# GMAT OFFICIAL GUIDE DS – Mary persuaded n friends to donate…

## Solution:

Mary persuaded n friends to donate \$500 each to her election campaign, and then each of these n friends persuaded n more people to donate \$500 each to Mary’s campaign. Since each of the first n people donated \$500, we know:

500n = amount donated by the first n people

Since each of these n people also persuaded n more people, we know that an additional n × n = n^2 people donated. Since each of these n^2 people also donated \$500, we know:

500n^2 = amount donated by the additional n^2 people

Thus,

500n + 500n^2 = total amount donated

We need to determine the value of n.

Note:  If you are having difficulty following the math above regarding the idea that “each of these n people also persuaded n more people,” assume that Mary initially persuaded n = 5 people to donate. Their contribution would be (\$500)(5). Now, each of these n = 5 people persuaded 5 additional people to donate, so this means that 5^2 = 25 additional people were persuaded, and each of them donated \$500. It follows that the additional people donated (\$500)(5^2), or (\$500)(25). Thus, the grand total donated by the (n + n^2) = (5 + 25) people is (\$500)(n + n^2) = (\$500)(n) + (\$500)(n^2). The grand total of donations in this example would be (\$500)(5 + 25), or \$15,000.

Statement One Alone:

The first n people donated 1/16 of the total amount donated.

From our given information we know that 500n + 500n^2 represents the total amount donated and that the first n people donated a total of 500n dollars. From statement one, we know that the first n people donated 1/16 of the total amount donated. Thus, we can create the following equation:

500n = 1/16(500n + 500n^2)

(500n)(16) = 500(n + n^2)

16n = n + n^2

n^2 – 15n = 0

n(n – 15) = 0

n = 0 or n = 15

Since n cannot be zero, n must be 15. Statement one alone is sufficient to answer the question. We can eliminate answer choices B, C, and E.

Statement Two Alone:

The total amount donated was \$120,000.

Since we know the total amount donated is 500n + 500n^2, we can create the following equation:

500n + 500n^2 =120,000

500(n + n^2) =120,000

n + n^2 = 120,000/500

n + n^2 = 240

n^2 + n – 240 = 0

(n – 15)(n + 16) = 0

n = 15 or n = -16

Since n cannot be negative, n must be 15. Statement two alone is also sufficient to answer the question.

Note: If we were unsure of how to break down the quadratic n^2 + n – 240 = 0, we could have concluded that we need one positive value for n and one negative value for n. We could have determined this because we know that in order to factor the quadratic n^2 + n – 240 = 0, we need two numbers that multiply to negative 240 and which sum to positive 1. The only way to achieve this would be from a negative number and a positive number. Since n must be positive in this problem, there is only one value of n that will answer the question.