Solve for x: x = 0.4 * 200 + 0.4x.

Step-by-step solution

1. 1

   Evaluate the numeric product. $0.4\times200=80$, so

   $x=80+0.4x$

   The unknown now appears on both sides, which is the only real difficulty.

2. 2

   See why you cannot simply divide. Reading the equation as $x=80/0.4=200$ would ignore the $0.4x$ on the right. Because $x$ is defined partly in terms of itself, the $x$ terms must be gathered before any division.

3. 3

   Move the 0.4x to the left. Subtract $0.4x$ from both sides and factor:

   $x-0.4x=80\qquad\Longrightarrow\qquad (1-0.4)x=80\qquad\Longrightarrow\qquad 0.6x=80$

   The operative coefficient is $1-0.4=0.6$, not $0.4$.

4. 4

   Divide by 0.6.

   $x=\frac{80}{0.6}=\frac{800}{6}=\frac{400}{3}=133.\overline{3}\approx133.33$

   The exact fraction $\tfrac{400}{3}$ is the cleaner answer, since the decimal repeats.

5. 5

   Compare with the sibling cases. The general solution of $x=c\cdot200+cx$ is $x=\dfrac{200c}{1-c}$: at $c=0.3$ it gives $\tfrac{600}{7}\approx85.71$, at $c=0.35$ it gives $\tfrac{1400}{13}\approx107.69$, and at $c=0.4$ it gives $\tfrac{400}{3}\approx133.33$. The growth accelerates as $c$ approaches $1$, where the equation has no solution at all.

6. 6

   Verify by substitution. With $x=\tfrac{400}{3}$: the right side is $80+0.4\cdot\tfrac{400}{3}=80+\tfrac{160}{3}=\tfrac{240+160}{3}=\tfrac{400}{3}$ ✓, identical to the left side in exact fraction arithmetic.