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PS: Mean

by GMAT Kolaveri » Mon Apr 30, 2012 8:17 pm
When a person aged 39 is added to a group of n people, the average age increases by 2. When a person aged 15 is added instead, the average age decreases by 1. What is the value of n?

(A) 7
(B) 8
(C) 9
(D) 10
(E) 11
Want to know alternate/time saving method to solve this problem



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by niketdoshi123 » Mon Apr 30, 2012 8:55 pm
Let the avg age be = Y
y+2 = (ny+39)/n+1
=> ny+2n+y+2=ny+39
=> 2n+y = 37 -----(1)

y-1 = (ny+15)/n+1
=> ny+y-n-1 = ny+15
=> y-n = 16 ----- (2)

Subtracting eq(2) from eq(1)
2n+y-y+n = 37-16
=> 3n = 21
=> n = 7

Hence the answer is A

The options are consecutive integers so guessing can't be done.
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by GMATGuruNY » Mon Apr 30, 2012 9:18 pm
GMAT Kolaveri wrote:When a person aged 39 is added to a group of n people, the average age increases by 2. When a person aged 15 is added instead, the average age decreases by 1. What is the value of n?

(A) 7
(B) 8
(C) 9
(D) 10
(E) 11
Want to know alternate/time saving method to solve this problem
Source: Veritas
Average = sum/number.
Let s = the current sum.
n = the current number of people.
Current average = s/n.

When the sum increases by 39 and the number of people increases to n+1, the average increases by 2:
(s+39)/(n+1) = s/n + 2.

When the sum increases by 15 and the number of people increase to n+1, the average decreases by 1:
(s+15)/(n+1) = s/n - 1.

Subtracting the second equation from the first, we get:
(s+39)/(n+1) - (s+15)/(n+1) = (s/n + 2) - (s/n - 1)

(s + 39 - s - 15) / (n+1) = 3

24/(n+1) = 3

24 = 3n +3

21 = 3n

n=7.

The correct answer is A.
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by GMAT Kolaveri » Tue May 01, 2012 12:19 am
@Mitch

Can you tell us alternate ways to solve this one.
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by aneesh.kg » Tue May 01, 2012 3:33 am
Popular Method:

Let T be the initial total of all ages.
Then
(T + 39) / (n + 1) = (T/n) + 2 -- (1)
(T + 15) / (n + 1) = (T/n) - 1 -- (2)
Subtracting (2) from (1),
24/ n+ 1 = 3
n = 7

(A) is the answer.

Alternate method:

If x is the initial average, then the rise in average is (39 - x)/(n + 1)
and the fall in average is (x - 15)/(n + 1)
since the ratio of rise:fall = 2:1
(39 - x)/(x - 15) = 2/1
or, we can also say that x must be dividing the separation between 39 and 15 in the ratio 2:1.
so x = 23 (8 ahead of 15, and 16 behind 39)

We can now use the value of x to find n.

(23n + 39)/(n + 1) = 23 + 2
n = 7
Last edited by aneesh.kg on Tue May 01, 2012 4:15 am, edited 1 time in total.
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by GMATGuruNY » Tue May 01, 2012 3:57 am
GMAT Kolaveri wrote:When a person aged 39 is added to a group of n people, the average age increases by 2. When a person aged 15 is added instead, the average age decreases by 1. What is the value of n?

(A) 7
(B) 8
(C) 9
(D) 10
(E) 11
Want to know alternate/time saving method to solve this problem
Source: Veritas
We could use alligation to determine the original mean.

Whereas adding 15 DECREASES the mean by 1, adding 39 INCREASES the mean by 2.
Thus, adding 39 changes the mean by TWICE AS MUCH as does adding 15.
This means that 39 is TWICE AS FAR from the mean as is 15:

15........x..........M..............2x.............39

Distance between 39 and 15 = 39-15 = 24.
Total distance = 3x.
Thus:
3x = 24
x = 8.
Thus, the original mean = 15+8 = 23.

Thus:
When there are n people, the average age is 23.
When the sum of the ages increases by 39, the average increases by 2 to 25.
Thus, the correct number of people must yield a multiple of 23 that is 39 less than a multiple of 25.

Looking at the answer choices, which represent the original number of people:
The smallest multiple of 23 will be yielded by A:
7*23 = 161.
161+39 = 200, which is a multiple of 25.
Success!

The correct answer is A.
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