Manhattan GMAT Challenge Problem of the Week - 28 Jan 10

by , Jan 28, 2010

Welcome back to this week's Challenge Problem! As always, the problem and solution below were written by one of our fantastic instructors. Each challenge problem represents a 700+ level question. If you are up for the challenge, however, set your timer for 2 minutes and go!

Question

The consumer price index in Zeropia in 2009 relative to the year 2000 was 1.75, meaning that for every Zeropian dollar spent on consumer goods in 2000, $1.75 on average had to be spent in 2009. In Zeropian dollars, what was the increase in the price of Brand Z running shoes from 2000 to 2009, if these shoes price increased precisely according to the consumer price index?

(1) The price of Brand Z running shoes was $91 in 2009.

(2) The ratio of the dollar increase in the price of Brand Z running shoes to the price of the shoes in 2009 was 3:7.

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

(D) EACH statement ALONE is sufficient.

(E) Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Solution

The consumer price index gives us a ratio between prices in 2000 and prices in 2009. We are told that "for every Zeropian dollar spent on consumer goods in 2000, $1.75 on average had to be spent in 2009." In other words, if something cost x dollars in 2000, it cost 1.75x dollars in 2009 (as long as the price increased exactly according to the index, which is just an average). In dollar terms, the increase in price would then be 1.75x x = 0.75x dollars.

We are asked for this dollar price increase for Brand Z running shoes. Representing the price of these shoes in 2000 as x, as we already have, we can rephrase the question as "What is 0.75x?" We can further rephrase this question to "What is x?"

(1) SUFFICIENT. We are told that the price of the shoes in 2009 is $91. We have represented the 2009 price as 1.75x dollars, staying consistent with our variable naming throughout the problem (never change variable designations midstream unless youre starting over completely). So we can write an equation:

1.75x = 91

We know we can solve for x, so we can answer the question. (Incidentally, if we had to solve for this x on a Problem-Solving problem, one fast way would be to convert 1.75 to a fraction. 1.75 = 7/4, so we can quickly write that x = (91)(4/7). Since 91/7 = 13, we get x = (13)(4) = 52.)

(2) INSUFFICIENT. We are told that the price increase in dollar terms, divided by the price of the shoes in 2009, is 3/7. However, this information is already completely implied by the stem. If the index is 1.75, then any goods price increase was 75%, or 75 cents for every 2000 dollar. Since the 2009 price is $1.75 for every 2000 dollar, the ratio of the price increase ($0.75) to the 2009 price ($1.75) will always be 0.75/1.75, or 3/7. This holds true no matter what the original 2000 price is, so we cannot determine x through this bit of redundant information.

The correct answer is (A) Statement 1 by itself is sufficient to answer the question, but Statement 2 is not sufficient.

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