Problem Solving

nakul_anand wrote:Last month, 15 homes were sold in town X. The average (Arithmetic Mean) sale price was $150000 and the median sale price was $130000. Which of the following is true?

(I) At least 1 home was sold for more than $165000
(II) At least 1 home was sold for more than $130000 but less than $150000.
(III) At least 1 home was sold for less than $130000


(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

nakul_anand wrote:Last month, 15 homes were sold in town X. The average (Arithmetic Mean) sale price was $150000 and the median sale price was $130000. Which of the following is true?

(I) At least 1 home was sold for more than $165000
(II) At least 1 home was sold for more than $130000 but less than $150000.
(III) At least 1 home was sold for less than $130000


(A) I only
(B) II only
(C) III only
(D) I and II
(E) I and III

The key word in this question is MUST. As in, which of the following MUST be true. So, if it's possible that one scenario is not true, we can eliminate it.

So, let's looks at one possible scenario and see which answer choices we can eliminate.

Aside: To make things simpler, let's divide all of the prices by 1000.

First, we'll use a nice rule that says: sum of all values = (mean)(number of values)
So, the sum of all 15 prices = ($150)(15) = $2250.

If the median is $130, then the middlemost value is $130

So, one possible scenario is:
130, 130, 130, 130, 130, 130, 130, 130, 130, 130, 130, 130, 130, 130, 130, 430

Aside: To find the last value (430), I took the sum of all 15 numbers (2250) and subtracted (14)(130)

Notice that this scenario tells us that II and III need not be true (since our scenario does not conform to either one).

Since answer choices B, C, D and E all include either II or III, we can eliminate them.

This leaves us with A, which must be the correct answer.

Cheers,
Brent