Solve for x: (x - 0.19)/x = 0.165.

Step-by-step solution

1. 1

   Recognise the structure. The left side is (something minus a fixed amount) divided by that something — the margin form. If $x$ is a selling price and $0.19$ a unit cost, the equation says the gross margin is $16.5\%$. Seeing this tells you in advance that the answer must be bigger than $0.19$.

2. 2

   State the domain. Since $x$ appears in the denominator, $x\neq0$; the value found at the end must be checked against this.

3. 3

   Clear the denominator. Multiply both sides by $x$:

   $x-0.19=0.165x$

4. 4

   Collect the x terms on one side. Subtract $0.165x$ and add nothing else — this is where the useful closed form appears:

   $x-0.165x=0.19\qquad\Longrightarrow\qquad (1-0.165)x=0.19\qquad\Longrightarrow\qquad 0.835x=0.19$

   The factor $1-0.165=0.835$ is the fraction of the price that the cost occupies.

5. 5

   Divide to finish, and read off the general rule.

   $x=\frac{0.19}{0.835}=0.227545\ldots\approx0.2275$

   In words: $\text{base}=\dfrac{\text{fixed amount}}{1-\text{margin}}$. That formula is what makes this problem reusable with any other cost and margin pair.

6. 6

   Verify to full precision. With $x=0.2275449\ldots$: $x-0.19=0.0375449$, and $\dfrac{0.0375449}{0.2275449}=0.165000$ exactly ✓. The answer is also comfortably larger than $0.19$, as predicted.