three times as old as the monkey was.
The weight of the rope and the sand bag is one and a half times the weight of the monkey. How long is the rope?
[Problem submitted by Roger Wolf, LACC Professor of Mathematics and Chairman of the Department of Mathematics and Computer Science.]
a) Let t be the age in years of the monkey when the monkey's mother was three times as old as the monkey. Then at that time the mother was 3t years old. Note that the difference between the mother's age and the monkey's age is 2t years, which remains the constant difference between their ages at any time.
b) When the monkey is three times the mother's age in a), the monkey will be 9t years old.
c) When the mother was half the monkey's age in b), the mother was
years old. Since the mother is always 2t years older than the monkey, the monkey was 
years old when the mother was years old.
d) The mother's current age is twice the age of the monkey in c). So, the mother is 5t years old. Then the monkey is 3t years old.
e) The sum of their ages is 4 years. So, 5t +3t = 4. Therefore, t =
, and the mother is 5t = 
= 
years old.
f) The monkey's weight in pounds is the same as the mother's age in years; so the monkey weighs
pounds which is equal to 40 ounces.
g) The sandbag weighs the same as the monkey, 40 ounces.
h) Let r be the weight of the rope in ounces and l the length of the rope in feet. The rope weighs 4 ounces per foot: 4l = r.
i) The weight of the rope and the sand bag is one and one half times the monkey's weight. r + 40 =
.
j) Substituting 4l for r: 4l + 40 =
. So, the rope is 5 feet long.