The ratio of paperback to hardcover is 22 : 3. The question asks how many paperback books are on the shelf.
(1) SUFFICIENT: The ratio of P to H is 22 : 3 and there are between 202 and 247 books on the bookshelf. Logic this out:
Case 1: Multiply each part of the ratio by 10 to get 220 : 30. This adds up to 250 books, but the statement specified that there are at most 247 books, so this case isn't possible—discard it but also think about how to use it to help find a valid case.
Case 2: Multiply each part of the ratio by 9. That's an annoying number, so start from the prior case and subtract 22 paperbacks from the P part of the ratio and subtract 3 paperbacks from the H part of the ratio: (220 – 22) : (30 – 3) or (198 : 27). That adds up to 225 books, which is in the right range for the total number of books, so one valid set of values is P = 198 and H = 27.
Case 3: Do it again! If you were to subtract 22 P books and 3 H books, you'd be subtracting a total of 25 books. But the total from Case 2 is only 225 books; if you subtract 25 more books from that, you'll have 200 books. That's impossible, since this statement says that there are at least 202 books. So this case is also invalid.
The only valid case is Case 2, so there must be exactly 198 paperbacks and 22 hardcovers. This statement is sufficient to answer the question.
(2) SUFFICIENT: Currently, the ratio of P to H is 22 : 3. But if you removed 18 of the P's and replaced them with H's instead, the ratio would then be 4 : 1. When a problem tells you that changing specific values will result in a new ratio, set up two proportions, one "before" and one "after" the change:
Before: P / H = 22 / 3
After: (P – 18) / (H + 18) = 4 / 1
Cross multiply to make the equations easier to examine.
Before: 3P = 22H
After: 1(P – 18) = 4(H + 18)
P – 18 = 4H + (4)(18)
P = 4H + (4)(18) + 18
From here, it's possible to plug one equation into the other to solve for one of the variables (and then plug that value back into one of the original equations to solve for the other variable). You don't actually have to do this work, since this is Data Sufficiency; it's enough to know that it can be done. This statement is also sufficient to get to one specific set of values for P and H.
The correct answer is (D): Each statement is sufficient by itself to answer the question.
