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
of them, and
each of them pledged 25 miles, so the outer ring has pledged
miles. That's
a total of
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
of the total.
That would mean
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
of the total
number of people. The total number of people would be
. That's one 24 plus twenty-four 24's - twenty-five
24's, so indeed one 24 is
of the total
number of people. Statement (1) is sufficient.
Statement (2) gives the total number of miles. So we have
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.
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:
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...
Therefore,
. Statement (2) is sufficient.
The correct answer is (D).
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