SOLUTION:

A cook went to a market to buy some eggs and paid $12. But since the eggs were quite small, he talked the seller into adding two more eggs, free of charge. As the two eggs were added, the price per dozen went down by a dollar. How many eggs did the cook bring home from the market?

A. 8

B. 12

C. 15

D. 16

E. 18

First, I noticed that this problem was about averages. Thus, I recalled the formula \(A = \frac{S}{n}\), and could quickly set up 2 equations.

equation #1 ... \(A = \frac{12}{n}\)

equation #2 ... \(A - \frac{1}{12} = \frac{12}{{n+2}}\) ( the price per dozen went down by 1 dollar --> the price per egg went down by \(\frac{1}{12}\) dollar)

Secondly, I combined the 2 equations, resulting in

\(\frac{12}{n} = \frac{12}{{n+2}} + \frac{1}{12}\)

\(144 * (n+2) - 144 * n = n * (n+2)\)

\(288 = n * (n+2)\)At this point I was done, and knew that the answer was 18 because of a "shortcut", which I remembered from calculation squares!

You might want to remember that "shortcut":

\(n^2 = (n - x) * (n + x) +x^2\)

Examples:

\(17^2= (17 - 1) * (17 + 1) + 1^2\) --> \(289 = 16 * 18 + 1\) --> \(288 = 16 * 18\)

\(17^2= (17 - 2) * (17 + 2) + 2^2\)

\(17^2= (17 - 3) * (17 + 3) + 3^2\)

(hint: It is the last one, which is generally useful IRL

)

Generally, using this "shortcut" has been very helpful in solving a couple of difficult questions!

Finally, I hope that my answer was helpful and would kindly welcome kudos.

Thanks,

Johannes