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## Official Guide Explanation:Data Sufficiency #123

Background

This is just one of hundreds of free explanations I've created to the quantitative questions in The Official Guide for GMAT Review (12th ed.). Click the links on the question number, difficulty level, and categories to find explanations for other problems.

These are the same explanations that are featured in my "Guides to the Official Guide" PDF booklets. However, because of the limitations of HTML and cross-browser compatibility, some mathematical concepts, such as fractions and roots, do not display as clearly online.

Question: 123
Page: 283
Difficulty: 5 (Moderate)
Category 1: Algebra > Linear Equations-Two Unk >
Category 2: Arithmetic > Properties of Integers > Other

Explanation: Algebraically, we can represent the total cost of the stamps as:

c = 0.15x + 0.29y

Statement (1) is sufficient. It doesn't initially appear to be sufficient, since all we can establish is the new equation:

4.40 = 0.15x + 0.29y

However, be careful any time the variables represent integers. There are infinite number of solutions to an equation like that one, but most of them have non - integer values. In this case, there's only one solution that includes integers for both variables. One warning sign is the unusual number "29" making an appearance.

For instance, no number of 29 cent stamps would cost 50 cents, or 100 cents. The only possible totals are multiples of 29:

29 cents, 58 cents, 87, etc.

Since any multiple of 15 ends in either a 5 or a 0 and 4.40 ends in a zero, the total price of the 29 cent stamps must end in a five or a zero as well. Thus, the number of 29 cent stamps must be a multiple of five:

0.29 * 5 = 1.45

0.29 * 10 = 2.90

0.29 * 15 = 4.35

We can quickly dismiss the last as impossible, because no number of 15 cent stamps could be added to 4.35 and result in 4.40.

Thus, we can check the other two. If the 29 cent stamps total \$1.45, that leaves \$2.95 for the 15 cent stamps. Two 15 cent stamps is 30 cents, which means that 20 would be \$3.00. We can't get to \$2.95 -- the only nearby options are \$2.85, \$3.00, and \$3.15.

That leaves only the middle choice: Ten 29 cent stamps. The 15 cent stamps must cost a total of \$1.50, which is the price of 10 such stamps. We can conclude that Joanna bought ten of each price of stamp.

Statement (2) is insufficient. This tells us that the total price must be a multiple of 44 cents (the total price of one of each stamp), but we don't know how many she bought. Choice (A) is correct.

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