Yesterday, Manhattan GMAT posted a GMAT question on our blog. Today, they have followed up with the answer:
This problem requires some “stripping of the fluff.” We have four musicians who play in different rhythms. Consider the pianist: after the initial downbeat, the pianist’s next downbeat comes at 9/8 of a measure. When would the next one come? At 2 x 9/8, or 18/8. The one after that would be at 3 x 9/8, etc. So we’re taking multiples of 9/8, to find the downbeat of the pianist.
Likewise, for the other three musicians, their downbeats occur at multiples of their time signatures (7/4, 5/8, and 4/4). To find the first common downbeat after the song has started, we must find the least common multiple of 9/8, 7/4, 5/8, and 4/4.
First, let’s make a common denominator for all of these fractions. We should use 8, so we get 9/8, 14/8, 5/8, and 8/8.
Now find the least common multiple of the numerators. The LCM of 9, 14, 5, and 8 is not trivial, because we have a lot of unique prime factors:
9 = 3 x 3
14 = 2 x 7
5 = 5
8 = 2 x 2 x 2
We can drop one 2 (overlap between 14 and 8), but otherwise we’re stuck. If we try to combine primes to avoid having to multiply two 2-digit numbers together, we get the following. (Hint: save the ugly 7 for last.)
3 x 3 x 2 x 7 x 5 x 2 x 2
= 3 x 3 x 2 x 7 x 10 x 2
= 3 x 3 x 2 x 7 x 20
= 3 x 6 x 7 x 20
= 3 x 7 x 120
= 7 x 360
= 2,520
So the common downbeat will happen at 2,520/8 measures. We must divide 2,520 by 8 now, which yields 315.
The correct answer is A.