Explanation

This question is pretty standard word problem material. If C is the number of units of corn and R is the number of units of rice, than Pat earns a total of dollars and the number of units combined is . We are looking for C. On to the statements, separately first.

Statement (1) tells us that . We don't have enough information to solve completely, since we don't have, say, the total number of units, giving two equations and two variables. Or do we? This question is a good example of why we recommend at least checking an answer by analysis of cases if you have the time. Because considering actual possible values of C and R encourages us to note that C and R must be integers. That's what this unopened package business is all about. A single equation is insufficient to solve for two variables when those variables can be any real number, but it may be sufficient if we know that both variables must take on integer values. So, we want to check cases for in which C and R are integers. We'll note two fairly different ways of getting to the answer from this point:

You might notice that the prices are distinct prime numbers. It's always worth keeping an eye out for prime numbers, because where they are present, there are fewer possible cases, and it may be possible to solve exhaustively by cases. To do that, we obtain the prime factorization of and . We can confirm that 47 is a prime number by noting that it's smaller than 49 and checking possible factors of 7 and below down to 2. That means that . All of the numbers in this equation are prime. 17 and 13 must be multiplied and added to a multiple of . Two 13's and a 17 doesn't yield 47, but two 17's and a 13 does. That means, in this instance, there must be twice as many 17's as 13's. We can then write and find that .

Suppose that you hadn't noticed or didn't take the prime factor approach. You might not have found possible integer solutions for R and C after trying a few values. One trick is to look at Statement (2) to get an idea for case. We are evaluating Statement (1) alone currently, so looking at Statement (2) violates the "separately first" maxim, but we can look at Statement (2) just for an idea of how to build a case, not in order to take Statement (2) as a piece of data. This gives the idea that , and that leads straight back into the case we just discussed.

Statement (2) tells us that , but without the benefit of Statement (1), we don't know the total value of these packages. The number of packages could be 6 and 12, or 5 and 10, yielding different answers to the question. We don't have sufficient information to answer definitively here, so Statement (2) is insufficient.

The correct answer is (A).