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

Question

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.

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

reto · Accepted Answer

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.

Skywalker18 · Answer

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

BrentGMATPrepNow · Answer

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

There are 48 people in total. 
So,  22/48 = the proportion of women 
And  26/48 = the proportion of men

Let  x = average age of the women

So, plugging the information into the formula, we get:  (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

Answer: D

Cheers,
Brent

reto · Answer

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.

GMATinsight · Answer

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

Answer: option D

v12345 · Answer

let average age of women be x

then, we have

22 * 38 + 26 * x = 48 * 35

=>  x = 32.46

Answer choice D

skeeterv8 · Answer

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).

ScottTargetTestPrep · Answer

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

ynk · Answer

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.

mihir0710 · Answer

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....

colorblind · Answer

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:

----------------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)

pvs29 · Answer

average age of man * number of men = summation of ages of men
average age of woman * number of women = summation of ages of women

summation of ages of men + summation of ages of women = summation of ages of men and women

Thus, (38*22) + (average age of woman*26) = 35*(22+26)

simplifying,
average age of woman = { -(38*22)}/26 = a little greater than 32 =>  32.5

Answer:  D

mimajit · Answer

How can we assume that 35 is the weighted average and not a simple avg of 48 people in total ?

BrentGMATPrepNow · Answer

The average = 35 = weighted average
They are the same thing. 
The "weight" here is dictated by the number of women compared to the number of men.

Cheers,
Brent

TheNightKing · Answer

I solved it in 49 seconds and here is the approach.

Average Age = 35.

22 people have contributed = +3 each. Which means we are 66 above the average.

26 people have to reduce the total of their age by 66 to keep the average intact at 35.

66/26=33/13=2.53=2.5

So each goes 2.5 below the average which is 35.

Hence 32.5

Answer D

ThatDudeKnows · Answer

Classic example of how GMAC crafts questions like this. You are welcome to do the "real" math if you'd like, but the vast majority of the time, you can get to the right answer without all that...faster and with minimal chance of making a careless mistake.

If there were 22 men and 22 women and the men's average age were 38 and the women's average age were 32, the overall average would be 35. But we have more women than men, so that would drive the overall average down. C is wrong. If we lower the women's average age even more, that would drive the overall average down even more, so makes A and B wrong. We are down to D and E. If the women average 33 years old and the men average 38, we would need 3 women for every 2 men to get to an average of 35. Our ratio is 26:22, which is not 3:2. E is wrong.

Answer choice D.

Thib33600 · Answer

all seems so simple, but am i the only one looking at your statement and wonder "how the hell can i do 844/26 and find around 32.5 wihtout calculator?"

I see a lot of GMAT questons that are easy to go through but are too difficult once the calculation needs to be done without calculator.
Any tips?

Thanks!!

KarishmaB · Answer

Thib33600 - You don't need to do that calculation. Simply use the scale method discussed here: https://youtu.be/_GOAU7moZ2Q ?? ----------------------- 35 -------------------------- 38 (26)................................................................(22) The ratio of the distances is inverse of the ratio of weights. So 22/26 = x/3 x = 33/13 (This is a calculation you will need to do by hand) x = 2.5 approx Hence age of women is 2.5 less than 35 i.e. at 32.5 Answer (D)

krishna_sunder_ · Answer

age->VALUES
number. of ppl->WEIGHTS

VALUE         x..................... 35.................... 38
WEIGHTS                22           :           26
                              11           :           13
                               
                                                         13 units => (38-35)
                                                                      =>3
                                                         11 units => ?
                                                      
                                                         ?=(3*13)/11 = in between 2 and 3
                              
                                                        therefore, 
                                                              x is in between (35-3) & (35-2)
                                                              x is in between 32 & 33
                                                        therefore,
                                                              the answer is 32.5 (option D)

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