Soccer league participation

[vishal.pathak]

Participation in soccer league is 10% higher this year. Participation of men increased by 5% and that of women increased by 20%. What is the % of female in the league now
a.1/3 b.4/11 c.2/5 d.4/9 e/1/2

This question can be solved using equation but since this is a question dealing with mixtures, so I tried solving it using the allegation

The proportion of each element in the mixture is equal to the distance between the average attributed to the other element in the mixture and the average attributed to the entire mixture.

Some how I couldn't understand the mechanism to solve this problem using the allegation.

Please help

[user123321]

last year say we have
m men,w women & m+w total
but this year
1.05m men,1.2w women & 1.1(m+w) total

since sum of men and women should tally to total..
1.05m + 1.2w = 1.1(m+w)
0.05m = 0.1w
w/m = 1/2
=>w/(w+m) = 1/3

so 1/3 or 33% are females in the league now

user123321

[vishal.pathak]

last year say we have
m men,w women & m+w total
but this year
1.05m men,1.2w women & 1.1(m+w) total

since sum of men and women should tally to total..
1.05m + 1.2w = 1.1(m+w)
0.05m = 0.1w
w/m = 1/2
=>w/(w+m) = 1/3

so 1/3 or 33% are females in the league now

user123321

This answer is correct buddy but I want to know the mechanism to solve this question using the below theorem

The proportion of each element in the mixture is equal to the distance between the average attributed to the other element in the mixture and the average attributed to the entire mixture.

[Ian Stewart]

last year say we have
m men,w women & m+w total
but this year
1.05m men,1.2w women & 1.1(m+w) total

since sum of men and women should tally to total..
1.05m + 1.2w = 1.1(m+w)
0.05m = 0.1w
w/m = 1/2
=>w/(w+m) = 1/3

so 1/3 or 33% are females in the league now

That's not quite right, and if the source says that's the right answer (where is it from?) then there's a problem with their solution. The question asks for the proportion of participants who are women *now* in the league. In the solution above, m and w represent the number of men and women *last year*, not this year. We need to apply the percent increases in the question first to get the answer: we have that m = 2w, so there are 1.2w women in the league now, and (1.05)(2w) = 2.1w men in the league now, so women make up 1.2w/(1.2w + 2.1w) = 12/33 = 4/11 of all participants this year.

You can certainly use alligation here provided you understand what the answer you get means. We have two groups, one of which increased by 5%, the other by 20%, and the overall increase was 10%. So on a number line we have:

--5-------10-------------------20---

The distances to the middle are 5 and 10, so the ratio of the two groups *before the increases* was 5 to 10, or 1 to 2. Since the overall increase is closer to 5%, we must have more men, so the league was 2/3 men and 1/3 women before the increases. Now we can get the answer by applying the percent increases in the question, just as I did above.

[vishal.pathak]

That's not quite right, and if the source says that's the right answer (where is it from?) then there's a problem with their solution. The question asks for the proportion of participants who are women *now* in the league. In the solution above, m and w represent the number of men and women *last year*, not this year. We need to apply the percent increases in the question first to get the answer: we have that m = 2w, so there are 1.2w women in the league now, and (1.05)(2w) = 2.1w men in the league now, so women make up 1.2w/(1.2w + 2.1w) = 12/33 = 4/11 of all participants this year.

You can certainly use alligation here provided you understand what the answer you get means. We have two groups, one of which increased by 5%, the other by 20%, and the overall increase was 10%. So on a number line we have:

--5-------10-------------------20---

The distances to the middle are 5 and 10, so the ratio of the two groups *before the increases* was 5 to 10, or 1 to 2. Since the overall increase is closer to 5%, we must have more men, so the league was 2/3 men and 1/3 women before the increases. Now we can get the answer by applying the percent increases in the question, just as I did above.

Thanks Ian. I read the approach of the answer given by user123321 and did not check the whole thing. Yes, the final step is missing and it is my mistake that I overlook the error

Thanks for the solution using the alligation.It certainly helped

Regards,
Vishal

[user123321]

@Ian

Thanks for correcting.

user123231