SOLUTION: At a leadership conference, 43% of the men and 73% of the women voted `yes' on a
particular resolution. If all the men and women voted, and together 65% of them
voted `yes', then
Question 1103952: At a leadership conference, 43% of the men and 73% of the women voted `yes' on a
particular resolution. If all the men and women voted, and together 65% of them
voted `yes', then what must be the ratio of men to women at the conference? Answer by greenestamps(13216) (Show Source):
A method for solving this using standard algebra might look something like this....
Let x be the number of men and y the number of women. Then the problem says
The ratio of men to women is 4:11.
This is essentially a mixture problem: there is one group of people with a 43% yes vote and another group with a 73% yes vote, and mixing them yields a group with a 65% yes vote.
And in this mixture problem, the objective is to find the ratio of the numbers of people in the two groups.
Alligation is one method of solving mixture problems. That method finds the solution to mixture problems by calculating the ratio in which two ingredients are mixed to produce the desired mixture. So that method is especially useful in this problem.
Here is what the solution using the method of alligation looks like for this problem.
The first column shows the percentages of men (1st row) and women (3rd row) who voted yes; the second column shows the percentage of the total who voted yes.
The numbers in the third column are the differences, computed diagonally, between the numbers in the first and second columns: 65-43=22; 73-65=8.
The method of alligation says those numbers in the third column are the ratio between the numbers of people in the two groups. In this problem the numbers in the third column say there are 8 men for every 22 women -- making the ratio 8:22, or 4:11.
