Question

You and a friend want to start a business and design t-shirts. You
decide to sell your shirts for $15 each and you paid $6.50 a piece plus
a $50 set-up fee and $25 for shipping. How many shirts do you have to
sell to break even? Round to the nearest whole number.

Answer 1

you're selling the t-shirts at $15 a pop, so if you sell say Q quantity of shirts, your Revenue is 15Q.

your costs are a one-time setup fee of 50 bucks plus 25 bucks for shipping, so 75 bucks fixed cost, and you paid $6.50 to make each t-shirt, so if you make Q quantity of shirts, your costs will then be the 75 bucks plus 6.5Q.

the break-even point is where the revenue = costs, namely where R(Q) = C(Q).

$\bf \begin{cases} R(Q)=15Q\\ C(Q)=6.5Q+75 \end{cases}\implies \stackrel{revenue}{15Q}~=~\stackrel{costs}{6.5Q+75} \\\\\\ 8.5Q=75\implies Q=\cfrac{75}{8.5}\implies Q\approx 8.8235\implies \stackrel{rounded~up}{Q=8}$

we used 8, because we can't use 9, since it's less than 9 shirts, is 8.8, then again 8.8 shirts is 8 shirts and then you grab another one and ripped it up by 0.8, which we're not going to do, well, unless one client wants that.