# How does knowing a ratio help me determine a total?

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.

• To find $x$ you'll have to divide by $7$. So you need an answer that is a multiple of $7$. – lhf Nov 5 '13 at 1:13

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 $7$.
• Yes, it's obvious now. I was looking at $\frac{7}{3}x$ and thinking it didn't tell me anything useful. But it tells me that the total is a multiple of 7. Thank you. – dumb question Nov 5 '13 at 1:21