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.

Explanatory Answer

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.