Last month, 15 homes were sold in Town $X$...... | Free GMAT Practice Question | GMAT Panda

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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.

Explanation	Calculations	Help
$We first name the individual sale prices in order from lowest to highest.$	$x_{1} \leq x_{2} \leq ... \leq x_{15}$	Theory & Tactics Method Card TRANS1-A Click to view full details
$We translate the text about the median sale price.$	$x_{8} = 130000$	Theory & Tactics Method Card TRANS1-B Click to view full details
$We translate the text about the average sale price.$	$\frac{x _{1} + x _{2} + ... + x _{15}}{15} = 150000$	Theory & Tactics Method Card TRANS1-B Click to view full details Theory & Tactics Method Card TRANS1-C Click to view full details
$We test Statement I by assuming no home sold for more than 165,000 USD and finding a contradiction with the required total.$	$x_{1} = ... = x_{8} = 130000 For 9 \leq i \leq 15 : x_{i} \leq 165000 8 \times 130000 = 1040000 7 \times 165000 = 1155000 1040000 + 1155000 = 2195000 2195000 < 2250000$	Theory & Tactics Method Card INEQ1-D Click to view full details
$We give an example showing Statement II need not hold.$	$x_{1} = ... = x_{8} = 130000 x_{9} = ... = x_{14} = 150000 x_{15} = 310000 8 \times 130000 + 6 \times 150000 + 310000 = 2250000$
$We note in this example that all sale prices are at least 130,000 USD, so Statement III need not hold.$	$x_{1} = ... = x_{8} = 130000 x_{9} = ... = x_{14} = 150000 x_{15} = 310000 For 1 \leq i \leq 15 : x_{i} \geq 130000$

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A

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.

Explanation	Calculations	Help
$We first name the individual sale prices in order from lowest to highest.$	$x_{1} \leq x_{2} \leq ... \leq x_{15}$	Theory & Tactics Method Card TRANS1-A Click to view full details
$We translate the text about the median sale price.$	$x_{8} = 130000$	Theory & Tactics Method Card TRANS1-B Click to view full details
$We translate the text about the average sale price.$	$\frac{x _{1} + x _{2} + ... + x _{15}}{15} = 150000$	Theory & Tactics Method Card TRANS1-B Click to view full details Theory & Tactics Method Card TRANS1-C Click to view full details
$We test Statement I by assuming no home sold for more than 165,000 USD and finding a contradiction with the required total.$	$x_{1} = ... = x_{8} = 130000 For 9 \leq i \leq 15 : x_{i} \leq 165000 8 \times 130000 = 1040000 7 \times 165000 = 1155000 1040000 + 1155000 = 2195000 2195000 < 2250000$	Theory & Tactics Method Card INEQ1-D Click to view full details
$We give an example showing Statement II need not hold.$	$x_{1} = ... = x_{8} = 130000 x_{9} = ... = x_{14} = 150000 x_{15} = 310000 8 \times 130000 + 6 \times 150000 + 310000 = 2250000$
$We note in this example that all sale prices are at least 130,000 USD, so Statement III need not hold.$	$x_{1} = ... = x_{8} = 130000 x_{9} = ... = x_{14} = 150000 x_{15} = 310000 For 1 \leq i \leq 15 : x_{i} \geq 130000$

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A