Yesterday, Manhattan GMAT posted a GMAT question on our blog. Today, they have followed up with the answer:
Initially, we can’t really rephrase the question. We are asked for the total number of eggs hidden for a hunt.
Statement 1: INSUFFICIENT. This statement tells us 2 facts. Using x to represent the unknown integer, we can write the following:
Red Eggs = x2
Total Eggs = 24x
However, we have no way of determining x, so this statement is not enough.
Statement 2: INSUFFICIENT. This statement tells us the following:
Non-Red Eggs = 143
By itself, we cannot hope to know how many eggs were hidden in all.
Statements 1 & 2 together: INSUFFICIENT. We know the following:
Red + Non-Red = Total
x2 + 143 = 24x
We can rearrange this quadratic equation, setting one side equal to 0:
x2 – 24x + 143 = 0
At this point, we can stop if we study the equation closely. The factored form of the equation must be as follows:
(x – …)(x – …) = 0
The reason is that the middle term (–24x) is negative, while the constant term (143) is positive. This means that the factored form on the left must have two minus signs.
As a result, we expect two positive solutions for x. In fact, we could have just one positive solution, if the equation factors into something like this: (x – …)2 = 0. However, that would require the constant term (in this case, 143) to be a perfect square, since x is an integer. (For instance, if the original equation were x2 – 24x + 144 = 0, it would factor to (x – 12)2 = 0, and x would have just one possible value, 12.) Thus, there are two possible values of x.
Alternatively, we could simply factor x2 – 24x + 143 = 0. Since 143 = 11 × 13, we have the following:
(x – 11)(x – 13) = 0
x = 11 or x = 13
Thus, there are two possible values for x, leading to two possible total numbers of eggs. Even together, the statements are not sufficient.
The correct answer is (E).