In a survey consisting of $8$ questions answered b... | Free GMAT Practice Question | GMAT Panda

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In a survey consisting of 8 questions answered by 8 respondents each, every response receives either 2 points or 3 points. For the n th question, exactly n respondents receive 2 points (and the rest receive 3 points), for n = 1, 2, \dots, 8 . What is the average score per response across all questions?

Explanation	Calculations	Help
$We first note how many respondents give 2 points and how many give 3 points for the nth question.$	$Number of 2-point responses for question n = n Number of 3-point responses for question n = 8 - n$
$We then calculate the total score for the nth question by multiplying counts by their point values.$	$2 n + 3 (8 - n) = 24 - n$
$The sequence of question sums from n=1 to n=8 is evenly spaced, so we apply Average and Median Property of Evenly Spaced Sets to find the average question sum.$	$First question sum = 24 - 1 = 23 Last question sum = 24 - 8 = 16 Average question sum = \frac{23 + 16}{2} = 19.5 Total sum of all responses = 19.5 \times 8 = 156$	Theory & Tactics Method Card EVEN1-C Click to view full details
$We finally divide the total sum by the total number of responses (8 questions \times 8 respondents = 64) to get the average per response.$	$\frac{156}{64} = \frac{4 \times 39}{4 \times 16} = \frac{39}{16} \frac{39}{16} = 2.4375 \approx 2.44$

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C

In a survey consisting of 8 questions answered by 8 respondents each, every response receives either 2 points or 3 points. For the n th question, exactly n respondents receive 2 points (and the rest receive 3 points), for n = 1, 2, \dots, 8 . What is the average score per response across all questions?

Explanation	Calculations	Help
$We first note how many respondents give 2 points and how many give 3 points for the nth question.$	$Number of 2-point responses for question n = n Number of 3-point responses for question n = 8 - n$
$We then calculate the total score for the nth question by multiplying counts by their point values.$	$2 n + 3 (8 - n) = 24 - n$
$The sequence of question sums from n=1 to n=8 is evenly spaced, so we apply Average and Median Property of Evenly Spaced Sets to find the average question sum.$	$First question sum = 24 - 1 = 23 Last question sum = 24 - 8 = 16 Average question sum = \frac{23 + 16}{2} = 19.5 Total sum of all responses = 19.5 \times 8 = 156$	Theory & Tactics Method Card EVEN1-C Click to view full details
$We finally divide the total sum by the total number of responses (8 questions \times 8 respondents = 64) to get the average per response.$	$\frac{156}{64} = \frac{4 \times 39}{4 \times 16} = \frac{39}{16} \frac{39}{16} = 2.4375 \approx 2.44$

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C