# How do I round a calculation involving both addition/subtraction and multiplication/division to the correct number of significant figures?

I'm just learning about significant figures (sig figs) in my chemistry class, and I'm confused about the rules. An example to my problem: if I had the following expression: $$\left ( \frac{2.378 - 1.2}{1.03} \right )$$ I've thought of 2 ways to approach this. Which one is correct (or is neither correct)?

# Approach 1:

Round each intermediate operation as you go. This is what my textbook does for its examples, but it itself suggests "Approach 2" below. I'm just listing this approach here as an option since addition/subtraction and multiplication/division have their own rules for rounding (hence, maybe do intermediate-operation rounding).

### 1) Round the subtraction part.

$$\left ( \frac{1.2}{1.03} \right )$$

### 2) Do the division and round.

$$1.2\,\,\, (2\ \mathrm{sig\ figs})$$

# Approach 2:

In actual practice, the book says:

In a series of calculations, carry the extra digits through to the final result, THEN round.

I think it means this:

### 1) Do the subtraction, but keep the *exact* number while noting that the rounded number was supposed to have 2 sig figs.

$$\left ( \frac{1.178}{1.03} \right )$$

### 2) Do the division with the exact number and the denominator, and since this step produces the final result, NOW you round, rounding to 2 sig figs per the note in Step 1.

$$1.1\,\,\,(\text{2 sig figs})$$

The problem here is that these two approaches produce 2 different results. So, to repeat, which one is correct (or is it neither)? The textbook doesn't provide an example like this. I found two different websites, but they too conflict with each other.

http://www.occc.edu/kmbailey/chem1115tutorials/Sig_Fig_and_Math_Answers.htm (From the solution to problem #3, I should use "Approach 1".)

http://fabice.com/misc/significant_figures.html (It quotes, "In a composed operation, intermediate results are not rounded." So that means I should use "Approach 2?".)

Compared to the other questions on this forum, this question seems silly, but I find the concept of sig figs important if I want to understand chemistry at all.

• According to me first one is better. – Freddy Sep 1 '14 at 9:55
• According to me the second one is better. But that depends very much on the problem at hand. In praxi for every result you obtain you should also do an error analysis, which in the end gives you a value range, how trustworthy your number actually is. In this case, both of you values would be in this range. Rounding has been and will always remain a problem in the sciences. – Martin - マーチン Sep 1 '14 at 10:58
• So based on what you said, Martin, there's more to consider than just the type of operation involved, and that uncertainty should be somehow considered with every operation. Thanks, I'll look more into that. – DragonautX Sep 1 '14 at 15:36
• This question appears to be off-topic because it is about math, not about chemistry. – Wildcat Oct 5 '14 at 11:00
• I'm voting to close this question as off-topic because this question is more about maths. – Jan Oct 2 '15 at 16:17

Approach 2 is better than approach 1.

This approach corresponds with methodology employed in both web links you provided.

The basic 'rule' is to do your calculation keeping all digits as you go right until to the very end (without rounding parts as you suggest in approach 1). This ensures your calculation retains maximum accuracy. You leave the 'rounding' til the very end, that is when you report your answer, only 'so many' digits are 'significant'.

To use your example, when calculating the subtraction part: 2.378-1.2 = 1.178, in approach 1, by 'rounding' this down to 2 significant digits (1.2), you are reducing the accuracy of your calculation before you finish.

The correct (most accurate) way to calculate the result is to keep all significant digits in your calculation until the very end, that is, 1.178 / 1.03 = 1.14368932 and then report only the 'significant' digits. In this case, since the number 1.2 only has 2 significant digits, your final answer can only have 2 significant digits. That is, the final answer is only as precise as the least precise measurement (in your case, 1.2), so in reporting the final answer for your calculation, you can only rely on 2 significant digits, so you report 1.1 as your calculated answer.

If you re-examine the answer provided in your link (http://www.occc.edu/kmbailey/chem1115tutorials/Sig_Fig_and_Math_Answers.htm) in which you state problem 3 follows approach 1, you will find that the solution provided in that link actually follows approach 2, similar to what I described in the above paragraph. That is, all decimal places are retained in the calculation until the very, when the final answer is reported to the number of significant figures corresponding to the least significant measurement.

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