Solve (1600 x 11.0006)/100 + 0.13x = x.

Step-by-step solution

1. 1

   Simplify the constant term. Multiply first, then divide: $1600 \cdot 11.0006 = 17{,}600.96$ and $\dfrac{17{,}600.96}{100} = 176.0096$. Dividing by $100$ just shifts the decimal point two places left. The equation is now $176.0096 + 0.13x = x.$

2. 2

   Identify why x cannot simply be isolated by inspection. The unknown appears on both sides with different coefficients ($0.13$ and $1$), so the terms must be combined before dividing - you cannot just move $176.0096$ across and stop.

3. 3

   Collect the x-terms. Subtract $0.13x$ from both sides: $176.0096 = x - 0.13x = (1-0.13)x = 0.87x.$ In words: the fixed part $176.0096$ makes up $87\%$ of the total.

4. 4

   Divide by 0.87 to finish. $x = \frac{176.0096}{0.87} \approx 202.3098851 \approx 202.31.$ Multiplying numerator and denominator by $10^4$ gives the exact fraction $\tfrac{1760096}{8700} = \tfrac{440024}{2175}$.

5. 5

   Verify. $0.13 \cdot 202.3098851 = 26.3002851$ and $176.0096 + 26.3002851 = 202.3098851$, which is the value of $x$ - so the substitution closes exactly.

6. 6

   Watch the common trap. Dividing $176.0096$ by $1.13$ instead of $0.87$ (a very common slip) would give $155.76$. The $13\%$ is a share of the answer, not of the known constant, which is why the divisor is $1-0.13$.