Manhattan GMAT Challenge Problem of the Week - 16 Apr 10

by , Apr 16, 2010

Welcome back to this week's Challenge Problem! As always, the problem and solution below were written by one of our fantastic instructors. Each challenge problem represents a 700+ level question. If you are up for the challenge, however, set your timer for 2 mins and go!

Question

A number of eggs dyed various colors were hidden for an egg hunt. How many eggs in total were hidden?

(1) The number of red eggs hidden is the square of an integer, while the total number of eggs hidden is 24 times that integer.

(2) Exactly 143 of the eggs hidden were not red.

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

(D) EACH statement ALONE is sufficient.

(E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Solution

Initially, we cant 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 = [pmath]x^2[/pmath]

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

[pmath]x^2[/pmath] + 143 = 24x

We can rearrange this quadratic equation, setting one side equal to 0:

[pmath]x^2[/pmath] 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 [pmath]x^2[/pmath] 24x + 144 = 0, it would factor to [pmath](x - 12)^2 = 0[/pmath], and x would have just one possible value, 12.) Thus, there are two possible values of x.

Alternatively, we could simply factor [pmath]x^2[/pmath] 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).

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