A dataset consists of 25 distinct integers arranged in increasing order. The mean of all 2… — GRE General Test

A dataset consists of $25$ distinct integers arranged in increasing order. The mean of all $25$ integers is $52$ . The mean of the smallest $12$ integers is $30$ , and the mean of the largest $12$ integers is $70$ .

If $5$ additional numbers, each equal to the median of the original dataset, are added to the dataset, what is the mean of the new set of $30$ numbers?

Answer: 60

Answer

The mean of the new set of 30 numbers is 60.

The sum of all 25 numbers is

25 \times 52 = 1300

. Since the numbers are ordered, the median is the 13th value. The 12 smallest numbers sum to

12 \times 30 = 360

and the 12 largest sum to

12 \times 70 = 840

. The sum of these 24 numbers is

360 + 840 = 1200

, so the 13th number (the median) must be

1300 - 1200 = 100

. Adding 5 copies of 100 increases the sum to

1300 + 500 = 1800

across

30

numbers. The new mean is

\frac{1800}{30} = 60

Step-by-Step Solution

Calculate the total sum of the original dataset of 25 numbers.

Sum =

25 \times 52 = 1300

The mean multiplied by the number of elements gives the total sum.

Calculate the combined sum of the 12 smallest and 12 largest integers.

Sum of 12 smallest =

12 \times 30 = 360

; Sum of 12 largest =

12 \times 70 = 840

; Total =

360 + 840 = 1200

The 25 numbers consist of the 12 smallest, the 1 median (13th element), and the 12 largest.

Determine the value of the median.

Median =

1300 - 1200 = 100

Subtracting the sum of the 24 non-median values from the total sum yields the 13th element, which is the median.

Find the sum and count of the modified dataset.

New Sum =

1300 + 5(100) = 1800

; New Count =

25 + 5 = 30

Adding 5 numbers each equal to 100 increases the sum by 500 and the count by 5.

Calculate the mean of the new dataset.

New Mean =

\frac{1800}{30} = 60

Divide the new total sum by the new total count of numbers.

Key Concept

Relationship between Mean, Median, and Total Sum in Partitioned Datasets

Estimated Time:2m 30s

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