When 15 is Appended To a List of Integers, The Mean is Increased by 2 GMAT Problem Solving

Question: When 15 is appended to a list of integers, the mean is increased by 2. When 1 is appended to the enlarged list, the mean of the enlarged list is decreased by 1. How many integers were in the original list?

1. 4
2. 5
3. 6
4. 7
5. 8

Correct Answer: A

Solution and Explanation
Approach Solution 1:
It is asked to find what integers were in the original list
If m is the mean, x is the list's initial sum, and n is the number of integers in the list, then
m = x/n ,
m + 2 = (x + 15) / (n + 1)
And
m + 1 = (x + 16) / (n + 2).
The initial equation is x = mn.
Consequently, changing x in the other two equations:
We will get
m + 2 = (mn + 15) / (n + 1) equation (1)
m + 1 = (mn + 16) / (n + 2).equation (2)
Now From equation (1):
(m + 2)(n + 1) = mn + 15
mn + m + 2n + 2 = mn + 15
m + 2n + 2 = 15
m + 2n = 13
m = 13 - 2n.
Same way when we substitute for m in equation (2):
13 - 2n + 1 = ( n(13 - 2n) + 16) / (n + 2)
14 - 2n = (13n - 2n2 + 16) / (n + 2) Multiplying through by n + 2:
(14 - 2n)(n + 2) = 13n - 2n2 + 16
14n + 28 - 2n2 - 4n = 13n - 2n2 + 16
10n + 28 = 13n + 16
12 = 3n
n = 4
There were 4 integers

Approach Solution 2: 
There is another approach to answering this question:
Let's tackle the problem using the fundamental idea of what mean is, which is- the value that can swap out every other value in a list.
The mean rises by 2 when the value 15 is added. This indicates that 15 has 2 extra (above mean) to offer to each person on the list as well as to itself.
The mean decreases by 1 each time a new value of 1 is added. It indicates that 1 has a sufficient deficit (below mean) to subtract 1 from each person on the list and still have 1 left over.
We can infer from the difference between 15 and 1 that the initial list of numbers was likely to contain 4 or 5.
For ex., if the initial list contained 4 no's, the original mean would be 5. As the no.15 includes ten extras (2 for each of the 4 members and 1 for itself). The mean changes to 7 when we add 15, and the list now has 5 numbers.
Now that 1 has been added, the list is 6 less than the mean (7), therefore all 5 numbers add 1 to the list to make it 6, bringing the average down to 6.
So It works.
Therefore, the list contained 4 numbers.

Approach Solution 3:
Let the number of integers initially be n, and their mean be x.

So, given:

nx+15/n+1= x+2
nx+15= (n+1)(x+2)= nx+2+x+2n
x+2n= 13

Also, on appending 1:
nx+15+1/n+2= x+2-1
nx+16/n+2= x+1
nx+16= (n+2)(x+1)- nx+2+n+2x
n+2x= 14
2n+4x= 28
2n+4x= 28
2n+x= 13
3x= 15=> x= 5
N= 14-2x= 14-10= 4

Therefore, number of integers in original list is n= 4.

“When 15 is appended to a list of integers, the mean is increased by 2.”- is a topic of the GMAT Quantitative reasoning section of GMAT. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. GMAT Quant practice papers improve the mathematical knowledge of the candidates as it represents multiple sorts of quantitative problems.

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