A group of men and women gathered to compete in a marathon.

Question

A group of men and women gathered to compete in a marathon. Before the competition, each competitor was weighed and the average weight of the female competitors was found to be 120 lbs. What percentage of the competitors were women?

(1) The average weight of the men was 150 lb.
(2) The average weight of the entire group was twice as close to the average weight of the men as it was to the average weight of the women.

KarishmaB · Accepted Answer

Responding to a pm:

Question: 
A group of men and women gathered to compete in a marathon. Before the competition, each competitor was weighed and the average weight of the female competitors was found to be 120 lbs. What percentage of the competitors were women?

(1) The average weight of the men was 150 lb.
(2) The average weight of the entire group was twice as close to the average weight of the men as it was to the average weight of the women.

Solution :

Avg weight of women = 120 lbs

Stmnt 1: Avg weight of men = 150 lbs
This statement doesn't tell us the average weight of the entire group. The avg weight of men is not helpful till we know the avg weight of the entire group. Not sufficient

Stmnt 2: Avg weight of group was twice as close to the average weight of the men as it was to the average weight of the women.
Since avg weight is twice as close to avg weight of men, the number of men must be twice of the number of women So women must be 1/3 of the total group. Think of the scale method we use. Ratio of the distances is 1:2 (avg is closer to men's avg than women's avg) so number of men: number of women = 2:1.
Sufficient.

Answer (B)

KarishmaB · Answer

Use scale method i.e. use a number line. We see that the group that has more number of people pulls the weighted average toward it.

The number of women:number of men = Distance of men's average from weighted average:Distance of weighted average from women's average.
If the distance of men's average from weighted average is half of the distance of weighted average from women's average (because men's average is twice as close), number of women:number of men = 1:2
So women must be (1/3)rd = 33.33% of total competitors.

You don't need to know their exact averages. Just knowing where the weighted average lies relative to their averages is enough to know the ratio of the number of people in each group.

Dreaming · Answer

Hi,

theres a derieved formula* that goes like this:
RATIO OF{( avg of group1-average of entire group) to (average of entire group- avg of group1)}=RATIO of {(number of units in group 2) to (number of units in group1).
(120-x)/(x-men(avg))=men/women

using the above, we can now analyze the 2 statements as follows:
1. men(avg)=150. replacing in formula: (120-x)/(x-150)=men/women. x is still unknown....cannot find the ratio to men and women. INSUFFECIENT
2. 'closeness' means difference......according to this statement (x-men(avg))=2*(120-x)
therefore, (120-x)/(x-men(avg))=men/women=1:2...percentage of women is 2/3*100=66.66%. SUFFECIENT
hence answer is B

* you can easily derive the formula from the basic formulae of averages. if you need help with that let me know

blink005 · Answer

This question deals with weighted averages.
The two subgroups are the men and women. Each subgroup has an average weight.
We need the averages of each subgroup along with the ratio of men to women to calculate the overall average weight of the group. The ratio of men to women would determine the weight to give to each subgroup's average.
This question is simply asking for the ratio of women to men (i.e. what percentage of the competitors were women).

(1) INSUFFICIENT
This statement merely provides us with the average of the other subgroup - the men. We don't know what weight to give to either subgroup; hence we don't know the ratio of the women to men.

(2) SUFFICIENT
If the average weight of the entire group were twice as close to the average weight of the men as it were to the average weight of the women, there must be twice as many men as women.
With a 2:1 ratio of men to women of, 33 1/3% (i.e. 1/3) of the competitors must have been women.

B

ankit0411 · Answer

how can it be B ?

To use the average weight of men we need to consider statement 1 .

So B alone is insufficient, any thoughts ?

Thanks,
Ankit

KarishmaB · Answer

The question asks for the % of women. If you have the ratio of men to women, you can answer this question. Statement II gives you the relation between weighted avg of the group and the weights of men and women. This information is enough to find the ratio of men and women. Check out the link I have given in the first post.

divineacclivity · Answer

Hi, thanks in advance.

Stmnt 2: Avg weight of group was twice as close to the average weight of the men as it was to the average weight of the women.
I couldn't really understand the second statement & how you arrived at this conclusion from the second statement:
Since avg weight is twice as close to avg weight of men, the number of men must be twice of the number of women

What I could gather from this statement is that the difference between the avg wt of the group & avg wt of men is twice the difference between avg wt of group and avg wt. of women
i.e. |avg wt. of group - avg wt. of men| = 2 x |avg wt. of group - avg wt. of women| where || represent mod

Could you please help me understand the language of the statement?

KarishmaB · Answer

Statement 2 is a direct reference to the number line method. Or you can say that it refers to the core concept of average weight. 
It says that the group avg is twice as close to the men's average as the women's average. We know that group average lies between the two averages. If men are more, the group average will be closer to men's average. If women are more, the group average will be closer to women's average

....................|.......................................|..........................|
.........Women's avg.....................Group Avg...................Men's Avg

The statement says that group average is twice as close to men's avg so it means that number of men is twice of number of women. If you look at the scale method discussed in the link, you will understand it even better. We are given that the distance between group avg and men's avg is half the distance between group avg and women's avg. So men must be twice of women.

gmat6nplus1 · Answer

Here you go hope it helps.

rustamz · Answer

I was confused about the explanation, but think about this simple example:

Women's average = 2
Group average = 4
Men's average = 5

Notice that the group average is twice as far from women's average than it is from men's.

To get group average of 4, we must have ONE Woman and TWO Men --> (1*2+2*5) / 3 = 4.

ersheet · Answer

For someone like me, whose not aware of the weighted avg rules, (i.e. if avg wt of the group is twice as close to the avg wt of men that women, then the number of men is twice that of women) u can use the following method:

lets say the difference between the avg wt of the group and the avg wt of men be x,
then the difference between the avg wt of the group and the avg wt of women will be 2x,
now we can write avg wt of group = 120 + 2x
and avg wt of men = 120 + 3x

now we can put these values in the weighted avg equation:
{120f + (120 + 3x)m}/(f +m) = 120 + 2x (where f and m are number of women and men respectively)
=> 120f + 120m +3xm = (120 + 2x)(f + m)
=> 120f + 120m +3xm = 120f + 120m + 2xf + 2xm
=> xm = 2xf
=> m = 2f

From this we can get the required % =   *100 =   *100 = (f/3f)*100 = 33.33%

I know this is a bit long, but if u r not aware of the rule, like i was, this can be helpful.

soodia · Answer

if the average of men and women were equal, we could say the ratio of the distance between total average with men and women indicate the proportion
but it could be so different! for example, the average weight of men is 500, so the total average will be the great number and close to men.

am I wrong?

KarishmaB · Answer

"if the average of men and women were equal,"

If the avg weight of men was 120 lbs too, the average of the entire group would be 120 too. It wouldn't matter what the proportion of women is in the group. The average of the two would always be 120 lbs. 
I am not sure what the rest of your comment means.

njabraham85 · Answer

How can you say that the number of men is more just because the group's avg is closer to that of men? Shouldn't it depend on both the number and the actual weight itself?

And how can u assume the women weigh less? Seriously!

Posted from my mobile device

SiffyB · Answer

In this question, we have to find the ratio of the number of female participants to the total number of participants, i.e.,  \frac{w}{w+m}

We don’t need to find any absolute values. Thus, we don’t require the actual values of average weights of the participants.

STATEMENT 1: Clearly insufficient as it does not talk about the number / ratio of participants as all.

STATEMENT 2: Gives us the ratio of the participants. It says that the mean is twice as close to the average weight of men as to the women.
The average is always skewed towards the group that has the larger number of participants.
The mean is twice as close to the number of men means that the number of men are twice as many as the number of women.
Thus, the required ratio will be  \frac{w}{(w+m)} , as m = 2w

=  \frac{w}{(w+2w)}  =  \frac{w}{3w}  =   \frac{1}{3}

Thus, women form a third of the total participants and men form 2/3rd of the total. 
 Sufficient. (B)

sambitspm · Answer

When I was solving I took the arithmetic approach to solve the question. Landed on B.
Later when I read through solutions for an easy approach, the distance concept could have landed me to the answer in few seconds.

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A group of men and women gathered to compete in a marathon.

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