Solve the equation x - 1 = 0.98x.

Step-by-step solution

1. 1

   Move both $x$ terms to one side. With $x$ on the left and $0.98x$ on the right, no division is possible yet. Subtract $0.98x$ from both sides:

   $x-0.98x-1=0\qquad\Longrightarrow\qquad0.02x-1=0$

   Here $1-0.98=0.02$; getting this subtraction right is the whole difficulty of the problem.

2. 2

   Isolate the variable term by adding $1$ to both sides:

   $0.02x=1$

3. 3

   Divide by the coefficient. Since $0.02=\tfrac{2}{100}=\tfrac{1}{50}$, dividing by it multiplies by $50$:

   $x=\frac{1}{0.02}=50$

   Doing it as a fraction — $\tfrac{1}{50}x=1\Rightarrow x=50$ — keeps the arithmetic exact.

4. 4

   Check both sides of the original. Left: $50-1=49$. Right: $0.98\times50=49$ ✓. Verified in exact rational arithmetic as well, so $x=50$ is precise.

5. 5

   Read the percentage structure. The equation says a number shrinks by exactly $1$ when cut by $2\%$, so $1$ is $2\%$ of it and the whole is $1/0.02=50$. The general pattern is $x-1=(1-p)x\Rightarrow x=1/p$: the smaller the percentage cut, the larger the number — which is why $0.99$ gives $100$ and $0.98$ gives $50$.