Solve for x: 6.58x + 14 = 6.63x.

Step-by-step solution

1. 1

   Read it as a catch-up problem. The right side grows at $6.63$ per unit and the left at $6.58$, but the left starts $14$ ahead. The question is how far it takes for the extra $0.05$ per unit to make up a head start of $14$.

2. 2

   Move the x terms to the right. Subtract $6.58x$ from both sides:

   $14=6.63x-6.58x$

3. 3

   Subtract the coefficients.

   $6.63-6.58=0.05\qquad\Longrightarrow\qquad 14=0.05x$

   The gap is only five hundredths, so the answer will be roughly twenty times $14$.

4. 4

   Divide by 0.05 using the reciprocal shortcut. Since $\dfrac{1}{0.05}=20$, dividing by $0.05$ is multiplying by $20$:

   $x=\frac{14}{0.05}=14\times20=280$

   Unlike the $0.17$ version of this problem, which gives the repeating $\tfrac{1400}{17}\approx82.35$, this one lands on a whole number because $0.05=\tfrac{1}{20}$ divides $14$ evenly.

5. 5

   Verify in the original equation. Left: $6.58\times280+14=1842.4+14=1856.4$. Right: $6.63\times280=1856.4$ ✓. Exact agreement, confirming $x=280$.