Solve 281.45 = x + 53.23 + (x x 0.05)/(1 - 0.05).

Step-by-step solution

1. 1

   Simplify the denominator. $1-0.05 = 0.95$, so the equation reads $281.45 = x + 53.23 + \frac{0.05x}{0.95}.$ The structure is a base amount $x$, a fixed charge $53.23$, and a loading computed as $5\%$ of the grossed-up figure.

2. 2

   Reduce the fraction to an exact value. $\frac{0.05}{0.95} = \frac{5}{95} = \frac{1}{19} = 0.052631578\ldots$ Keeping $\tfrac{1}{19}$ rather than the rounded $0.0526$ keeps the whole calculation exact.

3. 3

   Move the constant across. $281.45-53.23 = 228.22 = x+\frac{x}{19}.$

4. 4

   Factor out x. $228.22 = x\left(1+\frac{1}{19}\right) = \frac{20}{19}\,x.$ This is the key simplification: the coefficient is the tidy fraction $\tfrac{20}{19}$, not an awkward decimal.

5. 5

   Solve by multiplying by the reciprocal. $x = 228.22\cdot\frac{19}{20} = \frac{4336.18}{20} = 216.809.$

6. 6

   Verify. $216.809 + 53.23 = 270.039$, and the loading is $\tfrac{216.809}{19} = 11.411$; the total is $270.039+11.411 = 281.450$, matching the left-hand side exactly.