GMAT Practice Question: Last month, 15 homes were sold in Town X...
Question
Last month, 15 homes were sold in Town X. The average (arithmetic mean) sale price of the homes was 150,000 USD and the median sale price was 130,000 USD. Which of the following statements must be true? I. At least one of the homes was sold for more than 165,000 USD. II. At least one of the homes was sold for more than 130,000 USD and less than 150,000 USD. III. At least one of the homes was sold for less than 130,000 USD.
- I only
- II only
- III only
- I and II
- I and III
Topics: statistics, word problems, mean, median, must be true statements
Solution
Step 1
We first name the individual sale prices in order from lowest to highest.
x_1 \le x_2 \le ... \le x_{15}
Step 2
We translate the text about the median sale price.
x_8 = 130000
Step 3
We translate the text about the average sale price.
\frac{x_1 + x_2 + ... + x_{15}}{15} = 150000
Step 4
We test Statement I by assuming no home sold for more than 165,000 USD and finding a contradiction with the required total.
x_1 = ... = x_8 = 130000 \text{For }9 \le i \le 15:\; x_i \le 165000 8 \times 130000 = 1040000 7 \times 165000 = 1155000 1040000 + 1155000 = 2195000 2195000 < 2250000
Step 5
We give an example showing Statement II need not hold.
x_1 = ... = x_8 = 130000 x_9 = ... = x_{14} = 150000 x_{15} = 310000 8 \times 130000 + 6 \times 150000 + 310000 = 2250000
Step 6
We note in this example that all sale prices are at least 130,000 USD, so Statement III need not hold.
x_1 = ... = x_8 = 130000 x_9 = ... = x_{14} = 150000 x_{15} = 310000 \text{For }1 \le i \le 15:\; x_i \ge 130000
Answer
A
