# Statistics - Range - Problem Solving

10 distinct integers are arranged in ascending order. The range of the first 7 is 25 and that of last 7 is 30. What is the maximum range of the 10 numbers?
A. 30
B. 25
C. 55
D. 27
E. 52

Correct Answer is Choice E. 52.

Let the smallest of the 10 numbers be X1 and the largest be X10

The range for the 10 numbers is maximized when the smallest number is X1 is minimized and the largest number X10 is maximized.

Let us assume X1 to be 1.
The range of the first 7 numbers is 25. So, X7 = X1 + 25 = 1 + 25 = 26.

The range of the last 7 numbers is 30.
i.e., X10 = X4 + 30.
Our objective is to maximize X10.
X10 is 30 more than X4.
Therefore, if X4 is maximized, X10 will also be simultaneously maximized.

These 10 numbers are distinct integers.
Now let us find out what is the maximum value that X4 can take.

We know X7 is 26.
Hence, X6 cannot be more than 25; X5 cannot be more than 24 and X4 cannot be more than 23.
So, X4 can take a maximum value of 23.

So, max X10 = 23 + 30 = 53.

The range of the 10 numbers is X10 - X1 = 53 - 1 = 52.