Solve 16000000 x 11.0006 + 0.13x = x.

Step-by-step solution

1. 1

   Collapse the constant product first. The left-hand side hides a plain multiplication: $16{,}000{,}000 \cdot 11.0006 = 176{,}009{,}600$. Doing this before touching the $x$ keeps the equation readable: $176{,}009{,}600 + 0.13x = x.$

2. 2

   Recognise the shape of the problem. The unknown appears on both sides, so this is a gross-up: some fixed amount plus $13\%$ of the total equals the total. The standard move is to collect all $x$ terms on one side.

3. 3

   Subtract 0.13x from both sides. $176{,}009{,}600 = x - 0.13x = (1-0.13)x = 0.87x.$ Factoring $x$ out is the key step - $1 - 0.13 = 0.87$ is the fraction of the total that the fixed amount represents.

4. 4

   Divide by 0.87. $x = \frac{176{,}009{,}600}{0.87} = \frac{17{,}600{,}960{,}000}{87} \approx 202{,}309{,}885.06.$ The exact value is the fraction $\tfrac{17600960000}{87}$, which does not terminate, so round only at the end.

5. 5

   Check the answer by substitution. $0.13 \times 202{,}309{,}885.06 \approx 26{,}300{,}285.06$, and $176{,}009{,}600 + 26{,}300{,}285.06 \approx 202{,}309{,}885.06$ - the two sides agree.

6. 6

   Note the scaling shortcut. Because the equation is linear and homogeneous in the constant, scaling the constant scales $x$ by the same factor: the same equation with $\tfrac{1600\cdot 11.0006}{100} = 176.0096$ gives $x \approx 202.31$, exactly one millionth of this answer.