How to solve (0.1 - 10.3 + 5.132)/12.8 and round off correctly?

Question

I've recently learnt the rules about rounding off when adding/subtracting and when multiplying/dividing. I know that when you add/subtract, the number of decimal places in the result should equal the smallest number of decimal places of any term in the sum or difference.

For instance, 2.3 + 10.88 = 13.18 ≈ 13.2

I also know that when you multiply/divide,the number of significant figures in the final answer is the same as the number of significant figures in the quantity having the smallest number of significant figures.

For instance, 10 * 1.73 = 17.3 ≈ 17

How do you round off correctly when there's mixture of adding/subtracting and multiplying/diving in the same math task? Do you follow the rules of rounding off when adding/subtracting or multiplying/dividing, or both?

The concrete math task is this: "Perform the following calculation and round off in the appropriateway":

(0.1 - 10.3 + 5.132) / 12.8

I'm not sure whether to: 1. calculate the first bracket precisely, so that I get (-5.068)/12.8 and then perform the division and round of so that the answer has three significant figures. or 2. calculate the first bracket and round off to -5.1 so that the division would look like this: (-5.1)/12.8 and the answer would have two significant digits.

Answer 1

Here is the rule I give to both my Chemistry and my Physics classes.

When mixing types of operations, do all you can of one type of operation, round, and then do all you can of the other type, then round again.

Of course, this is done according to the order of operations. So in your problem, do the additions inside the parentheses (in the numerator of the fraction), round, do the division, then round. So,

$$\begin{align} \frac{0.1 - 10.3 + 5.132}{12.8} &= \frac{-5.068}{12.8} & \text{not yet rounded} \\[2 ex] &= \frac{-5.1}{12.8} & \text{rounded to $1$ decimal place, due to $0.1$} \\[2 ex] &=-0.3984375 & \text{not rounded} \\[2 ex] &=-0.40 & \text{rounded to 2 significant digits, due to $5.1$} \end{align}$$