How do I set up an equation to solve this linear system?

Question

Recently, I read a news article that mentioned some demographic statistics. Several percentages (of a population) were given, but I wanted to figure out the ratio of one demographic group to the other (which was not given in the article). Please note the source material is very controversial, so I've bowdlerized and anonymized the scenario (and changed the numeric values).

Please note that this is not a homework problem--I'm not a math student (anymore).

Theoretical Scenario

Consider a population with two demographics, "A" and "B", and a theoretical ballot referendum.

Demographic breakdown:

• 41% of "A" is "for" the referendum
• 78% of "B" is "for" the referendum

Total population breakdown:

• 51% of the total population is "for" the referendum

Note that since there are only two demographics, and only two possible outcomes of the referendum, then the "against" values can easily be calculated (see data summary below).

Math problem (not the question I'm asking here)

How much of the total population is demographic "A"?

I've figured out the answer through various mathematical trial-and-error techniques (with the aid of a calculator). But, what I really want to know is:

Question

How do I set this problem up?

Data Summary

        +---------+---------+
        |   FOR   | AGAINST |
+-------+---------+---------+
|  "A"  |   41%   |   59%   |
+-------+---------+---------+
|  "B"  |   78%   |   22%   |
+-------+---------+---------+
| Total |   51%   |   49%   |
+-------+---------+---------+

     +-------+--------+
     |  "A"  |  ????  |
     +-------+--------+
     |  "B"  |  ????  |
     +-------+--------+
     | Total |  100%  |
     +-------+--------+

Again, my question isn't about how much (percentage) of the total population are "A" and "B", but rather how to mathematically set up this problem.

P.S. The (math) answer is

72.9% of the total population is "A" (and 27.1% "B").

Answer 1

Let the total population be $p$. Let the size of A be $ap$ and the size of B be $bp$, so that $a + b = 1$. We also know that the 'for' population is $0.41ap + 0.78bp = 0.51p$ and the 'against' population is $0.59ap + 0.22bp = 0.49p$. The first equation for 'for' is actually sufficient for us. Simplifying, $$a + b = 1 \\ 0.41a + 0.78b = 0.51$$ which you can solve like any system of equations to get $a \approx 0.73 = 73\%$ and $b \approx 0.27 = 27\%$.

Answer 2

Let $A$ and $B$ be the populations. So, from the results of the vote for $$0.41A+0.78B=0.51(A+B)\implies 0.10A=0.27B\implies A=2.7B$$ So, if $A+B=100$ then$$3.7B=100\implies B=\frac{100}{3.7}\implies A=\frac{100\times 2.7}{3.7}=72.97$$