Bunuel wrote:
Forrest and Jenny distributed $100 each in charity. Forrest distributes 5 more people than Jenny and Jenny gives each $1 more than Forrest. How many people are the recipients of the charity ?
A. 20
B. 25
C. 30
D. 40
E. 45
Are You Up For the Challenge: 700 Level QuestionsFirst, this question has a couple of slight issues that an official question might handle differently. However, its core mechanism is a common one in word problems. Essentially, it's a rate or
price x
quantity problem: one scenario has $100 broken into smaller increments of money and given to more people, while the other scenario has $100 broken into larger increments and given to consequently fewer people. While a question like this can be handled in a few different ways, most algebraic paths require quadratics, generally leaving arithmetic options as the better bet. I'll lay out both below.
ArithmeticOften, the arithmetic approach on such questions involves testing the answers. This problem complicates that option by asking for a total in the question stem, as it's usually more practical to work backwards from the answers when they represent an individual value. And while the answers here do have a promising looking triad in 20 - 25 - 45, it's safer to consider the other arithmetic approach: divisibility. Most word problems on the GMAT deal in integer values, sometimes (such as with dollar figures) out of mathematical convenience, and other times (such as with numbers of people) out of sheer decency. And poblems that deal with multiplication or division of exclusively integer values tend to function as divisibility problems. In practical terms, this means that factoring is likely to be a productive option. Because 100 is going to be the product of the number of people who received charity from Forrest and Jenny and the average amount of charity each of those people received, break 100 down into its factors:
# ppl x
avg $ per person 1 x 100
2 x 50
4 x 25
5 x 20
10 x 10
20 x 5
25 x 4
50 x 2
100 x 1
When we find a place where a difference of 5 people accompanies a difference of $1 in the amount distributed on average to each. That happens when we go from 20 people to 25 people and the amount distributed drops from $5 to $4. If we try to tie those values to the information in the problem, it would mean that Forrest gave $4 to each of 25 people and Jenny gave $5 (that is, $1 more) to each of 20 people. It checks out! So the total number of people who received charity would be 20 + 25 = 45, or
answer (E).
AlgebraThere are a couple of ways the algebra can be setup here (such as the aforementioned
price x
quantity =
total framework), but given that the problem doesn't specifically state that each person receives the same amount of charity, the safest play might be to set up an equation for Forrest and Jenny's
average per person contribution. Jenny's average amount distributed per person (let's call this
d) is her total distribution of $100 divided by the number of people (we'll call this
p) to whom the money was distributed:
\(d = \frac{100}{p}\)
Forrest, on the other hand, distributed $1 on average less to each of 5 more people:
\(d - 1 = \frac{100}{p + 5}\)
This is the expected bad news moment: systems of equations with multiplication of or division by variables tend to lead to quadratics. And indeed, if we substitute the value of
d from Jenny's equation for the
d in Forrest's equation, and you've got an equation with variables in denominators:
\(\frac{100}{p} - 1 = \frac{100}{p + 5}\)
Now there's not much algebraic choice but to multiply all three terms in the equation by
p and again by
p + 5 to clear all the denominators from the question (this tends to be a fairly unavoidable step when you've got equations with variables in denominators). That plays out thus:
\((p)(p + 5)*\frac{100}{p} - (p)(p + 5) = (p)(p + 5)*\frac{100}{p + 5}\)
\(100p + 500 - p^2 - 5p = 100p\)
Rearranging the elements so that the \(p^2\) term is positive and there is only a 0 left on one side of the equation, we get
\(p^2 +5p - 500 = 0\)
As a sidenote here, 500 is rarely a number you're happy to see in a quadratics context, because now our job is to think about what numbers multiply to -500 while adding up to +5. If it wasn't already apparent that the factoring approach above is the more practical one, hopefully this seals the deal: the algebraic approach
still makes us factor, and factor an even harder number to boot. #betrayedbyalgebra
Back to the action:
\((p - 20)(p + 25) = 0\)
\(p - 20 = 0\) or \(p + 25 = 0\)
So either
p = 20 or
p = 25.
As the GMAT has a strict "No negative people allowed" policy, Jenny must have distributed charity to
p = 20 people and Forrest to
p + 5 = 25 people, for a total of 45 recipients and (again) an answer of (E).
I hope that does the trick!