“Water with a mass of 35.4g is added to an empty flask with a mass of 87.432g. The mass of the flask with the water is 146.72g after a rubber stopper is added. Express the mass of the stopper to the correct number of sig. fig.”
A friend of mine posed this question to me earlier tonight. I believe the answer to be 23.9, as the number with the fewest SDs is 35.4, with 3.
Anyone have a different answer? Thanks
I believe you’re correct. I was taught that the most significant figures you are allowed to use is the number of sigfigs in the number with the least sigfigs.
Or something.
Yeah . . . that’s what I remember being taught as well. Then again, I learned that soph year of HS. That was '96 to '97. Four years can do a lot to the understanding of sig figs. Thanks 
Yes, I get 1.64 millislugs. (It’s a different answer, at any rate).
I agree with your solution. Use the lowest precision of any measured quantity as the precision of your answer.
You probably understand the following as well, but I’d like to mention it since I saw an error of this sort in a book that purported to be about clarity in mathematics : the precision of significant figures only applies to measured or estimated quantities, not “abstract” numbers or ratios. For example, 20% of 85.96 is 17.19 (same precision), not 20 or 17. However, if the 20% were an estimate, such as, “we figure about a 20% increase”, then the precision of the estimate is important.
panama jack
The real problem of a critique of our own cultural models is to ask, when we see a unicorn, if by any chance it is not a rhinoceros. - Umberto Eco
I also agree with your answer of 23.9 grams.
However, my reasoning differs from friedo’s. From what I recall, the pure number of sigfigs is what you go by when multiplying or dividing. When adding/subtracting, however, you count places from the decimal point.
For example: 0.1 g of water into a 32.1 g flask. What’s the total weight? I say that it’s 32.2 g, where you have significance to the 10[sup]-1[/sup] place. Surely you wouldn’t say that it’s 30 g, where the only significant figure is the 3?
Here’s my way of thinking about it: “0.1 g” really means that the mass is somewhere in between 0.05 and 0.15 grams - that’s as precisely as we know it. Similarly, 32.1 really translates into somewhere between 32.05 and 32.15.
When we add those two masses together, we kind of close our eyes, and hope that the roundoff error cancels out on average, and say that we’re reasonably confident that the total weight of 0.1 g and 32.1 g is 32.2 g (+/- 0.05 g again).
This is a little hand-wavy, and I sort of made it up on the spot here, but hopefully it makes at least some sense.
brad is correct. iampunha is right, but for the wrong reason.
Number of sig figs is used when multiplying or dividing.
Position of sig figs is used when adding or subtracting.
Example: 3.123 + .0022 = 3.125. You take it out as far right of the decimal point as the least precise figure (3.123), even though .0022 only has 2 sig figs.
111.2 x .0034 = .38. .0034 has 2 sig figs, 111.2 has 4, therefore the answer has 2.
If you stop and think about it for a while, it really does make sense.
So, what would you guys do with 3000 grams plus .2 grams?
Following the rules strictly (and assuming that only the 3 in 3000 is significant), 3000 + 0.2 = 3000.
If only the thousands digit of my 3000 gram figure is significant, that means that, really, I only know the mass of that object to within 500 grams - it’s someplace between 2500 and 3500 grams. Another 0.2 grams is, quite literally, lost in the uncertainty of the larger mass.
If I have a really good scale, and I measure my 3 kilogram mass to the nearest 0.1 gram, and it comes up to be 3000.0 grams, then adding another 0.2 grams is of course significant. You could then reliably quote the sum of the masses as 3000.2 grams. A reader would infer, correctly in this case, that you know the combined mass to the nearest 0.1 gram or so.
If you wrote 3000.2 when only the first digit of the 3000 gram measurement was significant, you’d really mislead a reader of your work. You still only know the mass of the combination to the nearest 500 grams, but quoting that many decimal places gives a very different impression. The guy building a pedestal for the 3000.2 gram object will be very disappointed when it weighs in at 3468 grams (as it could, given the limits of our measurement) and crushes his elegant design…
My point was, your rule (count places from the decimal point when adding) and smeghead’s rule (position of sig figs is used when adding) are wrong.
I guess I’m being thick…I don’t understand what you’re saying. Could you elaborate how you feel smeghead and I are wrong? Was the 3000 + 0.2 example supposed to point out an obvious flaw in my/our reasoning? If so, I don’t see it.
For that matter, I thought smeghead and I were saying the same thing. Am I missing something?
No, in your rules.
Okay…
I’m still not sure what you’re finding objectionable. What’s the problem with the rule(s)?
RM_Mentock -
I’m sure you understand the rules well. But brad & smeghead haven’t made any errors. They did not say to only use “decimal places” or “number of sig figs” in your answer. They did, in fact, say that the least precise number according to that method ought to be used. So in 3000 + 0.2, the 3 in the thousands place is the least precise, and determines the precision of the answer.
Couple of notes on working with numbers, from what I’ve learned :
If you need to indicate the precision of an exactly even number (contains 0’s, in other words) you might have a couple of ways to write it.
The simplest and most consistent way is to use scientific notation, simply giving the number of significant digits : 3.000 x 10[sup]3[/sup], 3.0 x 10[sup]3[/sup].
If the precision is better than the decimal point, it is of course easy to indicate by writing the extra digits: 3000.00
If the precision happens to coincide with the decimal place, you have the option to simply place the decimal point : ‘3000.’ indicates the same accuracy as the scientific notation 3.000 x 10[sup]3[/sup].
If the precision is worse than that, the only way would be to use scientific notation, though.
panama jack
RM, perhaps you’ve heard this one:
There’s a story about a family at a museum, and the kid asks how old the dinosaur skeleton is. A janitor standing nearby pipes up and says that it’s 70 million and four years old. The parents ask how he knows this, and the janitor replies, “Oh, we had some scientist come in and he said it was 70 million years old, and that was four years ago.”
panamajack said:
Not exactly. I’m in Chemistry 3-4 and my teacher is real big on significant figures.
He says that another way to do it is to put a horizontal mark (think long vowel sound mark) over the last significant figure. He had a textbook and a teacher who did it that way, and it does seem to work pretty well. EXAMPLE:
_
To the nearest 1: 3000g
_
To the nearest 10: 3000g
_
To the nearest hundred: 3000g
I happen to like scientific notation, myself.
And, from someone who’s had it more recently, Smeghead has given the complete, correct answer.
Neat little annoying thing I learned: the 3000 grams has but one sig fig in it. My chem teacher used to have us write 3000.0 grams, as that shows it’s been measured to the gram. Also she’d write 0.20 grams, again to how it’d been measured.
A kid in my class once got a 99 on a test (I am not making this up) because he failed to put the aforementioned line over the 0 in 30.
So 3000 + .2 grams I wouldn’t shake a stick at. 3000.0 + 0.20, OTOH, would be 3000.20. As per sig figs, I’d do 3000 with a line over the last 0. You’re rounding down in this case.
I’ll hopefully have an update as to whether my friend’s answers were right or not. And we’ll continue this thread with yet more chem Qs.
Garfield, the line thing you mentioned is EXACTLY what my teacher taught me. Made things a lot easier.
RM Mentock, you seem to be implying that there is a problem with these rules because it can’t handle adding .2 grams to something that weighs 3000 grams. It can indeed handle this. You just have to specify how many of the zeroes in the 3000 are significant, as has been elucidated already. If you know that it weighs 3000 grams to within a thousandth of a gram, you would call it 3000.000 grams, in which case the number has 7 sig figs. So if you have something that weighs 3000.000 grams, and you want to add .2 grams, the answer would be 3000.2 grams.
As has been said, the key is precision. If you have a pile of bricks that weighs “about” 3000 pounds and you add .2 pounds more, it still weights “about” 3000 pounds. Zeroes to the left of the decimal can be confusing, but that confusion can be overcome with scientific notation as has been pointed out. There is no application in which using sig figs will give you a wrong answer.
Not really. You only go to the LEAST accurate number’s amount of decimal places. In this case, 3000.0 goes to the tenths place, where .20 goes to hundredths. So, you round to tenths: 3000.2
Garfield:
Yeah . . . I know. I know. I know. Damnit. Minor fingerslip. Now I am mad at myself for that. Oh well.
No one seems to have made any errors in calculating significant figures (well, I guess iampunha has), but I don’t see where “least precise” appears in either set of rules.
Chronos
I almost posted that joke myself. Except my version has the janitor say the age of the dinosaur skeleton was 5 million years and 4 years and 3 months.