What is the median of a set of 10 positive integers?

Question

(1) The numbers are in an AP with the first term being 10
(2) The arithmetic mean of these 10 numbers is 45

Skywalker18 · Answer

Median of set 10 positive integers(sorted in order) will be average of 5th and 6th term
(1) The numbers are in an AP with the first term being 10
First term , a = 10
We do not have any information about the common difference of the AP
Not sufficient

(2) The arithmetic mean of these 10 numbers is 45
Sum of 10 numbers = 45*10 = 450
Not sufficient

Combining 1 and 2 , we get
Last term of AP= 80
Last term , an = a+ (n-1)d
=>80 = 10 + 9d
=> 9d = 70
We can calculate the terms of AP -5th and 6th
Hence , we can calculate the median.

Answer C

nalinnair · Answer

If d = 70/9, then other 8 numbers (apart from 10 and 80) cannot be integers. 
This is not in line with what is given in the question stem.

MathRevolution · Answer

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

What is the median of a set of 10 positive integers?

(1) The numbers are in an AP with the first term being 10
(2) The arithmetic mean of these 10 numbers is 45

In the original condition, there are 10 variables, which should match with the number of equations. So you need 10 equations. For 1) 1 equation, for 2) 1 equation, which is likely to make E the answer.
When 1) & 2), L=80 is derived from (10+L)/2=45. Then, you can figure out the 10 variables, which is sufficient. C is the answer.
In fact, this is an integer question, which is one of the key questions and apply the mistake type 4. However, you can't apply the mistake type in this case.
Therefore, the answer is C.

 For cases where we need 3 more equations, such as original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 80% chance that E is the answer (especially about 90% of 2 by 2 questions where there are more than 3 variables), while C has 15% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since E is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, C or D.

nalinnair · Answer

MathRevolution  and  Skywalker18 
I agree that using both the statements we can figure out that: 1) the 10 positive integers are in AP; and 2) the first and last terms are 10 and 80 respectively. We also know that when a set of numbers is in AP, then arithmetic mean of the set = median of the set. So, the answer is C.
However, if first term = 10 and last term = 80, then the common difference = 70/9. This implies that other numbers in the Set will not necessarily be integers. This goes against what is given in the question stem (i.e. all 10 numbers are integers). Can you please explain this discrepancy? Is there any inadvertent error in the question?

MathRevolution · Answer

Sorry, probably numbers of 1) and 2) should be different.

jn30 · Answer

Can you please explain what AP is? I am new to the boards and haven't seen this abbreviation. Thanks!

nalinnair · Answer

AP is short for Arithmetic Progression also known as an evenly spaced set. You can read about it here:  https://gmatclub.com/forum/math-sequences-progressions-101891.html

PrakharGMAT · Answer

Hi Experts / chetan2u / Bunuel ,

S1 and S2 are not sufficient individually
Combining 1 and 2 we get few things-
1) First term= 10
2)Last term = 80 (10th term)

a+ 9d= 10th term = 80
we get d = 70/9 which is not integer

Ideally, If there is an AP in that case Mean = Median
So, Median should also be 45
But, if we talk about actual series,what would that be...?

Whats going on in this question.

chetan2u · Answer

Hi,
you have correctly found the following terms--
 First term=10
last term=80..
median=45
and d= 70/9.. 
 now where does this median lie-- between 5th and 6th term.. 
5th term= 10+4*70/9=370/9..
6th term= 10+5*70/9=440/9..
so median = (5th +6th)/2= (370/9 + 440/9)/2= (810/9)/2=90/2 =45..
So here too ans is 45..

NOTE as a afterthought- the set is of positive numbers and not positive integers..

PrakharGMAT · Answer

Hi  chetan2u ,

But, if we look on question
What is the median of  a set of 10 positive integers ?

Here, its not the set of +ve integers.

Isn't the question contradicts itself...??
Bcz..GMAT standard question doesn't do this..

Do you agree with me ..??

chetan2u · Answer

Hi,

You are absolutely correct on this point....

the median is in CENTER of 5th and 6th number..
so if I take each gap between two numbers as 2, we have 9 gaps between the first number and the average of this AP..

1)so if we take FIRST term as 10, average should be 10 + 9*x where x is a positive integer.. 
here 46 would have worked fine since 10+9*4=46..
so each succeeding number will be 4*2 more..
the set would have been 10, 18, 26, 34, 42, 50, 58, 66, 74, 82..
average= (42+50)/2=46

2) Incase the average is taken as 45.. 
the first term should be 45-9*x, where x is an integer
so closer to 10 would be 45-9*4=9..
series= 9, 17, 25, 33, 41, 49, 57, 65, 73, 81
median= (41+49)/2=45

Bunuel , Please change positive integers to positive number as the set does not consist of integers..

nisthagupta28 · Answer

Can someone explain as to how the last term = 80 is calculated?

What is the median of a set of 10 positive integers?

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Data Sufficiency (DS) Questions

What is the median of a set of 10 positive integers?

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards