Solve (12812 x 1.0398 - x)/x = 10%.

Step-by-step solution

1. 1

   Turn the percent into a decimal. $10\% = \tfrac{10}{100} = 0.10$, so the equation reads $\dfrac{12812\cdot1.0398-x}{x} = 0.10$. Leaving the percent sign in place is the most common source of a factor-of-100 error here.

2. 2

   Evaluate the constant in the numerator. $12812 \cdot 1.0398 = 12812 + 12812(0.0398)$, and $12812 \cdot 0.0398 = \dfrac{12812\cdot 398}{10000} = \dfrac{5{,}099{,}176}{10000} = 509.9176$. Hence $12812\cdot1.0398 = 13{,}321.9176$.

3. 3

   Clear the denominator. Multiplying both sides by $x$ (valid since $x=0$ would make the original expression undefined) gives $13{,}321.9176 - x = 0.10x.$

4. 4

   Collect the x-terms. Add $x$ to both sides: $13{,}321.9176 = 1.10x.$ Note the divisor is $1.10$, not $0.90$: the ratio in the problem measures the surplus relative to x, so $x$ plus $10\%$ of $x$ makes up the whole numerator constant.

5. 5

   Divide by 1.1. $x = \frac{13{,}321.9176}{1.10} = \frac{16{,}652{,}397}{1375} \approx 12{,}110.83.$

6. 6

   Check the ratio directly. $13{,}321.9176 - 12{,}110.8342 = 1{,}211.0834$, and $\dfrac{1{,}211.0834}{12{,}110.8342} = 0.10$ exactly as required.