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## Official Guide Explanation:Problem Solving #157

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: 157
Page: 174
Difficulty: 6 (Moderately Difficult)
Category 1: Algebra > Functions > other
Category 2: Arithmetic > Properties of Integers > Evens and Odds

Explanation: While one can use the given formula to find the sum of the even integers between 99 and 301, it isn't the easiest way.

For any set of evenly - spaced integers, the mean of the set is equal to the mean of the endpoints. So if the smallest and largest integers are 100 and 300, the mean is 200. To find the sum of a set, we need the mean and the number of terms, so we're already halfway there.

Next, the number of terms. Between 100 and 199, inclusive, for instance, there are exactly 100 integers, 50 of which are even. If we look at 100 to 200, inclusive, there are 51 evens. By the same principle, between 100 and 299, inclusive, there are exactly 200 integers, 100 of which are even. Include 300, and there are 101.

Thus, there are 101 terms, the mean of which is 200. The sum, then, is:

200(101) = 20,200, choice (B).

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