Solve for x: x/(10 + x) = 0.98.

Step-by-step solution

1. 1

   Recognise the structure and predict the size. The left side is a fraction of a total: $x$ out of $10+x$. Requiring that share to be $98\%$ means the fixed $10$ can only be $2\%$ of the total, so the total must be about $500$ and $x$ about $490$. Having that estimate first makes the algebra self-checking.

2. 2

   Convert the decimal to a fraction. Exact arithmetic beats decimals here:

   $0.98=\frac{98}{100}=\frac{49}{50}$

   so the equation is $\dfrac{x}{10+x}=\dfrac{49}{50}$. Note also the domain restriction $x\neq-10$.

3. 3

   Cross-multiply.

   $50x=49(10+x)$

   This is valid because $10+x\neq0$ at the solution.

4. 4

   Expand and collect the x terms.

   $50x=490+49x\qquad\Longrightarrow\qquad 50x-49x=490\qquad\Longrightarrow\qquad x=490$

   The coefficients almost cancel, leaving $1\cdot x$ — which is why no division is needed at the end.

5. 5

   Verify by substitution.

   $\frac{490}{10+490}=\frac{490}{500}=0.98\ \checkmark$

   matching the estimate of about $490$ from step 1 ✓.

6. 6

   Note the general formula. For $\dfrac{x}{a+x}=p$ the solution is $x=\dfrac{ap}{1-p}$. Here $\dfrac{10(0.98)}{0.02}=\dfrac{9.8}{0.02}=490$ ✓. The $1-p$ in the denominator is why the answer blows up as the target share approaches $100\%$.