Solve for a: 450a - (70 + 240a + 171 + 69.50) - (450a * 13% - 15a - 30.4) = 0.

Step-by-step solution

1. 1

   Convert the percentage to a decimal first. The only non-obvious term is the commission:

   $450a\cdot 13\%=450a\cdot 0.13=58.5a$

   Leaving it as $13\%$ inside the algebra is the most common source of a factor-of-$100$ error.

2. 2

   Remove both sets of brackets, flipping every sign inside.

   $450a-70-240a-171-69.50-58.5a+15a+30.4=0$

   Note the two negative terms inside the second bracket become positive: $-(-15a)=+15a$ and $-(-30.4)=+30.4$.

3. 3

   Collect the $a$ terms.

   $450-240-58.5+15=166.5\;\Longrightarrow\;166.5a$

   Financially this is the net margin per unit: a price of $450$, less $240$ of variable cost, less the $58.5$ commission, plus a $15$ rebate.

4. 4

   Collect the constants.

   $-70-171-69.50+30.4=-280.1$

   so the equation reduces to $166.5a-280.1=0$ — the fixed costs that must be covered.

5. 5

   Solve and interpret.

   $a=\frac{280.1}{166.5}\approx 1.6823$

   Substituting back: $166.5(1.6822823)=280.1\;\checkmark$. Since $a$ counts units, breaking even needs $a\approx 1.68$, so in practice $2$ units are required to cover fixed costs of $280.1$.