Math Problem 4: Selling Apples
An apple vendor regularly sells his apples at a local farmer's market. He receives his apples from two different providers at different prices, though the apples are the same. In order to make a reasonable profit, the vendor has to sell the apples from the first provider at 10 apples per dollar, and sell the apples from the second provider at 15 apples per dollar.
On the first day of sales, the apple vendor has 150 apples from each provider. They are put in different boxes. The apple vendor figures out how much he expects to make. Dividing the 150 apples in the first box into 15 bags of 10 apples each, he would sell each bag for one dollar and make $15. Dividing the 150 apples in the second box into 10 bags of 15 apples each, he would sell each bag for one dollar and make $10. So, the apple vendor expects to make a total of $25.
However, the apples are the same, so the vendor decides to sell them together. For example, taking 10 apples from the first box and 15 apples from the second box will make 25 apples, which he can then put in one bag and sell for $2. So, he puts all the apples together, divides them into bags of 25, and sells each bag of apples for $2.
When he counts up his money at the end of the day, he finds he has only made $24, not the expected $25. He asks his wife what the problem was, and she calculates that dividing 300 apples into bags of 25 would result in only 12 bags. These bags were sold for $2 apiece - sure enough, only $24.
Then she smacks him for being stupid. Where did the other dollar go?
Don't focus on the price of the apples. A better way to look at the problem is to consider how many apples are in each bag.
The number of apples per bag in the first box is different than the number of apples per bag in the second box.
Consider the number of apples remaining in each box after a bag is removed from each.
The vendor, if he had actually combined the bags one at a time instead of all together, would have run out of apples in the second box. By combining all the apples at the beginning, he was actually substituting some of less expensive apples with the more expensive apples.
If the vendor would have combined the bags one at a time until the second box was empty, and then sold the remaining apples in the first box at the original price (10 apples per dollar), he would have made the expected $25.