sidhu4u wrote:

Last month 15 homes were sold in Town X. The average (arithmetic mean) sale price of the homes was $150,000 and the median sale price was $130,000. Which of the following statements must be true?

I. At least one of the homes was sold for more than $165,000.

II. At least one of the homes was sold for more than $130,0000 and less than $150,000

III. At least one of the homes was sold for less than $130,000.

A. I only

B. II only

C. III only

D. I and II

E. I and III

We are given that 15 homes were sold in Town X last month. The average (arithmetic mean) sale price of the homes was $150,000 and the median sale price was $130,000. We can start by reducing $150,000 and $130,000 by dividing each number by 1,000.

We now have that the mean sale price of the homes was $150 and the median sale price was $130. Let’s now analyze the statements to determine which MUST be true.

I. At least one of the homes was sold for more than $165,000.

(We can reinterpret Roman numeral I as: At least one of the homes was sold for more than $165.)

To determine whether the above statement MUST be true, let’s see if we can find a scenario in which none of the homes is priced at more than $165. Furthermore, since the median price is $130, let’s create a scenario in which the eighth value of the 15 values (i.e., the middle number) is 130, the first seven values are all 130 and the last seven values are all 165. That is:

130, 130, 130, 130, 130, 130, 130, 130, 165, 165, 165, 165, 165, 165, 165, 165

If this is the case, then the sum would be 130 x 8 + 165 x 7 = 1,040 + 1,155 = 2,195. However, since the average of price of the homes is $150, the sum of the prices of the homes is 150 x 15 = 2,250. This means that at least one of the numbers in the list above has to be changed to a number greater than 165 to make up the difference between 2,195 and 2,250. (For example, since the difference is 55, we can change the last number 165 to 220 to make up this difference.)

So Roman numeral I is correct.

II. At least one of the homes was sold for more than $130,000 and less than $150,000.

(We can reinterpret II as: At least one of the homes was sold for more than $130 and less than $150.)

In the analysis of Roman numeral I, we showed that none of the homes had to be priced between $130 and $165. Roman numeral II is not correct.

III. At least one of the homes was sold for less than $130,000.

(We can reinterpret III as: At least one of the homes was sold for less than $130.)

In the analysis of Roman numeral I, we showed that none of the homes had to be priced less than $130. Roman numeral III is not correct.

Answer: A

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Scott Woodbury-Stewart

Founder and CEO

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