Rounding appropriately

[Probability]

How to get your legs kicked from under you!

I have this old math book which is over 10 years old. I did the course back then but do like a little revision now and again to try and keep the brain active.

In this book we were told that when rounding number like 91.2 or 91.5 then we should round appropriately. We were told that numbers less than 0.5 should be rounded down, while number 0.5 or more should be rounded up. Later we were shown how to round distance measurements to 9 s.f, which is very accurate in deed! Then we were advised about rounding to the nearest mile. So 91.2 would become 91 while 91.5 would become 92.

Using a much less accurate conversion factor we were then asked to convert 465 km to miles using 0.62 as a conversion factor, hence you can only attain the answer 288.3 miles. We were then asked to round this number appropriately, so given the previous understanding I'd of gone for 288 miles!

Previously in this course material we were advised and shown that it was not correct to write conversions like this; 2.3971 = 2.40 They are not equal.

Part of the question asked to explain why the answer given 288 miles was not accurate. Obviously because the conversion factor being 0.62 against the original conversion factor having 9 significant places would do it.

I then looked up the authors solution and found that 288.3 miles was agreed as I'd calculated it, however my rounded answer of 288 miles was somewhat different to the authors who had said that rounding the answer to 2 s.f gave 289, he said the answer found here agrees with the answer found in the calculation, hence; 289 = 290 to 2 s.f.

While I do fully agree with the coursework book being correct, and it is, I'm somewhat at a loss at the moment what the author was thinking when the solution was written. I know you'd need full facts but I can't post it was copyright material. It seems learning from a book (when under pressure) has its downfalls.

 

[JeffM]

First, you can and should quote the book. That is the very important "fair use" exception to the US law of copyright. Based on what you have said about the book, I suspect it may contain oversimplifications appropriate to high school students.

Second, there is no universal rule about rounding or truncation. If you read the section on rounding to an integer in the wikipedia article on rounding

Rounding - Wikipedia

en.wikipedia.org

you will find a multitude of rounding techniques and their rationales. It looks as though your book assumed that "commercial rounding" (also called "half-adjusting" among programmers) is universal, which it is not because it is biased.

Third, in mathematics, it is common to use the symbol [MATH]\approx[/MATH] to indicate that an answer is not numerically exact. So
you can truthfully say

[MATH]\sqrt{2} \approx 1.4[/MATH]
or

[MATH]\sqrt{2} \approx 1.414214[/MATH]
because both give fair warning that neither answer is exact.

Fourth, physical scientists have their own conventions about indicating accuracy and precision. I have not done anything with the physical sciences in over fifty years and so will not comment on those because they may have changed over the decades and my memory may fail me over that length of time..

In short, your book may be guilty of oversimplification of a complex issue. We can never know without your giving us exact quotations.

 

[Probability]

Thanks for replying Jeff. I'm not in the US so am not bound by the same laws. Here in the UK we cannot reproduce by any means copyright material. Removed information I've wrote just in case of a possible breach of copyright laws from the book material.

 

[JeffM]

It is impossible to comment accurately on something that has not been disclosed.

US law on exceptions to copyright (fair use) represents a fairly broad expansion of the exceptions permitted under UK law (fair dealing). But even in the UK, according to the UK's own Intellectual Property Office's website
https://www.gov.uk/guidance/exceptions-to-copyright#non-commercial-research-and-private-study, the exception is broader than you may think:

You are allowed to copy limited extracts of works when the use is non-commercial research or private study, but you must be genuinely studying (like you would if you were taking a college course). Such use is only permitted when it is ‘fair dealing’ and copying the whole work would not generally be considered fair dealing.

The purpose of this exception is to allow students and researchers to make limited copies of all types of copyright works for non-commercial research or private study. In assessing whether your use of the work is permitted or not you must assess if there is any financial impact on the copyright owner because of your use. Where the impact is not significant, the use may be acceptable.

If your use is for non-commercial research you must ensure that the work you reproduce is supported by a sufficient acknowledgement.

The question you have paraphrased does not make sense. Total distance in kilometers = 69 + 95 + 83 = 247. You appear to be given that fuel consumption is 12 kilometers per liter. So you have the information to solve the problem. Why do you say you have nothing? The answer is

[MATH]20 < \dfrac{247}{12} < 21 \implies 21 \text { liters will be enough.}[/MATH]
Now if you are studying mental arithmetic, you could do in your head that distance is approximately 70 + 100 + 80 = 250 and divide by 10, which suggests that 25 liters will be enough. Personally, I do not like that crude an approach because, although it worked this time, it will not always work. Instead, I would ensure that distance is overestimated and fuel consumption underestimated.

Distance < 70 + 100 + 90 = 260.

Fuel consumption > 10

Therefore 260/10 = 26 liters is perfectly safe.

Again, you are giving your version of problems, not the actual wording, and you are not telling us the context of what the problem is teaching. I have no idea whether my responses are at all relevant.

 

[Probability]

Thanks Jeff. Copying and reproducing copyright material is not something I'm familiar with regards the law except I'm aware that I've always been lead to understand we're not supposed to use it on open forums etc. I thought afterwards when I looked back on the material that the question advised about 12 km per litre used and I'd not thought about the idea of rounding that figure off to 10, which is what the author had done. Now if the mileages from A to B to C have been calculated in the question, such that they advised 69 + 95 + 83, then I can understand rounding them off and gaining results of 70 + 100 + 80, however I'm not sure in the context of the question that I'd be in the right to add 10 miles to a given distance? I see where you are coming from but in the question I'd assume that I must stick to their factual information for their example? I'm sorry if I've not been as clear as could be, but for security purposes and piece of mind maybe these types of problems would be better shared if the forum had of had a private section not viewable to the world at large, with some type of security in place then copyright material could be shared without legal comebacks on the poster.

 

[JeffM]

Hey, paranoia is not always irrational. I greatly doubt that, even in the UK, a student's quotation on a public help site of a problem in a book ten years old would be considered a legal infraction, but I am not the one who will get sued if I am wrong. You can always quote the exact problem to me in a private message, and I promise that I shall not disclose it to the world at large. In any case, what is probably more important than the exact words is context.

As I have said, I cannot be sure of context. But it seems that what is being discussed is not rounding but rather approximation more generally. Rounding is just one form of approximation. In general, when approximating, the aim is to avoid having errors cause harm. If I underestimate the gas needed and, as a result, I get stuck in a rainy Glaswegian slum at night with a squalling grandbaby, my wife will be seriously displeased. Then I might get haggis for supper the next evening. So I want to overestimate the gas needed. That is, even slight errors in approximations may have serious consequences, and whatever technique of approximation I select, I want to greatly reduce the probability that serious adverse consequences occur. A great deal of life involves making decisions that are not, and cannot be, exact. The idea that you should think about how to minimize the consequences of error is simply common sense, not mathematics. Good techniques of approximation depend on the purpose and circumstances of approximation, not a rule book.

From the standpoint of a beginning textbook, I can see why it may say that the "right" answer is 25 liters. And if you were taking a test, I'd say to follow whatever nonsense the book spouts. But, if I understand correctly, you are trying to learn, not pass the A levels. Once you abandon the exact information supplied, every answer is technically wrong. Moreover, the information supplied may not be the true information, in which case finding the "correct" answer is a matter of pure luck. That is, how confident are you that the specific car that you will be driving on that specific route under the driving conditions of that specific day will get exactly 12 kilometers per liter and that the distance between the specific destinations to be visited is exactly 247 kilometers? Unless all that information is known with certainty, there simply is no way to find "exact" answer. If I buy 26 liters of gasoline rather than 25, what is the harm? I shall use it the next time I drive the car.

There actually is a branch of math that deals with this kind of problem. It is not hard, but it requires understanding that numbers are always exact only in an ideal world, but no one actually lives in that kind of world.

Decades ago, when I was much younger and had small children dependent on my continued employment, my job was bidding to acquire failed banks. If I bid too high, my employer would have lost millions, but I would have faced a catastrophe. I recommended reasonable bids, but never at the high end of reasonable. And when we won a bid, everyone was eventually pleased. A wise director at the time called them "no tears bids." But I was still scared witless when we won a bid.