Multipling / Dividing with Significant Figures

Question

I have:

$$\frac{(10.3) + (0.01345) }{ (10.3) \cdot (0.01345)}$$

The answer need to be given in scientific notation. I know the rules for addition/subtraction and multiplication/division, however, something is messing me up. Do I round before putting it in scientific notation or after?

For example, $10.3 + 0.01345$ is $10.3145$. Rounding it before would yield $10.3$ thus $1.03 \cdot 10^1$. Rounding it after would yield $1.0 \cdot 10^1$. Depending on what numbers I get here will affect my answer, and so far I've gotten $3$ different answers.

Answer 1

You should round according to the rules of significant figures in the order of operations after performing each operation. For the numerator, you should add them and then round to the appropriate number of significant figures. I can't speak for actual scientists, but that's how they want you to do it in school most of the time.

Answer 2

If you use what I said...which is almost certainly not what you "should" do...then we would have:

$$ f(x) = \frac{x + y}{xy} = \frac{1}{x} + \frac{1}{y}\\ \frac{\partial f}{\partial x} = -\frac{1}{x^2} \\ \frac{\partial f}{\partial y} = -\frac{1}{y^2} $$

This would give (assuming that each last significant digit is correct to $\pm 0.5$ of the next):

$$ \Delta f \approx \frac{1}{10.3^2}(0.05) + \frac{1}{0.01345^2}(0.000\ 005) \approx 0.028\ 110\ 495\ 86 $$

The actual result would then be:

$$ f(x) \approx \frac{10.3 + 0.01345}{10.3*0.01345} \approx 74.446\ 529\ 757\ 8 $$

This would suggest that the result is accurate to the "second" decimal:

$$ f(x) \approx 74.4(4) $$

This notation means that the results is approximately $74.44$ but that the final digit is uncertain. If, instead, you use the "rules", you will find that:

$$ 10.3 + 0.01345 \approx 10.3 $$

Then that $10.3 * 0.1345 \approx 0.139$ and finally that:

$$ \frac{10.3}{0.139} \approx 74.1 $$

If, instead, we were to use: $\frac{1}{x} + \frac{1}{y}$, we would find that:

$$ \frac{1}{10.3} \approx 0.0971\\ \frac{1}{0.01345} \approx 74.35 $$

We would then go ahead and add them, to get:

$$ 74.35 + 0.0971 \approx 74.45 $$