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 X_{1} and the largest be X_{10}.

The range for the 10 numbers is maximized when the smallest number is X_{1} is minimized and the largest number X_{10} is maximized.

Let us assume X_{1} to be 1.

The range of the first 7 numbers is 25. So, X_{7} = X_{1} + 25 = 1 + 25 = 26.

The range of the last 7 numbers is 30.

i.e., X_{10} = X_{4} + 30.

Our objective is to maximize X_{10}.

X_{10} is 30 more than X_{4}.

Therefore, if X_{4} is maximized, X_{10} will also be simultaneously maximized.

These 10 numbers are distinct integers.

Now let us find out what is the maximum value that X_{4} can take.

We know X_{7} is 26.

Hence, X_{6} cannot be more than 25; X_{5} cannot be more than 24 and X_{4} cannot be more than 23.

So, X_{4} can take a maximum value of 23.

So, max X_{10} = 23 + 30 = 53.

The range of the 10 numbers is X_{10} - X_{1} = 53 - 1 = 52.

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 X

The range for the 10 numbers is maximized when the smallest number is X

Let us assume X

The range of the first 7 numbers is 25. So, X

The range of the last 7 numbers is 30.

i.e., X

Our objective is to maximize X

X

Therefore, if X

These 10 numbers are distinct integers.

Now let us find out what is the maximum value that X

We know X

Hence, X

So, X

So, max X

The range of the 10 numbers is X

Labels: GMAT Descriptive Statistics, GMAT Range, GMAT Statistics