finding the ratio of students in one class to another using average test scores

John's class had an average score of 75.
Amy's class had an average score of 78.
Both classes together had an average of 77.

Find the ratio of the number of students in John's class to Amy's class.

Ok so...

If John was the only person in his class and Amy was the only person in her class, the average of both classes would be 76.5.

Therefore, there must be more people in Amy's class than in John's class.

I will guess and check.

If the scores of both classes put together were 75, 78, 78, then the average of both classes would be (75+77+78)/3=76.5

Testing larger, equivalent ratios to 1/2 finds that this continues to hold true.

So the answer is 1 student in John's class for every 2 student's in Amy's class.

Is there an easier way to do this?

Try algebra, which is a standard alternative to guess and check.

Suppose there are x students in John's class, and y students in Amy's class. Can you write an equation expressing the average of all the students? Then see if you can solve for x/y.
 
Very good! The only minor criticism I have is that you're missing some grouping symbols. For now, it's clear what you meant from context, but in the future it may not be as clear and could result in wrong answers. For instance, you wrote "75x + 78y / x+y = 77." Taken literally, exactly as written, this evaluates to \(\displaystyle 75x + \dfrac{78y}{x} + y = 77\), which I don't think is what you meant. Instead you should have written (75x + 78y)/(x + y) which evaluates to \(\displaystyle \dfrac{75x+78y}{x+y}\).
 
Average of all students = Sum of Scores in John's Class + Sum of Scores in Amy's Class / x +y.

Average of all students = Sigma(J) + Sigma(A) / x+y = 77

Sigma J / x = 75

Sigma A / y = 78.

But, now I have four variables and only three equations. So I can't solve it.

I could rewrite Average of all students as:

75x + 78y / x+y = 77.

But, now I have two variables and only 1 equation. So I still can't solve it.

I could rewrite it as: 75x +78y = 77x + 77y.

I could rewrite that as -2x = -1y.

I could rewrite that as x/y = -1/-2 = 1/2.

Wow, I didn't actually expect to get the answer. That seems like a lot of work though. But, I guess that's that.

Thanks for the nudge.

Nice work! Yes, at first you think there isn't enough information, with only one equation; but since you don't have to solve for each variable, only for a combination of them, it turns out to be possible. You just need to be willing to try.

And it's not really a lot of work, when you boil it down; I actually did it in my head before writing. I saw 75x + 78y = 77(x+y) by equating two representations of the sum of all students' scores, then distributed and subtracted: 75x + 78y = 77x + 77y, so y = 2x, and y/x = 2. Of course, I've seen this sort of thing before, so I was expecting it to work out, and knew the general direction to go. The amount of work you did, feeling your way along, is about right for your first time through.
 
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