There are five distinct positive integers such that

Question

There are five distinct positive integers such that their median is 9 and mean is 8. If the average of 2nd smallest and 2nd largest number is 8 then find the range of these five integers.

A. 24
B. 22
C. 15
D. 13
E. 10

shriwasakshat · Answer

Let the numbers are a, b, 9, d, e.

According to the question: (a+b+9+d+e)/5 = 8

a + b + 9 + d + e = 40 or a+b+d+e= 31......(1)

According to the question: (b + d) / 2 = 8 or b+d= 16....(2)

Equation (1) equation (2) gives: a + e = 15

To maximize the range we need to maximize the difference between e and a and we can do so by maximize the value of e and minimize the value of a.

As the numbers are positive integers, so the minimum possible value of a = 1 and maximum possible value of e = 15 - 1 = 14 Maximum possible range= e-a = 14-1=13.

GraemeGmatPanda · Answer

List the five integers in increasing order and identify the median.
 a < b < 9 < d < e

Express the sum of the five integers using the mean of 8.
 a + b + 9 + d + e = 5 	imes 8

Simplify the sum equation.
 a + b + 9 + d + e = 40

Use the given average of the second smallest and second largest integers.
 \frac{b + d}{2} = 8

Simplify to express  b + d .
 b + d = 16

Isolate the sum of the smallest and largest integers by removing the median from the total.
 a + b + d + e = 40 - 9

Simplify this expression.
 a + b + d + e = 31

Substitute  b + d = 16  to find  a + e .
 a + e = 31 - (b + d) = 31 - 16

Simplify to get the sum of the smallest and largest.
 a + e = 15

Determine the smallest integer under the ordering constraints (ensuring  d>9  when  b+d=16 ).
 a = 1

Find the largest integer using  a+e=15 .
 e = 15 - 1 = 14

Compute the range as the difference between the largest and smallest integers.
 	ext{Range} = e - a = 14 - 1

Finalize the numerical value of the range.
 	ext{Range} = 13

Answer 13, since the only valid set of numbers yields smallest integer 1 and largest integer 14, giving range 14 − 1 = 13.

ʕ•ᴥ•ʔ

Regor60 · Answer

Since the sum of the smallest and largest equals 15 for the reasons others describe, the range can't be >15, so the given answer is incorrect.

Additionally, since therefore the smallest is 15-largest, the range is:

Largest - (15-largest) =

2*largest - 15 = odd number.

The only answer choice that fits is 13

laborumplaceat · Answer

Let's assume the five distinct positive integers in increasing order be

a<b<9<d<e

Given conditions

Median = 9
Mean = 8
total sum = 5×8 = 40.

Average of 2nd smallest and 2nd largest is 8

(b+d)/2=8
⇒b+d=16

Use the sum
a+b+9+d+e=40

Substitute - b+d=16:
a+e+16+9=40
⇒a+e=15

Apply ordering constraints

b<9<d and b+d=16

This gives valid pairs:

(b,d)=(6,10),(5,11),(4,12),(3,13)

For each case,

a<b and e>d with a+e=15

Checking all valid combinations yields the following possible ranges:
e−a∈{7,9,11,13}

gchandana · Answer

Bunuel , can you kindly once check this question and help with a solution?
Out of the options mentioned, only 13 seem feasible. I am not sure what is happening here.

Bunuel · Answer

Yes, the OA is 13. Fixed the options.

AndyRugger · Answer

I won't post my whole solution, but there are a few tricks that made this (eventually) work for me. I didn't need to use max/min logic or test a bunch of ordered pairs.

One, recognize that you can find the sum of a set by multiplying Mean*Count, even if (as in this case) the set is skewed.

Two, after some plug n' chug work, you come to the equation: a + e = 15. What we need is, of course, e-a, the range.

Just subtract 2a from both sides, and you get e - a = 15-2a, which we know from the prompt must be an integer, and 2*a must also be an integer, and the range must necessarily be less than 15!

That only leaves two answer options: 10 or 13. A little backsolvong makes it clear: If the range = 10, then 2*a =5, and there is no integer solution to that equation!

So range = 13 it is.

cledgard · Answer

The question is NOT what is the maximum possible range. It is "what is the range?"
There are four possible answers for that: 7,9,11,13. 
13 is the only one that is in the selections.
However, the question should be changed.
Either to "What is the maximum possible range?" or to ¿Which of the following answers could be the range?"

There are five distinct positive integers such that

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