My boyfriend wants to pay the same percentage out of each of our salaries for rent (eg; 18% of each person's salaries go to rent.) What would that amount be according to our salaries?
Using x as the percentage that would be the same for each person, we need to look at yearly rent vs yearly salary

104,000\*x + 45,700\* x = 19,200

149,700\*x = 19,200

x = 19,200 / 149,700 = 0.12826= 12.83%

Edit: updated the correct rent amount.
Rent is 19200 per year. Your boyfriend makes 104/45.7 in terms of ratio compared to you. Let's call what you must pay x. Then x + x*104/45.7 = 19200. (If you struggle to see where this comes from, think of this as if your boyfriend earned twice your salary then you would have had x + 2x = 19200).

Solving for this means that x = 5861 is what you must pay annually. Multiply that with 104/45.7 for what your boyfriend must pay annually.

Divide both results by 12 to see what you must each pay monthly.
Just in case you want to re-calculate when one of these numbers change, or you decide you want to use post-tax numbers.

If your boyfriend makes B a year, you (OP) make O a year and you pay a combined R in rent a month.

Boyfriend Monthly Rent Payment = R\*B/(B+O).

OP Monthly Rent Payment = R\*O/(B+O).

In this case we get 1600\*104/(104+45.7) = $1111.55 Your monthly rent payment would be the leftover which is 1600-111.55 =$488.45
First, imagine pooling all of your money together. $104k +$45.7k = $149.7k What percentage of the total is your bf's income?$104k/$149.7k = 0.695 = 69.5% What percentage of the total is your income?$45.7k/149.7k = 0.305 = 30.5%

So, to have rent be an equal percentage for both of you:
He would pay 0.695 \* $1600 =$1112
You would pay 0.305 \* $1600 =$488

This would mean both of you pay an equal 12.83% of your pre tax income.
$1112 \* 12 /$104,000 = 0.1283 = 12.83%
$488 \* 12 /$45,700 = 0.1283 = 12.83%
If you calculate what percent of your total household income is contributed by each of you, this percentage can be used to figure out the fair share no matter what the expense is.

Your total household income is just the sum of your income, so $104,000 +$45,700 = $149,700. Your boyfriend contributes$104,000 of the $149,700, which is about 69.5%. You contribute the difference, or 30.5%. So, to pay fair shares of a$1,600 rent payment, your boyfriend would pay 69.5% of it, or $1112, and you would pay your 30.5%, or$488.

Edit to add: I just rounded the percentage to the nearest tenth, to make it simpler.  This means that the share isn't *exactly* proportional, but I feel like most people would agree it is close enough.  If you prefer an exact proportion, just don't convert to a percentage at all.  Multiply the rent payment by the original ratio of contributed money to total income.  For example, an exact proportion would mean that your boyfriend would pay ($104,000 /$149,700) × $1600 =$1111.56.  (This isn't entirely accurate either, because the actual number of cents is 55.6446, but of course you can't pay in fractions of a cent, so $1111.56 is as close as you're going to get to a perfect proportional split.) You obviously would pay the difference here, or$488.44.  My approximation is simpler and within a dollar.  If that isn't precise enough, using the exact ratio is as close as you are going to be able to get.
Your share is 45.7/(45.7+104.0) * 1600.

The other share is 104.0/(45.7 + 104.0) * 1600