*Submitted by Dr. Ted Stanford, NMSU Mathematician.
Ocotillo High School held a dance last week to celebrate the start of a new school year. Some of the students came in boy-girl pairs, and some came as singles. If 40% of the boys came with a date, and 50% of the girls came with a date, what percentage of all students came with a date?
Answer: This problem was written a few years ago, back when most people would assume that dating pairs would all be boy-girl. Attitudes are changing, and fewer people would make that assumption now. However, to solve this problem algebraically, the pairs have to be boy-girl.
The issue is that we are not told how many boys there are or how many girls there are. In this kind of situation, one strategy is to pick a number. Let’s say that 100 girls are at the dance. Then 50 girls came with a date, which means 50 boys also came with a date. To find how many total boys there are, we can ask, 40% of what number is 50? The answer is 125 boys. So now we know there are 225 students at the dance. Since 100 students came with a date, the proportion of students who came with a date is 100/225, or 4/9, or 44%.
This answer is reasonable, because the total percentage of students with a date should be in between 40% (the boys who have a date) and 50% (the girls who have a date).
The “pick a number” strategy works here, because it doesn’t matter how many total students there are. We could have chosen 200 girls, or 1000 girls instead, and the answer would still come out to be 4/9.
If you want to use a variable instead of a number, you could do something like this. Suppose there are X girls and Y boys at the dance. Then (.5) X girls have a date, and (.4) Y boys have a date. Since (.5) X = (.4) Y, we see that Y = (1.25) X, and the total number of students is X + (1.25) X = (2.25) X. The number of students with dates is (2)(.5)(X) = X. So the proportion of students with dates is X / (2.25 X) = 4/9.