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A total of 22 men and 26 women were at a party, and the average

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A total of 22 men and 26 women were at a party, and the average (arithmetic mean) age of all of the adults at the party was exactly 35 years. If the average age of the men was exactly 38 years, which of the following was closest to the average age, in years, of the women?

(A) 31
(B) 31.5
(C) 32
(D) 32.5
(E) 33

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[Reveal] Spoiler: OA

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Re: A total of 22 men and 26 women were at a party, and the average [#permalink]

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Let Aw = average age of women
Average = (Number of men X average age of men + Number of women X average age of women)/total number of people

35 * 48 = 22 * 38 + 26 * Aw
=>1680 = 836 + 26 Aw
=> 26 Aw = 844
=>Aw = 32.5

Answer D


Scale method -
If number of men and women were equal , the average would have been at the center of the average age of the individual groups
Average age of women = 32 as 35 will be equidistant from 32 and 38 (average age of men)
At this point we can eliminate A , B and C

But since , number of women are more , average age will be more than 32
Ratio of women to men = 26:22 = 13:11
If Aw= 32.5
Distance from the average
35- 32.5 :38-35
2.5 :3
5:6
which is rougly equal to inverse of 13:11

Answer D
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Re: A total of 22 men and 26 women were at a party, and the average [#permalink]

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Bunuel wrote:
A total of 22 men and 26 women were at a party, and the average (arithmetic mean) age of all of the adults at the party was exactly 35 years. If the average age of the men was exactly 38 years, which of the following was closest to the average age, in years, of the women?

(A) 31
(B) 31.5
(C) 32
(D) 32.5
(E) 33

Kudos for a correct solution.


One can use the weighted average method:

\(\frac{Weight 1}{Weight 2} = \frac{Average2-Average}{Average-Average1}\)

For or example:

\(\frac{26}{22} = \frac{38-35}{35-Average1}\)

Solving for Average1 you will get exactly 32.46154. Answer D.
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Re: A total of 22 men and 26 women were at a party, and the average [#permalink]

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New post 20 Oct 2015, 07:09
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Bunuel wrote:
A total of 22 men and 26 women were at a party, and the average (arithmetic mean) age of all of the adults at the party was exactly 35 years. If the average age of the men was exactly 38 years, which of the following was closest to the average age, in years, of the women?

(A) 31
(B) 31.5
(C) 32
(D) 32.5
(E) 33

Kudos for a correct solution.


You can also think of it like this on a scale method base:

x(Average female)..........(35-x)............Average(35)..........(3)............(38)Average male
#26 (No. of females)..........................................................................#22 (No. of males)

So again this shows that we have to groups which mixed together have an a weighted average of 35. The "weighs" in this example are the number of people. For men 22 and for woman 26. You can now equate the known weighs with the differences to average:

\(\frac{26}{22}=\frac{3}{(35-x)}\)

Then solve for x to get x= 32.4615

Answer D.
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Re: A total of 22 men and 26 women were at a party, and the average [#permalink]

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New post 20 Oct 2015, 07:40
Bunuel wrote:
A total of 22 men and 26 women were at a party, and the average (arithmetic mean) age of all of the adults at the party was exactly 35 years. If the average age of the men was exactly 38 years, which of the following was closest to the average age, in years, of the women?

(A) 31
(B) 31.5
(C) 32
(D) 32.5
(E) 33

Kudos for a correct solution.




38*22 + W*26 = 35*(22+26)
i.e. W = 844/26 = 32.46 = 32.5

Answer: option D
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Re: A total of 22 men and 26 women were at a party, and the average [#permalink]

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Bunuel wrote:
A total of 22 men and 26 women were at a party, and the average (arithmetic mean) age of all of the adults at the party was exactly 35 years. If the average age of the men was exactly 38 years, which of the following was closest to the average age, in years, of the women?

(A) 31
(B) 31.5
(C) 32
(D) 32.5
(E) 33

Kudos for a correct solution.


Weighted average of groups combined = (group A proportion)(group A average) + (group B proportion)(group B average) + (group C proportion)(group C average) + ...

For more information on weighted averages, you can watch this free GMAT Prep Now video: http://www.gmatprepnow.com/module/gmat-statistics?id=805

Let x = average age of the women
So, we have (22/48)(38) + (26/48)(x) = 35
Simplify: (11/24)(38) + (13/24)(x) = 35
Multiply both sides by 24 to get: (11)(38) + 13x = (35)(24)
Simplify: 418 + 13x = 840
Rearrange: 13x = 422
Solve: x = 422/13 ≈ 32.5

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Re: A total of 22 men and 26 women were at a party, and the average [#permalink]

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New post 21 Oct 2015, 04:54
Bunuel wrote:
A total of 22 men and 26 women were at a party, and the average (arithmetic mean) age of all of the adults at the party was exactly 35 years. If the average age of the men was exactly 38 years, which of the following was closest to the average age, in years, of the women?

(A) 31
(B) 31.5
(C) 32
(D) 32.5
(E) 33

Kudos for a correct solution.


let average age of women be x

then, we have

\(22 * 38 + 26 * x = 48 * 35\)

=> \(x = 32.46\)

Answer choice D

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Re: A total of 22 men and 26 women were at a party, and the average [#permalink]

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I followed a similar approach to the scale. Given that m:w is 11:13 and we know that the AVG(m) = 38 and AVG(m+w)=35. We already know the AVG(w) has to be less than 35 given the higher weighting to w. I ended up multiplying the (11/13) * 38. To simplify, I rounded to redo the formula as (11/13)*39 = 33. Therefore, the answer must be just under 33 or 32.5. Also, keep in mind the key word of "closest", which implies that you do not need an EXACT answer (no crazy math).

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Re: A total of 22 men and 26 women were at a party, and the average [#permalink]

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Bunuel wrote:
A total of 22 men and 26 women were at a party, and the average (arithmetic mean) age of all of the adults at the party was exactly 35 years. If the average age of the men was exactly 38 years, which of the following was closest to the average age, in years, of the women?

(A) 31
(B) 31.5
(C) 32
(D) 32.5
(E) 33

Kudos for a correct solution.


1) Since there are 48 people (22 men and 26 women) and the average age of all of them is 35, the total ages of all the people at the party is 48(35).

2) Since there are 22 men and their average age is 38, the sum of all the men's ages is 22(38).

3) There are 26 women; letting their average age be x, the sum of all the women's ages is 26x.

Since the sum of the ages of all 48 people must be equal to the sum of the ages of the 22 men plus the sum of the ages of the 26 women, we have

48(35) = 22(38) + 26x

1680 = 836 + 26x

26x = 844

x = 844/26

x = 32 12/26 ≈ 32.5

Answer: D
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A total of 22 men and 26 women were at a party, and the average [#permalink]

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New post 04 May 2016, 10:23
Bunuel wrote:
A total of 22 men and 26 women were at a party, and the average (arithmetic mean) age of all of the adults at the party was exactly 35 years. If the average age of the men was exactly 38 years, which of the following was closest to the average age, in years, of the women?

(A) 31
(B) 31.5
(C) 32
(D) 32.5
(E) 33

Kudos for a correct solution.


Total age of men and women = 48*35 => 1,680

Total age of men is = 22*38 => 836

So, total age of women in = 1680 - 836 => 844

Average age of women is 844/26 => 32.46

Hence answer will be (D) 32.5 :P
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Re: A total of 22 men and 26 women were at a party, and the average [#permalink]

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Here is the MGMAT Solution to it.

Avg. age of men is +3 than the avg. age of the group.

Now,

3(22)+w(26)=0....(Make this differential cancel out,hence equal to 0)

22=no. of men
26=no. of women
3=differential from mean.

Hence w=-2.5

Thus group mean (35)+Differential (-2.5)=32.5

Hope its simple than all the long calculations.

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Re: A total of 22 men and 26 women were at a party, and the average [#permalink]

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New post 11 Aug 2016, 06:42
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Bunuel wrote:
A total of 22 men and 26 women were at a party, and the average (arithmetic mean) age of all of the adults at the party was exactly 35 years. If the average age of the men was exactly 38 years, which of the following was closest to the average age, in years, of the women?

(A) 31
(B) 31.5
(C) 32
(D) 32.5
(E) 33

Kudos for a correct solution.


I uesd the Std Deviation method :
Ave of Man = 38 (which is 38-35 = 3 more than the ave of Man & Women together)
and No of man = 22
Therefore Deviation More than the Mean = 3 * 22 = 66

Also, we know that No of women = 26
Now, For any given set of numbers, Deviation More than the Mean is ALWAYS = Deviation Less than the Mean
Therefore, each women can have a deviation of = 66/26 ~= 2.5 from the Average Value
Therefore, Ave age of Women = 35-2.5 = 32.5

Kudos if u like the method....

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Re: A total of 22 men and 26 women were at a party, and the average [#permalink]

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New post 09 Oct 2016, 16:02
Bunuel wrote:
A total of 22 men and 26 women were at a party, and the average (arithmetic mean) age of all of the adults at the party was exactly 35 years. If the average age of the men was exactly 38 years, which of the following was closest to the average age, in years, of the women?

(A) 31
(B) 31.5
(C) 32
(D) 32.5
(E) 33

Kudos for a correct solution.


It initially took mme ~2mins to solve this, but tried a couple of ways and think weighted avg would be fastest as lesser big numbers to deal with:

[Women-avg: (w)]----------------35--------------------38

\(\frac{26}{22} = \frac{3}{(35-w)}\)
\(w = 35 - \frac{33}{13}\)
13 * 2 = 26; hence \(\frac{33}{13} = 2.x\) (I wont even bother getting the decimal)
w = 35 - 2.x = 32.x (~32.5)
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Re: A total of 22 men and 26 women were at a party, and the average [#permalink]

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Re: A total of 22 men and 26 women were at a party, and the average   [#permalink] 18 Oct 2017, 10:55
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