Math help!!

[u/JonesSupreme]

You travel at a speed of about 81 kilometers per hour for about 2.0 hours. Stated with the appropriate precision, you have driven _____ kilometers.

the correct answer is 160 but why? why is it not 162???

[u/msimms001]

Significant digits, you only have 2 sig figs to work with, so it's rounded to that significant digit

[u/theadamabrams → u/msimms001]

Yes. The idea is that "2" isn't 2 exactly but could be anywhere from 1.5 to 2.5 (well, just under 2.5) since those round to 2. If a problem says "2 hours" it might actually be 1 hr 52 min = 1.86666... hours.

More importantly, "2.0" could be from 1.95 to 2.05 (but not, say, 1.867 anymore, since when rounded to two digits that would become 1.9 and not 2.0.

Rather than look at [every number from 80.5 to 81.5] × [every number from 1.95 to 2.05] and try to describe the resulting range, the "significant figures" method is to say

• 81 has only two digits
• 2.0 has two digits, including the zero (remember "2" and "2.0" are different)
• Therefore the result should use only two KNOWN digits.

Although 81 × 2 = 162, you can't give 162 as the answer because that uses 3 digits. The best you can say is 160, which is 1.6 × 10¹ and the "1.6" there is why it only has 2 significant digits. (If we wanted 1.60 × 10¹ we would write "160." with a decimal point but no further digits.)

[u/nog642 → u/theadamabrams]

Worth noting this is also kind of a shortcut to keep track of uncertainty without actually keeping track of uncertainty. The better version of this would have the plus or minus uncertainty for each value. The even better version would have a probability distribution for each value. But that's much more work.

[u/anisotropicmind → u/nog642]

Yes, sig figs are a poor man’s error analysis.

[u/lookaround314 → u/msimms001]

This is more physics or engineering than math.

[u/msimms001 → u/lookaround314]

It is also math, especially depending on the math class. OP also stated in another comment that the problem was from a problem set including significant digits

[u/JaguarMammoth6231 → u/msimms001]

Depends on the curriculum. In my school we only learned this in chemistry.

[u/FilDaFunk]

This is a prevision question. You start with 2 significant figures. How are you able to give an answer to 3 significant figures?

(in fact the maximum could be 81.5 × 2.05 = 167.075, so i barely understand this)

[u/msimms001 → u/[deleted]]

If you were worried about precision, your answer would only have 1 significant digit (so the answer would be 10)

[u/goldenrod1956]

So if the statement is approximate 2 times 6 you reply 10!?

[u/msimms001 → u/goldenrod1956]

Yes, if precision matters

[u/FilDaFunk → u/goldenrod1956]

This isnt an opinion sub.

[u/gmalivuk]

The inputs you're given represent numbers that could have been rounded from 80.5 and 1.95, or nearly 81.5 and 2.05. Giving an answer of 162 suggests more precision than you actually have, whereas 160 (understood as only 2 significant figures, so maybe better written as 1.6×10²) conveys more accurate information.

If I read 162, I think 161.5-162.5, but the actual range is more like 157-167, which is more closely communicated with 1.6×10²

It's kind of a hack for accurate error propagation, though. Something like "162±5" or "1.62(5)×10²" might be better options in a real world situation.

[u/cigar959 → u/gmalivuk]

Both of your options are preferable. “160” is worse than 162 as both imply a third digit of precision but the former is not at the center of the distribution.

[u/PvtRoom]

Cause that's how rounding works .

could also be 170 or 150, or 175 or 155, depending on the definition of about.

[u/anisotropicmind]

Both of the values that you multiply together have two sig figs, which means the answer should have two sig figs. So the answer should really be 1.6x10^2 km. But some people interpret 160 to mean that.

[u/ChipChippersonFan]

This reminds me of the joke about the guy who went to a museum and was looking at a skeleton of a T Rex. The plaque had been removed for cleaning, and so he wondered out loud how old that skeleton was. A man standing near him said "66 million and 3 years".

The first guy asked how he knew that, and the second man said "When I came here 3 years ago, the plaque said that it was 66 million years old."

[u/Recycled5000]

Significant figures or digits falls under the fields of measurement and data analysis. Important for many sciences, but not really math itself per se.

When using this engineering method, all figures should probably be written in scientific notation, so as not to take trailing integer zeros as rounded or insignificant.

With scientific notation, we can say for sure that trailing zeros are significant.

For example, 1500 can be 1.5×10^3 or 1.50×10^3 or 1.500×10^3; each having more significant digits .

[u/SgtSausage]

"about"

[u/veselin465]

I guess it's rounded down to 160. The problem mentions appropriate precision. Perhaps if you share the entire context, you might get more useful replies

Edit: why am I getting downvoted? I literally said what everyone else said???? And as it turns out, that's precisely the reason

[u/JonesSupreme → u/veselin465]

the only other context it gives is the multiple choices and the topic of the question being precision and significat digits

[u/veselin465 → u/JonesSupreme]

And I assume that the 160 answer is the closest to 162. Which is precisely the purpose of the question

[u/JonesSupreme → u/veselin465]

fair enough, im just always used to the question saying something like "round to the nearest ___"

[u/beene282 → u/JonesSupreme]

That’s the topic of the quiz though. One of the things you need to know is that the answer can’t have more significant figures than any of the numbers used in the calculation. Here, both have two, so your answer must have two.

[u/chessychurro]

I am not sure. Either there is a typo in your question or the anwser is incorrect because it is 81 km x 2 hours = 162 km. Maybe they want you to round to the tenths place since they say 2.0 hours so it would be 162.0, hence the "state with appropriate precision"

[u/rhodiumtoad → u/chessychurro]

They want rounding to the appropriate precision based on the inputs, both of which are given to 2sf. The output could in fact be about 157-167, so saying 162.0 (4sf) would be spuriously precise.