Solve the equation x - 1 = 0.99x.

Step-by-step solution

1. 1

   Get both $x$ terms on the same side. The variable appears on the left and the right, so nothing can be divided out yet. Subtract $0.99x$ from both sides:

   $x-0.99x-1=0\qquad\Longrightarrow\qquad0.01x-1=0$

   The coefficient is $1-0.99=0.01$ — a small number, which is precisely why the answer will be large.

2. 2

   Isolate the term in $x$. Add $1$ to both sides:

   $0.01x=1$

3. 3

   Divide by the coefficient. Dividing by $0.01$ is the same as multiplying by $100$, since $0.01=\tfrac{1}{100}$:

   $x=\frac{1}{0.01}=100$

   Working in fractions avoids any decimal drift: $\tfrac{1}{100}x=1\Rightarrow x=100$.

4. 4

   Verify in the original equation. Left side: $100-1=99$. Right side: $0.99\times100=99$ ✓. The check was also run in exact rational arithmetic, so the value is $100$ on the nose, not a rounded approximation.

5. 5

   Understand what the equation says. It asks for the number that drops by exactly $1$ when reduced by $1\%$ — so $1$ must be that $1\%$, making the whole $100$. Reading the structure this way turns the algebra into a one-line percentage argument, and it also predicts that $x-1=0.98x$ would give $50$, since then $1$ is $2\%$ of the whole.