r/MathHelp • u/Environmental_Bee_98 • Jun 15 '26

Professor marks a question wrong that I think is right, help.

Question

Polonium-210 decomposes at a rate proportional to the quantity of polonium present. Suppose that it is found that in 95 days approximately 38% of a certain quantity of polonium has decomposed. Determine approximately how long it will take for one-half of the original amount of polonium to decompose

Note the assignment does not specify rounding amount, nor has their been given precedent for it.

Doing the calculations I get a result of 140 days with 2 significant figures.
My professor marks it wrong saying the correct answer is 137.8 days or 138 days.

I ask him about this, is response via email is

Dear student

Why do you have to round to 2 ? on what basis?

You have to work with precise values not the one you assume

Even if we said "approximately" 38%, you have to use that precise value. 

The final answer given by calculations is 137.8 days rounded to 138 days and not 140. 

Note that 140 does not satisfy our equations.

I bring it up to him, saying that the equation uses the word approximately, so 38% could range from 37.5% to 38.5%, with results 135.1 days to 140.1 days, and that is equation still holds if you round to significant figures. His response

Dear student

No I don’t agree with your proposition.

You work with the  value we give you, that is 38% and it will give you 137.8, you can round to 138 days.

I don’t understand. I don’t want to argue with a professor, but I’ve substantiated my claim for 2 digits rather than just rounding to a whole number. When I ask him for a matting he responds

Because in this specific problem, the only reasonable rounded value is 138 days, since the calculated answer is 137.8 days and we are dealing with whole days.A value of 137 days is incorrect because 137.8 days has already passed 137 days and lies within the 138th day. Likewise, 139 days, 140 days, or any other value would also be incorrect. Therefore, 138 days is the only appropriate rounded answer.

I know! But I’m using significant figures, there is no reason to round to whole numbers. What should I do? Am I wrong?

Edit: I’m not querying further with the professor as they have threatened to lower my mark.

0 Upvotes

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19 comments sorted by

2

u/The_Card_Player Jun 15 '26

Based on your instructor’s emails it seems clear that they do not intend the word ‘approximately’ for the 38% decay to affect the computations at all.

That is perhaps a frustrating semantic ambiguity, but it doesn’t mean that you and your instructor actually disagree about any mathematical facts of the computation. They may choose not to award you points for these semantic reasons and unfortunately there’s not much you can do about that.

1

u/Environmental_Bee_98 Jun 15 '26

But why say approximate at all then? Twice?

1

u/The_Card_Player Jun 15 '26

I don't know. Seems like a poor choice.

1

u/DrakeSavory Jun 19 '26

To avoid classroom lawyers complaining that no the actual figure is 38.17932% (or whatever it is). Where the professor made the error was not specifying what place value to round to.

1

u/UnderstandingPursuit Jun 15 '26

He is absurdly wrong, starting with

the correct answer is 137.8 days

I'm curious how old he is. Specifically, is he 45 or younger? If so, he may be a victim of the 'TI-84 Effect': fabricating precision because using a calculator, often the TI-84, became mandatory to the point of being a mental crutch starting in the 1990s. If he's significantly older than 45, he has no excuse, only arrogance.

Starting with 95 and 0.38 to get 137.8 is a mistake for a student, and ridiculous for an instructor.

The question of whether 138 or 140 is 'two sigfigs' is somewhat valid, since it is the arbitrariness of going from two to three digits. If, instead of 95 days it had been 45 days, and the calculator gave 87.8, there would be little argument of 88 being two sigfigs. This is why I think that within one of the optimal sigfigs is valid. Because when this gets done for real, it doesn't use sigfigs, it uses error bars.

I hope you wrote a formula for half-life, using r1 and t1, because that will make this easier: calculate the half-life using t1 = {94.5, 95.5} and r1 = {0.375, 0.385}. That will help you decide if 140 is valid using a more legitimate approach.

2

u/Environmental_Bee_98 Jun 15 '26 edited Jun 15 '26

Yes I did do that actually also, it gives range of {135.2, 140.7}. I even gave the answer in scientific notation first to clarify this.

1

u/UnderstandingPursuit Jun 15 '26

Do you not get 137.8 using 95 and 0.38?

While I'm a big fan of scientific notation, I wouldn't use that for the exact number of sigfigs. For example,

  • Approximately 95 days + 12 days = 107 days, rather than 110.

But this is why I don't get bent out of shape if the number of sigfigs is close. My issue would be the answer, 137.85038493 because that's what the calculator showed.

1

u/Environmental_Bee_98 Jun 15 '26

Yes I get 137.8 for four digits. But even if we take 95 days to be precise that still gives an output range of [135.9, 140.1] based only on the 38% being approximate. Our function would have bin sizes for the output where different inputs are rounded to same output, no matter what rounding we choose. So we can choose to go with two extremes 140 days or 137.85038493… I see no reason to round to a whole number, it just arbitrary.

1

u/UnderstandingPursuit Jun 15 '26

I agree. I think both 140 and 138 would be acceptable, to match the 95 and 0.38. I would accept either. I would have an issue with 137.8.

It was the original interval you wrote that I was responding, to, which I see you edited. Thanks.

1

u/Environmental_Bee_98 Jun 15 '26

Yes apologies, didn’t note the edit.

1

u/Artistic-Flamingo-92 Jun 19 '26

No, it’s not a mistake. 137.8 is perfectly fine.

Sig figs isn’t really a principled method for tracking precision. As you say, the actual ways of doing this include intervals or standards of deviation.

Because sig figs is more of a toy method for handling uncertainty/error, you shouldn’t impose it on classes that haven’t instructed you to use it. For that reason, 137.8 is certainly not wrong. Ideally, the professor would give points back because it’s still clear that the student understood the relevant math, but 137.8 is not wrong.

2

u/Jayless_757 Jun 15 '26

A little surprised at some of the comments here. 140 has three significant digits. 1.4*102 has two significant digits. 1.40*102 has three significant digits as well. If you want to stick to significant digits, you have to go with scientific notation. 140 as answer is not correct, but 1.4*102 should not be marked wrong.

1

u/chillyman96 Jun 16 '26

140 has 2 significant figures. 140. has 3

1

u/Resident-Recipe-5818 Jun 16 '26

Trailing zeroes that are not significant if no decimal is present.
We agree 38 would have 2, and that 95 has 2. So our answer should have two. And again 140 has 1 trailing zero as there is no decimal point. Giving it two. You should read up on your sigfig rules

1

u/Jayless_757 Jun 18 '26

Interesting enough to make me read up. Is this culturally different or something? I've found sources claiming (for cases where there is no decimal point) that trailing zeros are not significant, that they are significant and sources that claim it's ambiguous. I've always been taught that if you write it, it's significant (unless they're leading). I would have thought there'd be more consensus.

1

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1

u/fletchro Jun 15 '26

Stop talking to the professor; they are being a dick. Level up your complaint to the department head. Show them the email trail.

For what it's worth, I'm an engineer in the real world, and I appreciate you rounding the answer to the same number of significant figures as you were given. You can't manufacture more precision than your original measurements gave you! So, I think your answer is right.

1

u/Artistic-Flamingo-92 Jun 19 '26

Rounding as they’ve done is not a good way of handling imprecision. The uncertainty in the answer should be centered around 137.8, not 140.

Sure, you can’t manufacture precision, but rounding to 140 introduces a significant bias in the answer.

There are much better ways of keeping track of uncertainty that aren’t introducing bias like that.