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Average- do it as fast as possible

by hey_thr67 » Fri Jun 08, 2012 9:14 am
In a group of 11 people, x is 32 years old and y is 4 years younger than x. If x and y are replaced by two other people, the average age of the group drops by 1 year. Find the average age of the two people replacing x and y.

1) 26
2) 28
3) 29
4) 30
5) 24.5
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by eagleeye » Fri Jun 08, 2012 9:27 am
First way:
Total age of x and y = 32+(32-4)=60.
Average age changes by 1, so total age changes by 11*1=11
Hence total of 2 replacement people's age must be 60-11 =49 which averages to 49/2 = 24.5


Second way:
Average of x (32) and y(4 years less) = 30 (mid way between 32, 28)
When those two are replaced, the average of group changes by 1 year, hence total age decreases by 11, so each person contributes 11/2 = 5.5 decrease.
Hence, average age of 2 people replacing x and y = 30-5.5=24.5


Observe that, you can do all calculations mentally, in fact they are more of observations than calculations.

Let me know if this helps :)
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by Birottam Dutta » Fri Jun 08, 2012 9:28 am
Let sum of ages of the other nine people be N.

Age of x=32 and age of y=32-4=28.

Average of 11 people = (N+32+28)/11=J(say)-------(1)

Let the ages of the two people replacing x and y be a and b.

New average = (N+ a+b)

By the question,(N+a+b)/11 = J-1------ (2)


Solving (1) and (2), we get (a+b) =49 which implies (a+b)/2 = 24.5

So, answer is E!
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by Mike@Magoosh » Fri Jun 08, 2012 9:29 am
hey_thr67 wrote:In a group of 11 people, x is 32 years old and y is 4 years younger than x. If x and y are replaced by two other people, the average age of the group drops by 1 year. Find the average age of the two people replacing x and y.

1) 26
2) 28
3) 29
4) 30
5) 24.5
Hi, there. I'm happy to help with this. :)

First of all, here's a blog I wrote about a general approach to "average questions" like the one you posted here:
https://magoosh.com/gmat/2012/gmat-avera ... -formulas/

Here's a solution to your question.
X is 32 years old
Y is 28 years old
so X + Y = 60.

Let's say the average of the 11 people is M. We know M = (sum of all 11 ages)/11, or
sum1 = 11M
(I'm calling this "sum1" because it's before we replace X & Y)

Now, replace X & Y with two younger people --- call them P & Q --- and the average drops by one:
sum2 = 11(M - 1) = 11M - 11 = sum1 - 11

sum1 - sum2 = 11

(X + Y + nine others) - (P + Q + same nine other) = 11

(X + Y) - (P + Q) = 11

60 - (P + Q) = 11

(P + Q) = 49

(P + Q)/2 = 49/2 = 24.5

Thus, the average ages of the two new people is 24.5, answer E.

Does all this make sense?

Here's a free practice question about averages,
https://gmat.magoosh.com/questions/349
When you submit your answer, the following page will have the complete video explanation.

Let me know if you have any further questions.

Mike :)
Magoosh GMAT Instructor
https://gmat.magoosh.com/
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