Alice has 3 times the number of stamps t that Doris does an

Before each person increased her number of stamps, let's let the number of stamps that Alice has be A, let's let the number of stamps that Doris has be D, and let's let the number of stamps that Jane has be J.

Since Alice has 3 times the number of stamps that Doris does, . Since Jane has 7 stamps more than Doris does, . We have more variables than equations, so these two equations cannot be solved for all the variables. We should keep this in mind while looking at the statements.

Let's consider statement I. We know that and . Let's see if it must be true that after each person increases her number of stamps by 4, Alice has more stamps than Jane. If , then , and . After each person increases her number of stamps by 4, Alice has stamps, Doris has stamps, and Jane has stamps. In this case, where Alice has 7 stamps and Jane has 12 stamps, it is not true that Alice has more stamps than Jane. Eliminate choices (C) and (E) which contain statement I.

Next, consider statement II. We saw in statement I that if before each person increases her number of stamps Doris has 1 stamp, then after each person increases her number of stamps by 4, Alice has 7 stamps, Doris has 5 stamps, and Jane has 12 stamps. So Jane has stamps more than Doris, and statement II, which says that Jane has 3 more stamps than Doris, is not true. Eliminate choice (D) which contains statement II.

We can see that since before each person increases her number of stamps by 4, Jane has 7 stamps more than Doris, after each of Jane and Doris increase her number of stamps by 4, Jane will still have 7 more stamps than Doris. Again, statement II is not true.

Now let's consider statement III. We know that and . If , then and . After each person increases her number of stamps by 4, Alice will have stamps and Doris will have stamps. After each person increases her number of stamps by 4, the total number of stamps that Alice and Doris will have is 7 + 5 = 12, which is a multiple of 4, since 12 = 3 4. If we let , then and . After each person increases her number of stamps by 4, Alice will have 6 + 4 = 10 stamps and Doris will have 2 + 4 = 6 stamps. After each person increases her number of stamps by 4, the total number of stamps that Alice and Doris will have is 10 + 6 = 16, which is a multiple of 4, since 16 = 4 4. If we select any other set of values for A, D, and J that are consistent with the question stem, we will find that the sum of the numbers of stamps that Alice and Doris have after each person increases her number of stamps by 4 is a multiple of 4. Statement III must be true. Choice (B), III only, is correct.

It can be shown algebraically that statement III must be true. Before each person increases her number of stamps by 4, Alice has 3D stamps and Doris has D stamps. After Alice increases her number of stamps by 4, she will have 3D + 4 stamps. After Doris increases her number of stamps by 4, she will have D + 4 stamps. The total number of stamps that Alice and Doris have after each person increases her number of stamps by 4 is . Since D is an integer, 4D is a multiple of 4. The integer 8 is a multiple of 4, since 8 = 2 4. Since 4D and 8 are both multiples of 4, 4D + 8 must be a multiple of 4 because the sum of two multiples of 4 must be a multiple of 4.