Solve x divided by (0.33668 + x) equals 0.85, for x.

Step-by-step solution

1. 1

   Interpret the equation as a share. $x$ is $85\%$ of the total $0.33668+x$, so the constant $0.33668$ is the other $15\%$. The answer must therefore be $\frac{85}{15}=\frac{17}{3}$ times $0.33668$ — a fraction that does not terminate, which forecasts a repeating decimal.

2. 2

   Multiply both sides by the denominator.

   $x=0.85(0.33668+x)$

3. 3

   Distribute the 0.85.

   $x=0.286178+0.85x$

   because $0.85\times 0.33668=0.286178$.

4. 4

   Move the $x$ terms together.

   $x-0.85x=0.286178\quad\Rightarrow\quad 0.15x=0.286178$

5. 5

   Divide by 0.15.

   $x=\frac{0.286178}{0.15}=1.907853\overline{3}$

   Exactly, $x=\frac{17}{3}\times 0.33668=\frac{5.72356}{3}$, whose decimal expansion repeats — the $\frac13$ predicted in step 1.

6. 6

   Verify. The total becomes $0.33668+1.9078533=2.2445333$, and

   $\frac{1.9078533}{2.2445333}=0.85$

   so $x\approx 1.90785$ (exactly $\frac{17}{3}\cdot 0.33668$).