Calculate total number of votes cast given percentage of winners I know that the top three candidates received 14.14%, 13.13% and 12.12% of a vote. I also know that the total number of votes cannot be more that 300, and the numbers of votes cast and the total must be integers, I also believe that the percentages are rounded to two decimal places.
How can I calculate how many votes were cast in total?
My workings so far:
I have managed to create the equations:
$\frac{a}{x*100} = 14.14$
$\frac{b}{x*100} = 13.13$
$\frac{c}{x*100} = 12.12$
which gives:
$\frac{a}{x*100} +\frac{b}{x*100} + \frac{c}{x*100} = 14.14+13.13+12.12$
that is:
$\frac{a+b+c}{3x} = 39.39$
and I don't know where to go from there..
Another route is from intuitively:
$A+B+C+D=100$ 
in this case $D = 60.61$
so we can say $\frac{d}{x*100}=60.61$
although I don't know how this helps...
Now from looking at simultaneous equations on Wikipedia I realise we should solve each equation for one variable which means we can get to: $a = x1414$, $b = x1313$ and $c = x1212$ or better put as: $\frac{a}{1414} =x$, $\frac{b}{1313} =x$ and $\frac{c}{1212} =x$
so: $\frac{a}{1414} =\frac{b}{1313} =\frac{c}{1212}$
and: $\frac{a}{1414} =\frac{b}{1313}$ means that: $a=b1.076923$
and $\frac{b}{1313} =\frac{c}{1212}$ means that: $b=c1.08333$
and $\frac{a}{1414} =\frac{c}{1212}$ means that: $a=c.166666$
therefore: $c1.166666 = (c1.08333)1.076923$ and this breaks when we expand the brackets i.e.: $c1.166666 = (c1.076923)(1.166666)$ hence my question.
 A: Assuming that those percentages should have infinitely repeating decimals, then 99 votes with amounts of 14, 13 and 12 to the top three candidates works. But so does doubling or tripling those numbers.
If the percentages as given are exact then there is no solution with total votes less than 300. The smallest solution has 10000 total votes with candidates getting 1414, 1313 and 1212.
A: As a general approach to find integers  $m,n$ whose ratio closely approximates a decimal fraction, we can use the method of continued fraction expansion.  I will illustrate it for the number $13.13\%$ or $0.1313$ appearing among the vote percentages.
First we take the absolute value, in case we were given a negative number.  Here the number is positive and nothing needs to be done.  We will construct a sequence of positive integer pairs $m_i,n_i$ such for their size give the best possible rational approximation. 
Pull off the integer (whole number) part to get started, setting $m_0=0$ in this case and put that over a denominator $n_0=1$.  That leaves a remainder $0.1313 - 0/1$ as our approximation "error".
At each successive step we have a nonegative error of approximation which is less than one, e.g. $0.1313$ in this instance.  If zero error, quit.  Otherwise taking the reciprocal of the error gives a new value greater than one:
$$ \frac{1}{0.1313} = 7 + 0.61614623\ldots $$
Considered in terms of using the single digit $7$ as an approximation to the reciprocal error, we have:
$$ 0.1313 \approx 0 + 1/7 $$
and if we continue in this fashion we will get a succession of improved approximations until we hit exactly:
$$ 0.1313 = 1313/10000 $$
For your problem you would carry the approxidation procedure out as long as the denominator remained under $300$ until a value rounding to two places in the percentage (four places without such a percentage sign).  The procedure would fail if the denominator went over $300$, but in the immediate case we succeed with:
$$ 0.1313 \approx 13/99 $$
