Solve 1 over 2.7821 equals 1 over 3.3 plus 1 over x, for x.

Step-by-step solution

1. 1

   Move the known reciprocal across.

   $\frac1x=\frac{1}{2.7821}-\frac{1}{3.3}$

   Since $2.7821<3.3$ the result is positive, so a positive $x$ is expected.

2. 2

   Use the difference-of-reciprocals identity.

   $\frac1a-\frac1b=\frac{b-a}{ab}\quad\Rightarrow\quad \frac1x=\frac{3.3-2.7821}{2.7821\times 3.3}$

   One subtraction and one multiplication replace two long divisions, which keeps the rounding error out of the calculation.

3. 3

   Evaluate both parts.

   $3.3-2.7821=0.5179,\qquad 2.7821\times 3.3=9.18093$

   (Check: $2.7821\times 3=8.3463$ plus $2.7821\times 0.3=0.83463$ gives $9.18093$.)

4. 4

   Take the reciprocal.

   $x=\frac{9.18093}{0.5179}=17.7272253\approx 17.7272$

   Notice how sensitive this is: the numerator $0.5179$ differs from the previous problem's $0.5579$ by only $0.04$, yet the answer moves by about $1.5$.

5. 5

   Check the arithmetic backwards. $\frac{1}{3.3}=0.3030303$ and $\frac{1}{17.7272253}=0.0564104$, summing to $0.3594407$; and $\frac{1}{2.7821}=0.3594407$ ✓. So $x\approx 17.7272$.