# Math Multiple Choice

If *a* and *b* are integers greater than 100 such that *a* + *b* = 300, which of the following could be the exact ratio of *a* to *b* ?

- (A) 9 to 1
- (B) 5 to 2
- (C) 5 to 3
- (D) 4 to 1
- (E)
**3 to 2**CORRECT ANSWER

##### Explanation:

To solve this question, you need to look at the answer choices. For any of the answer choices to be the ratio of *a* to *b*, some multiple of the sum of the two numbers must evenly divide 300. For example, if the ratio of *a* to *b* equaled 9 to 1, then *a* would equal 9*x* and *b* would equal *x* for some number *x*. Furthermore, 9*x* + *x* would have to equal 300. This is possible since 10*x* = 300 yields an integer solution, namely *x* = 30. However, if *x* = 30, then *a* would equal 270 and *b* would equal 30. Although the sum of these numbers equals 300, they do not satisfy the other condition in the problem. That is, both of these numbers are not greater than 100. Therefore, choice (A) can be eliminated.

Answer choices (B) and (C) can be eliminated since neither the sum of the two numbers in (B) nor the sum of the two numbers in (C) evenly divided 300. (5*x* + 2*x* = 300 does not yield an integer solution, nor does 5*x* + 3*x* = 300.)

Although answer choices (D) and (E) are possible ratios of *a* to *b* (both 4*x* + *x* = 300 and

3*x* + 2*x* = 300 yield integer solutions), (D) results in *a* = 240 and *b* = 60 and can be eliminated since 60 is not greater than 100.

Only choice (E) gives a correct ratio of *a* to *b* that satisfies all of the conditions in the problem. For (E), *a* = 180 and *b* = 120, and both integers are greater than 100.

(from the October 12, 1999 test)