Maths and Melons

I recently came across this simple example which shows clearly the difference between what seems intuitively to be correct, and what a very small amount of mathematics can show is actually the case.

This problem involves a merchant selling fruit at a market. The merchant sells melons at 2 for 80 cents each and mangoes at 3 for 80 cents. However, wanting to reduce the arithmetic he must do on each sale, he decides to commingle melons and mangoes and to sell any five pieces of fruit for 160 cents (32 cents each). Now if he sells 2 melons and 3 apples, he sells five pieces of fruit and receives 10 cents. This is therefore a reasonable move. Right?

The merchant, it turns out, is cheating himself.

Initially, the price per melon was 40 cents, and the price per mango was 80/3 = 26 2/3 cents. The average price for any piece of fruit was therefore

Let’s assume the dealer has 12 melons and 12 mangoes. If he sells the melons normally at 2 for 80 cents he receives $4.80 for the 12 melons. If he sells all mangoes at 3 for 80 cents he receives $3.20 for the 12 mangoes, and a total of $8.00 for all 24 pieces of fruit. However, selling all 24 pieces of fruit at 32 cents each, he receives only 24 x 32 = $7.68.

What accounts for this loss?

Well, intuitively, it seemed that the average price of the mangoes and melons should be 32 cents each. However, the price per melon was initially 40 cents, and the price per mango was 80/3 = 26 2/3 cents. The average price for any piece of fruit was therefore   (40 + 26 2/3)/2 = 33 1/3 cents, and not the 32 cents assumed by the merchant.

What initially seemed completely reasonable has been dismantled – and all by a simple bit of melon mathematics.

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