All the Quant (2019) #1 on page 270

[sshin]

Hi,

My question is on Statement 2 of Problem Set #1 on page 270 of the All the Quant strategy guide.

1) A bookshelf holds both paperback and hardcover books. The ratio of paperback books to hardcover books is 22 to 3. How many paperback books are on the shelf?

Statement #2: If 18 paperback books were removed from the shelf and replaced with 18 hardcover books, the resulting ratio of paperback books to hardcover books on the shelf would be 4 to 1.

I tried using the ratio method on this problem (P:B is 22:3). After replacing 18 paperback books with 18 hardcover books, the new ratio is (P:B is 4:1). So then I saw that for paperback ratio, the removal of 18 books is equivalent to a change of 18 in the ratio (22 to 4), which means that the multiplier is 1. Similarly, hardcover books replacements are 18 books, which is equivalent to a change of 2 in the ratio (3 to 1), which means that the multiplier is 9.

If we use the multiplier of 1, then the total number of books would be 22(1) + 3(1) = 25. If we use the multiplier of 9, it's 22(9) + 3(9), which is 225 total books. Since there are two possibilities of how many paperback books can be as it can be either 22(1) or 22(9), shouldn't Statement #2 be Not Sufficient?

Thank you

[Sage Pearce-Higgins]

Thanks for expressing your working so clearly. Although we don't have to calculate the numerical solutions for DS problems, it can often be a useful exercise when reviewing problems. Let's take those two solutions:

If the number of paperbacks is 22 (the multiplier of 1), then there would be 3 hardbacks. If we remove 18 paperbacks and replace them with 18 hardbacks, then we'd get 4 paperbacks and 21 hardbacks. This doesn't agree with statement 2 and is therefore not a possible case.

If the number of paperbacks is 198 (the multiplier of 9), then there would be 27 hardbacks. If we remove 18 paperbacks and replace them with 18 hardbacks, then we'd get 180 paperbacks and 55 hardbacks. This is, indeed, a ratio of 4:1 and agrees with statement 2. It's the only case to do so.

Now to find where your thinking went wrong. You wrote that

So then I saw that for paperback ratio, the removal of 18 books is equivalent to a change of 18 in the ratio (22 to 4)

That's not correct! Remember that a ratio is a relationship, so that you can't just change one side of the ratio. A ratio of 22:3 means that we could have 22pb and 3hb, or 44pb and 6hb, etc. A ratio gives us real numbers only when we have the multiplier. There is an algebraic way to think about ratios (assign the letter x as the multiplier, so that the number of paperbacks is 22x and the number of hardbacks is 3x). This is a strategy we call the unknown multiplier in All the Quant. However, for a problem like this one it's not important.

I would encourage you to test cases here. As soon as you see the ratio 22:3, think "okay, so it could be 22 and 3, or 44 and 6, etc.". Further, the presence of real numbers in statement 2 is a hint that we're very likely to be able to calculate the exact ratio, even if we're not totally confident of the Math involved.