I have a math problem and I hope somebody can help me out:

Example

I have a product that costs me 80 dollar to buy. I want to sell it with a 10% profit. The tax authorities will charge 21% tax over selling the product. Now comes the part that makes it difficult for me: Another company will charge me 15% of the selling price including tax.

How can I calculate my selling price to make sure I still have a 10% profit?

Thanks for your answer!

Update

Thanks for all the answers until so far! I hope by describing how far I got shows where I am struggling.

Buy in: 80 dollar

plus profit: 80 x 1.1 = 88

plus taxes: 88 x 1.21 = 106.48

for me: 106.48 x 0.85 = 90.508
for other company: 106.48 x 0.15 = 15.972

My check for the profit: (for me) / 121 x 100 - 80

My check seems to be right if I add 15.5 something to the profit with a selling price of about 125 dollar, but how do I get to that price?

Let's label each important number in this problem. TC is total cost. SP is selling price.

Hence, $$TC = 80 + SPcdot0.21 + SPcdot(1+0.21)cdot0.15$$

Why does this formula make sense? The tax is obviously $0.21$ of the Selling Price, but the company cut is $15%$ of the SP with tax, which is why we multiply $0.15$ by $(1+0.21)cdot SP$.

So, $$TC = 80 + 0.3915cdot SP$$

We also know that we want there to be a 10% profit, so

$$TCcdot(1+0.1) = SP$$

We substitute to get

$$TC = 80+0.43065cdot TC$$
$$.56935cdot TC = 80$$
$$TC = $140.51$$
$$SP = 1.1cdot TC = $154.56$$

So, the selling price should be $154.56.

Edit

Under the assumption that @saulspatz is correct, the tax is slightly different, and takes the form $$0.21(0.85P−80)$$ since the tax is taken on the profit made. Make sure to adjust this process for this formula!

Well, saying that you want to make $10%$ profit means that you want to end up with $80+0.1cdot80=$88$ in your pocket after the sale.

Let's say the selling price is $X$ dollars. If I'm understanding your question correctly, you're going to lose $0.21X$ to tax and $0.15X$ to the other company. So all in all you'll have to pay out $0.21X+0.15X=0.36X$, and you'll get to keep $X-0.36X=0.64X$ in your pocket. So the equation to determine the selling price $X$ is
$$0.64X=88.$$

NOTE: If I'm misinterpreting how the tax and the other company's charge are calculated, let me know in comments, and we can adjust the calculation.