The Quest for 700: Weekly GMAT Challenge (Answer)

Yesterday, Manhattan GMAT posted a GMAT question on our blog. Today, they have followed up with the answer:

Let’s call the number of people in the group n, and let’s call each contribution $x. Then we know from the stem that 36 = nx. We are asked for n, which is equivalent to asking for x (because of the equation we are given).

(1) SUFFICIENT. This statement tells us that n = x. We can substitute into the given equation:

36 = n²

Since n represents a number of people, only the positive root makes sense, and n must be equal to 6.

(2) SUFFICIENT. This statement requires more work. We are told that if the number of people increased by 3, the contribution would fall by $2.

In other words, the new number of people is n + 3, and the new contribution is x – 2. The product will still be $36.

Thus, we know that 36 = (n + 3)(x – 2). We also still know that 36 = nx, or 36/n = x. Let’s expand the new equation and swap out x.

36 = (n + 3)(x – 2) = nx + 3x – 2n – 6

Since nx equals 36, we can substitute in 36 for nx as follows:

36 = 36 + 3x – 2n – 6

6 = 3x – 2n

Now substitute in 36/n for x:

6 = 3(36/n) – 2n

6 = 108/n – 2n

2n² + 6n – 108 = 0

Now, we need to try to factor this quadratic equation. At first glance, we can set up the following template:

(2n …)(n …) = 0

Since the 108 in the distributed quadratic equation has a minus sign in front of it, we know that one of the missing numbers is negative and that the other one is positive. This means that one of the solutions for n will be positive, while the other one will be negative. We could stop here, since only one positive solution exists.

If we wanted to keep going with the factoring, we could observe that 108 = 3×36 = 3³×2². We need a pair of factors relatively close in value to each other, since their difference (after multiplying one of the factors by 2) is only 6. The pair of factors that works is {9, 12}, as we can see by trial and error.

(2n – 12)(n + 9) = 0

n = 6 or n = –9

The negative solution is impossible, so we know that n is 6.

The correct answer is (D). EACH statement ALONE is sufficient.