Yesterday, Manhattan GMAT posted a GMAT question on our blog. Today, they have followed up with the answer:
Since we are asked “what is the approximate probability that x/y is an integer,” we should suspect that the exact answer will not be easy to calculate and that we’ll need to estimate somewhere along the way.
First, let’s figure the total number of possibilities. Each of the two variables is chosen at random from 100 possibilities (integers between 1 and 100 inclusive). Thus, the total possibilities for the x/y expression (unreduced) are 100 × 100 = 10,000 in number.
Next, how many of these possibilities result in an integer? Let’s take easy cases first, looking at the denominator and developing a pattern.
y = 1: Every x works, so we have 100 cases already.
y = 2: All even x’s work, so we have another 50 cases. We can now eliminate (A) definitively, since we’re already at 1.5%.
y = 3: We have exactly 33 cases (x = 3 through x = 99, inclusive). The pattern should be clear now: we divide 100 by y and round down to the nearest integer. Alternatively, we find the greatest multiple of y that is less than or equal to 100.
y = 4: 25 cases. Cumulatively, we are now at 208, or slightly over 2%.
5: 20 cases
6: 16 (since 6 × 16 = 96)
7: 14 (since 7 × 14 = 98)
8: 12 (since 8 × 12 = 96)
9: 11 (since 9 × 11 = 99)
10: 10
11: 9
12: 8 (we might now observe the way the pattern has flipped)
Pausing here, we can do some more sums. 20 + 16 + 14 = 50. 12 + 11 + 10 + 9 + 8 = 50 also. So we have another 100 cases exactly, bringing us to slightly over 3%.
13 and 14: 7 cases each, or 14 total
15 and 16: 6 cases each, or 12 total
17 through 20: 5 cases each for 4 numbers, or 20 total
21 through 25: 4 cases each for 5 numbers, or 20 total
26 through 33: 3 cases each for 8 numbers (33 – 26 + 1), or 24 total
34 through 50: 2 cases each for 17 numbers (50 – 34 + 1), or 34 total
51 through 100: 1 case each for 50 numbers, or 50 total
Glancing over these last numbers, we should see that the sum of the last 3 (50 + 34 + 24) is slightly greater than 100, and then the others (20, 20, 12, and 14) sum to ~70. So we can estimate that we have about 500 cases of integer quotients. Out of 10,000 cases, that works out to 5%. The exact number is 4.82%, but you shouldn’t calculate that precisely.
There’s no real shortcut here, by the way, short of calculus (and that’s not really a shortcut, to be honest!). The key here is to quickly ascertain that you need to crunch a lot of numbers that fall into something of a pattern, but not so much of one that you can entirely avoid grunt work. With just a little work, you can eliminate A, and then as the numbers progress, you should start to get the sense that you’ll wind up with either B or C.
The correct answer is C.
