BTGModeratorVI wrote: ↑Wed Dec 16, 2020 12:36 pm
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
Answer:
A
Source: official guide
Solution:
We are given that 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. We can simplify our considerations by reducing $150,000 and $130,000 to $150 and $130, respectively, by dividing each number by 1,000.
We now have that the mean sale price of the homes was $150 and that 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 rephrase 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 are priced at more than $165. Furthermore, since the median price is $130, let’s create a scenario in which the 8th value of the 15 values (i.e., the middle number) is 130, the first 7 values are all 130, and the last 7 values are all 165:
130, 130, 130, 130, 130, 130, 130, 130, 165, 165, 165, 165, 165, 165, 165, 165
If this were the case, the sum would be 130 x 8 + 165 x 7 = 1,040 + 1,155 = 2,195. However, since the average home price is $150, the sum of the home prices is 150 x 15 = 2,250. This means 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 rephrase II as: At least one of the homes was sold for more than $130 and less than $150)
When analyzing 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 rephrase III as: At least one of the homes was sold for less than $130.)
When analyzing Roman numeral I, we showed that none of the homes had to be priced at less than $130. Roman numeral III is not correct.
Answer: A
