Solve 1 over 2.7421 equals 1 over 2.7 plus 1 over x, for x.

Step-by-step solution

1. 1

   Isolate the reciprocal of the unknown. Subtract $\frac{1}{2.7}$ from both sides:

   $\frac1x=\frac{1}{2.7421}-\frac{1}{2.7}$

   Solving for $\frac1x$ first, and only inverting at the very end, avoids trying to combine three reciprocals at once.

2. 2

   Predict the sign before computing. Reciprocals reverse order for positive numbers: since $2.7421>2.7$, we have $\frac{1}{2.7421}<\frac{1}{2.7}$, so the difference is negative and $x$ must be negative too. In the physical reading (this is the shape of a thin-lens or parallel-resistance formula) that is the signal that no positive second element can push the combination above one of its parts.

3. 3

   Combine over a common denominator. Using $\frac1a-\frac1b=\frac{b-a}{ab}$ with $a=2.7421$, $b=2.7$:

   $\frac1x=\frac{2.7-2.7421}{2.7421\times 2.7}=\frac{-0.0421}{7.40367}$

4. 4

   Invert to get $x$.

   $x=\frac{7.40367}{-0.0421}=-175.8591$

   The answer is large in magnitude because the numerator $-0.0421$ is small: a tiny gap between the two given numbers forces a huge $x$.

5. 5

   Check by substitution. $\frac{1}{2.7}=0.3703704$ and $\frac{1}{-175.8591}=-0.0056864$; their sum is $0.3646840$, and $\frac{1}{2.7421}=0.3646840$ ✓. So $x\approx -175.86$.