For a charity, each of m people pledged to run 25 kilometers in a fund

Official Explanation

On this question, let's express the pledge in algebra. The total number of miles pledged was \(25m\) for the initial set of people. Then there is the secondary group of people, the ring of friends. There are \(m\) people in this ring for each of the original \(m\) people, so there are \(m^2\) of them, and each of them pledged 25 miles, so the outer ring has pledged \(25m^2\) miles. That's a total of \(25m + 25m^2\) miles. We can consider a case to confirm the logic. If there were 6 people initially, and each of them convinced 6 people to pledge, it would be \(25(6) + 25(36)\) miles. Seems right.

Statement (1) says that the pledge of the initial set is \(\frac{1}{25}\) of the total. That would mean

\(25m = \frac{25m + 25m^2}{25}\)

\(25m = m + m^2\)

\(25m = 1 + m\)

\(24 = m\)

It's a little confusing, because the number of people pledged is so similar to the number of miles each pledged. But the statement has uniquely determined \(m\); it's saying that 24 people each pledged 25 miles and each also convinced 24 more people each to pledge 25 miles. To double-check, we think it over in a new light; since everyone pledged the same amount, the statement is saying that m people are \(\frac{1}{25}\) of the total number of people. The total number of people would be \(24 +24(24)\) . That's one 24 plus twenty-four 24's - twenty-five 24's, so indeed one 24 is \(\frac{1}{25}\) of the total number of people. Statement (1) is sufficient.

Statement (2) gives the total number of miles. So we have

\(25m = 25m^2 = 15000\)

Since this is a quadratic equation, it will in general have two roots, although the two roots may turn out to be the same. So we need to proceed further to determine whether the number of possible values for m based on this information is 1 or 2.

\(25m = 25m^2 - 15000 = 0\)

Actually, we could already stop here. The fact that the signs are opposite in this quadratic means that one root will be positive and one will be negative. Since a number of people must be a positive number, there will be only one possible value and this will be sufficient. Supposing that we might have forgotten that fact or been unsure of it, we can continue:

\(m^2 + m - 600 = 0\)

\((m + ?)(m - ?) = 0\)

The numbers at the question marks must multiply to negative. If you hadn't seen the fact that one solution is positive and one is negative before, it's clear now...

\((m + 25) (m - 24) = 0\)

Therefore, m = 24 . Statement (2) is sufficient.

The correct answer is (D).