A university department tracked the number of research articles published by its 8 faculty… — GRE General Test

A university department tracked the number of research articles published by its 8 faculty members over a five-year period. The numbers of publications for 7 of the faculty members were $4, 7, 9, 12, 15, 18,$ and $23$ . If the arithmetic mean of the number of publications for all 8 faculty members is equal to $1.25$ times their median, what is the number of publications for the 8th faculty member?

Answer: 47 publications

Answer

The sum of the 7 known values is

88

, making the total sum

88 + x

and the arithmetic mean

\frac{88 + x}{8}

. Assuming

x \ge 15

, the 4th and 5th numbers when sorted are

12

and

15

, yielding a median of

13.5

. Setting the mean equal to

1.25 \times 13.5 = 16.875

gives

\frac{88 + x}{8} = 16.875

, which simplifies to

88 + x = 135

and yields

x = 47

Step-by-Step Solution

Calculate the sum of the 7 known data points and express the mean in terms of the unknown 8th value

x

Sum of 7 known values =

4 + 7 + 9 + 12 + 15 + 18 + 23 = 88

. Total mean =

\frac{88 + x}{8}

The arithmetic mean of

n

values is the sum of all values divided by

n

Analyze the position of

x

in sorted order to determine the median.

Assuming

x \ge 15

, the 4th and 5th values in ascending order are

12

and

15

, giving a median of

\frac{12 + 15}{2} = 13.5

For an even number of data points (

8

), the median is the average of the 4th and 5th terms in sorted order.

Formulate and solve the equation linking the mean and median.

\frac{88 + x}{8} = 1.25 \times 13.5 = 16.875 \implies 88 + x = 135 \implies x = 47

The problem specifies that the mean is equal to

1.25

times the median.

Key Concept

Calculating mean and median of a dataset containing an unknown value.

Estimated Time:1m 30s

Question

Answer

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