maggie27 wrote:
Mathematics, physics, and chemistry books are stored on a library shelf that can accommodate 25 books. Currently, 20% of the shelf spots remain empty. There are twice as many mathematics books as physics books and the number of physics books is 4 greater than that of the chemistry books. Ricardo selects 1 book at random from the shelf, reads it in the library, and then returns it to the shelf. Then he again chooses 1 book at random from the shelf and checks it out in order to read at home. What is the probability Ricardo reads 1 book on mathematics and 1 on chemistry?
A) 3%
B) 6%
C) 12%
D) 20%
E) 24%
VERITAS PREP OFFICIAL SOLUTION:The first phase of this problem requires you to determine how many mathematics and chemistry books are even on the shelf. To do so, you have the equations:
m + p + c = 20 (since 4/5 of the 25 spots are full of books)
m = 2p
p = 4 + c
From that, you can use Substitution to get everything down to one variable (and since all three equations include p, that's probably the easiest to work on). If
c = p - 4
m = 2p
p = p
Then (p - 4) + 2p + p = 20, so 4p = 24 and p = 6. That means that there are 12 math, 6 physics, and 2 chemistry books on the shelf.
Here's where this becomes a probability problem: the probability of reading math, then chemistry is (12/20)(2/20), which simplifies to 6/100. That makes B a tempting answer, but keep in mind that Ricardo could read them in the opposite order, chemistry then math. In that case, the probability works out to the same, but since 6% of the time he reads C then M and 6% of the time he reads M then C, then 12% of the time his outcome is that he reads one of each, so the answer is C.