I am working with this question:
A received 1/3 more votes than B. Which of the following could have been the total number of votes cast for the two candidates? Answer options are 12, 13, 14, 15, 16.
I can see that however many votes B got, A got the same number plus 1/3 more, so the total number of votes cast is $x + \frac{4}{3}x = \frac{7}{3}x$. I can also see that x has to be an integer, and that it should be evenly divisible by three.
I got that far, then couldn't see how to get anywhere with algebra. I solved the problem by testing multiples of three (first $x = 9$, then $x = 6$) to see if $2\frac{1}{3}x$ was in the list of answers. Of course it works for $x = 6$, but I feel like this was a very haphazard way of getting the answer.
Is there some way that knowing the total number of votes is $\frac{7}{3}x$ can help me determine the correct answer directly rather than testing values? I don't feel like this was a hard question, but I do feel like I'm missing some obvious connection between the ratio and the total.
You have figured out that $x$ must be divisible by three. Say $x = 3y$.
Then the total number of votes is $7 y$. So we see that the total is divisible by $7$. Now all you have to do is look at the answers and see that only one of them is divisible by $7$.