Solve the equation (2981 - x) / x = 4.3043 for x.

Step-by-step solution

1. 1

   Turn the ratio into a formula. Writing $r=4.3043$, the equation $\frac{T-x}{x}=r$ rearranges once and for all to

   $T-x=rx\;\Longrightarrow\;T=(1+r)x\;\Longrightarrow\;x=\frac{T}{1+r}$

   Knowing this closed form means the arithmetic below is just substitution, and it makes the $1+r$ (not $r$) unmistakable.

2. 2

   Substitute the numbers. With $T=2981$ and $r=4.3043$,

   $1+r=5.3043,\qquad x=\frac{2981}{5.3043}$

3. 3

   Carry out the division with enough digits.

   $x=561.99687\ldots$

   The digits after the decimal point are $99687$, so to two places $x=562.00$ — truncating to $561.99$ rounds the wrong way.

4. 4

   Recover the leftover part.

   $2981-561.99687=2419.00313$

   so the split of $2981$ is roughly $562.00$ and $2419.00$.

5. 5

   Verify the ratio.

   $\frac{2419.00313}{561.99687}=4.30430$

   This reproduces $r$ to six significant figures, confirming $x\approx 562.00$.