Solution to the Apples for Sale puzzle
To remind you, hereâs the puzzle again:
A version of this clever problem first appeared in England a century ago.
Two women were selling apples at the market. They each had an equal number of apples, but they were selling them at different prices. Mrs. Jones sold her apples at a price of 2 for a penny. Mrs. Smith sold hers at a price of 3 for a penny.
Mrs. Smith had to leave, so Mrs. Jones agreed to sell her apples for her. Mrs. Jones combined the two equal supplies of apples, and sold them at a price of 5 apples for 2 cents, which led to a slight reduction in the total proceeds.
When Mrs. Smith returned, the apples had all been sold. The proceeds were exactly 7 cents less than they would have been if Mrs. Jones had sold each supply of apples at its original price.
They divided the money equally. How much did Mrs. Jones lose compared to what she would have made if she had sold her own apples at their original price?
The problem gives you 3 prices. The original prices charged by Mrs. Smith and Mrs. Jones, and the price all the apples were actually sold at. Start by converting all the prices into âcents per appleâ format, so you can compare the prices on an âapples to applesâ basis (so to speak).
Mrs. Jones sold them at 2 for a penny, so her price was 1/2 cent/apple.
Mrs. Smith sold them at 3 for a penny, so her price was 1/3 cent/apple.
The price they were actually sold at was 2 for 5 cents, so that price was 2/5 cent/apple.
  Jones â 1/2 cent/apple.
  Smith â 1/3 cent/apple.
  Actual price â 2/5 cent/apple.
To compare fractions with different denominators we need to determine the least common denominator and convert them. In this case, the least common denominator is 30. Once we convert them, the prices are:
  Jones â 15/30 cent/apple.
  Smith â 10/30 cent/apple.
  Actual price â 12/30 cent/apple.
Now that we made the 3 prices easy to compare, we immediately see whatâs going on. The actual price they were sold at is higher than the Smith price and lower than the Jones price. So on every apple sold, Smith made extra money and Jones lost some. The next step is to calculate how much.
We can do this by calculating the price all the apples should have been sold at to collect the exact amount of money that would have earned at the original prices. Since there are an equal number of âSmith applesâ and âJones applesâ the correct price to have sold them at would have been the average of the two prices (10/30 and 15/30). To make it easy to average the two numbers, letâs convert the denominator from 30 to 60. Now we take the average of 20/60 and 30/60 and quickly determine that the apples should have sold at a price of 25/60 cents/apple.
One final step and then we can solve it. Letâs convert the price the apples were actually sold at â 12/30 cent â to the same denominator. The price the apples were sold at was 24/60 cent per apple.
Once we take that step, the key to the solution now jumps off the page and whacks us on the head. The apples should have sold at 25/60 cent/apple. They were sold at 24/60 cent/apple. So for every apple sold, 1/60th of a penny was lost.
The rest is easy. Weâre told in the problem that they lost a total of 7 cents. Since they lost 1/60th of a penny per apple, we divide 1/60th into 7 and determine that they started with a total of 420 apples.
Since they had an equal number of apples, we know Mrs. Jones had 210 apples. Now all we need to do is calculate how much money she lost. Since her original price was 30/60th of a cent, and they were sold at a price of 24/60th of a cent, we know she lost 6/60th (or 1/10th) of a cent per apple. Multiply 1/10th of a cent by her 210 apples, and we see that Mrs. Jones lost 21 cents! (Since overall they lost 7 cents, then Mrs. Smith made 21-7 or 14 cents.)
 Answer: Mrs. Jones lost 21 cents.