Bunuel wrote:

A library branch originally contained 18360 volumes, 30% of which were fiction novels. 1/3 of the volumes were transferred to another location and 1/3 of the volumes transferred were fiction novels. What percent of the remaining collection was fiction novels?

A. 2.5%

B. 17.67%

C. 28.3%

D. 45.2%

E. 73.6%

VERITAS PREP OFFICIAL SOLUTION:Students who notice the difficult, ugly numbers in this problem and the large spread in the answer choices can take a shortcut to the correct answer. Doing so relies on estimation and on our understanding of addition and subtraction with ratios.

When adding to -- or subtracting from -- a ratio, the ratio will be unchanged if and only if we add or subtract in the same ratio as was initially given. In other words, had the portion of novels among the removed books been 30% (equal to the original share of novels), the ratio of novels to other books would not have changed, remaining at 30%. But the percentage of novels removed was slightly higher than its "normal share" -- it was 33.3% and not 30%. So the share of remaining novels has to be slightly less than the original 30%. The only answer close is 28.3%, which is C.

Another option is to treat this problem as a mixture problem and do a weighted average. The entire collection of the library branch can be seen as the combination of two subsets -- the set of books that were removed, and the set of books that remained. The concentration of novels in the removed books was 33.3%, the concentration of novels in the remaining books is unknown (call it x), and the concentration of novels in the combined collection (the weighted average) was 30%. We can set up a tug-of-war:

x -------------------- 30% ---------------------------------------- 33.3%

Since there are twice as many books in the remaining set (2/3 of all the books) as in the removed set (1/3 of all the books), the distance between x and 30% should be half the distance between 30% and 33.3%. The latter distance is roughly 3.3%, so the former distance is roughly 1.7%. Finally, x must be 30%−1.7%=28.3%, which is C.

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