The Fruit Salad

Problem #17
Deadline to submit the solution: October 30th, 2000

You are vacationing on a quiet tropical island. There is only one old-fashioned grocery store nearby where you can buy food. You enter the store and want to buy a nice juicy cantaloupe. The owner has left the store for a short break, so you have to weigh the cantaloupe by yourself. You can’t find the weights for the scales, so you invent your own way of measuring the fruit. Here is what you find:

1. 1 cantaloupe + 1 peach = 1 pineapple
2. 1 cantaloupe = 1 grapefruit + 1 peach
3. 2 pineapples = 3 grapefruits

Q1: Assuming that all the fruits of the same kind are of equal size and weight, if one peach weighs 3 oz, what is the weight of the cantaloupe?

The solution

One cantaloupe weighs 15 oz.

The known facts are:

1. 1 cantaloupe + 1 peach = 1 pineapple
2. 1 cantaloupe = 1 grapefruit + 1 peach
3. 2 pineapples = 3 grapefruits

Hence, substituting for the cantaloupe in a. with the relation in b., we get:

1 grapefruit + 1 peach + 1 peach = 1 pineapple

1 grapefruit + 2 peaches = 1 pineapple

Or, by doubling the both sides of the equation, we get:

2 grapefruits + 4 peaches = 2 pineapples.

From c. we have 2 pineapples = 3 grapefruits,

Hence,

2 grapfruits + 4 peaches = 3 grapfruits.

Take away 2 grapefruits from both sides of the above equation, we get:

4 peaches = 1 grapefruit.

Since 1 peach weighs 3 oz, then 1 grapefruit weighs 12 oz.

From b. :

1 cantaloupe = 1 grapefruit + 1 peach = 12 oz + 3 oz = 15 oz.

One cantaloupe weighs 15 oz.