The given question is a GMAT hard math data sufficiency question testing concepts in Statistics. Concepts covered: Mean, Median, and Range.

This data sufficiency problem consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements, plus your knowledge of mathematics and everyday facts (such as the number of days in a leap year or the meaning of the word counterclockwise), you must indicate whether -

- Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
- Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
- BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
- EACH statement ALONE is sufficient to answer the question asked.
- Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

All numbers used are real numbers.

A figure accompanying a data sufficiency question will conform to the information given in the question but will not necessarily conform to the additional information given in statements (1) and (2)

Lines shown as straight can be assumed to be straight and lines that appear jagged can also be assumed to be straight

You may assume that the positions of points, angles, regions, etc. exist in the order shown and that angle measures are greater than zero.

All figures lie in a plane unless otherwise indicated.

In data sufficiency problems that ask for the value of a quantity, the data given in the statement are sufficient only when it is possible to determine exactly one numerical value for the quantity.

Question 27: What is the range of 5 distinct single digit positive integers if their average is 5?

**Statement 1**: Their median is 6.

**Statement 2**: The average of the 3 largest among the 5 numbers is 7.

@ INR

**Q1. What kind of an answer will the question fetch?**

The question asks us to find the range of 5 distinct single digit positive integers if their average is 5.

The data provided in the statements will be considered sufficient if, from the statements, we are able to find a unique answer for the range of 5 positive integers.

**Q2. What information do we have about x from the question stem ?**

1) The given numbers are distinct single digit positive integers. So, these numbers take values from 1 to 9, inclusive.

2) Let the 5 numbers in ascending order be a, b, c, d, and e.

3) The average of these numbers is 5. The sum of the numbers a + b + c + d + e = 25

If we can find a unique value for (e – a) the data is sufficient.

__Note:__ Getting a unique value for (e – a) does not necessarily mean that we have to get a unique value for each of ‘a’ and ‘e’.

**Statement 1**: Their median is 6.

So, a, b, 6, d, e are the 5 numbers. Therefore, a + b + d + e = 19

Because d and e are greater than 6, the following possibilities exist : (d, e) could be (7, 8), (7, 9), and (8, 9)

__Possibility 1__: If (d, e) = (7, 8): a + b + 7 + 8 = 19 or a + b = 4

The only value that (a, b) can take is (1, 3)

Range of the 5 numbers is 8 - 1 = 7

__Possibility 2__: If (d, e) = (7,9): a + b + 7 + 9 = 19 or a + b = 3

The only value that (a, b) can take is (1, 2).

Range of the 5 numbers is 9 - 1 = 8

__Possibility 3__: If (d, e) = (8, 9): a + b + 8 + 9 = 19 or a + b = 2

No values of (a, b) that are distinct positive integers will satisfy this case.

So, possibility 3 is infeasible.

We are **not able to find a UNIQUE value** for the range using Statement 1.

Hence, statement 1 is not sufficient.

__Eliminate answer options A and D__.

**Statement 2**: The average of the 3 largest among the 5 numbers is 7.

c, d, and e are the 3 largest of the 5 numbers.

Therefore, c + d + e = 21 and a + b = 4

Only possible value for (a, b) = (1, 3). So, a = 1.

However, c, d, and e can take different values. Let us list down possibilities.

__Possibility 1__: c = 5, d = 7, e = 9. Range is 9 - 1 = 8

__Possibility 2__: c = 6, d = 7, e = 8. Range is 8 - 1 = 7

__Possibility 3__: c = 4, d = 8, e = 9. Range is 8 - 1 = 7

We are **not able to find a UNIQUE value** for the range using Statement 2.

Hence, statement 2 is not sufficient.

__Eliminate answer option B__.

**Statements**: **From Statement 1**: Their median is 6.

**From Statement 2**: The average of the 3 largest among the 5 numbers is 7.

**Key inferences**: From statement 1: 'c' has to be 6.

From statement 2: (a, b) has to be (1, 3)

So, 1 + 3 + 6 + d + e = 25 or d + e = 15

d > 6 and e > d. The only possible values are d = 7 and e = 8.

Hence, the range is 8 - 1 = 7.

We are **able to find a UNIQUE value** for the range using the statements together.

Statements together are sufficient.

__Eliminate answer option E__.

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