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I encountered the following problem from a FCML match:

20% of the students who applied to State U. from Rufus High School were accepted. 26% of the students who applied to State U. from Buford High School were accepted. If there were 150 students in total who applied from the two high schools, Y% of the total number of students who applied were accepted, and Y is an integer, what are all possible values for the number of students who were accepted to State U. from Buford High School.

I couldn't figure out how to solve this.

If you assign R to the number of people who applied from Rufus and B to the number of people who applied from Buford, then I can start with: $B+R=150$.

And I guess you could write the equation $\frac{Y}{100}\times150=0.2R+0.26B$. But I don't know where to go from here. I don't know how to solve this, and I don't know if it's right.

The answer sheet says that the answers are 13 and 26, but it doesn't give a solution. Please help?

Edit: Using the answers, I backtracked and the answers make sense now. I realized that I could have used some intelligent guess-and-check to figure it out. But is there a mathematical way to achieve the answers?

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I think it is actually pretty simple if you notice that you can't have fractions of students. We have the relation that

$$ \frac{1}{5}R + \frac{13}{50}B = 150.$$

This means that the number of accepted students from school $B$, $\frac{13}{50}B$, needs to be an integer. The only two ways of assigning a number of the form $(50n, n \in \mathbb{N})$ to $B$ are $\{50,100\}$. Therefore the number of students accepted are $\{13,26\}$.

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The problem with your equation is there are 3 variables & only one equation. As the rule, there must be 3 or more than 3 equations to solve this type of questions if equationed property. I think data is inadequate in this question. At least ratio between R&B would have given...

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