Solve for x: x = 0.18x + 604.8.

Step-by-step solution

1. 1

   Read what the equation models. A total $x$ splits into an $18\%$ portion of itself plus a fixed remainder of $604.8$. So $604.8$ must be the other $82\%$ — which already tells you the answer is a bit above $700$.

2. 2

   Collect the x terms on the left. Subtract $0.18x$ from both sides and factor:

   $x-0.18x=604.8\qquad\Longrightarrow\qquad (1-0.18)x=604.8\qquad\Longrightarrow\qquad 0.82x=604.8$

   The operative coefficient is the complement $1-0.18=0.82$, not $0.18$.

3. 3

   Divide by 0.82.

   $x=\frac{604.8}{0.82}=\frac{60480}{82}=\frac{30240}{41}$

   Multiplying top and bottom by $100$ clears the decimal, then dividing by the common factor $2$ reduces it. Since $41$ is prime and does not divide $30240$, this is lowest terms.

4. 4

   Evaluate the decimal.

   $x=737.5609756\ldots\approx737.56$

   Matching the estimate of just over $700$ from step 1 ✓.

5. 5

   State the general rule. For $x=cx+k$ the solution is $x=\dfrac{k}{1-c}$. Here $\dfrac{604.8}{0.82}$ — and the same formula covers the whole family of self-referential percentage equations, blowing up as $c\to1$.

6. 6

   Verify by substitution. $0.18\times737.5609756+604.8=132.7609756+604.8=737.5609756$ ✓, identical to the left side to seven decimals.