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I was just tutoring someone and we went through some sort of diagnostic test thing, when the following question came up.

Question:

Here is some information about $50$ people who took the driving test: $18$ of the $50$ people are teenagers. One quarter of the adults failed the driving test. The number of adults who passed the driving test was $8$ more than the number of teenagers who passed the test.

The final part of the multi-part question was:

A person is chosen at random. What is the probability that they passed the driving test? [1 mark]

The student answered: $\ \frac{40}{50}.\ $

I said, "that's correct, but you should simplify the answer because it's always good to simplify fractions unless there is a good reason not to."

They replied, "but it's the last part of the question (and so there isn't another part of the question that relies on simplification of this fraction), simplifying takes up time that they could be spending doing other questions in the test." So basically, they disagreed with me.

I then stopped for pause and admitted, maybe they're right, but I'm not sure if they would get the mark if they do not fully simplify the fraction or give the answer as a decimal. But I'm not sure I am correct about this. Are they correct and I am mistaken? What is the correct response to my student?

Source of the test, for anyone interested: https://qualifications.pearson.com/content/dam/pdf/GCSE/mathematics/GCSE%20Maths%20Online%20Study%20Course_Diagnostic%20Assessment.pdf Note that the test doesn't give instructions to what form the answer should be given.

And the context is that this is a GCSE student I have just begun tutoring, and I basically gave them this assessment to broadly test their ability of fundamentals (arithmetic, shapes, problem solving...) just to get a feel for the level they are at, because students often over- or occasionally under-estimate their own abilities, as well as testing their speed and their accuracy in test conditions. As already stated, I expect students to simplify fractions where possible unless there is a good reason not to do so because that is what I have been taught to do and it has fared me well. This entire question is basically me calling into question whether or not simplifying fractions is a reasonable expectation in this context.

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    $\begingroup$ Related, from math SE: math.stackexchange.com/questions/3900917/… $\endgroup$
    – Ethan Bolker
    Commented Nov 13, 2021 at 16:05
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    $\begingroup$ Related but mostly about radicals rather than rationals. I do think that cancelling common factors should be nearly automatic in the final answer (when guaranteed never to vanish, should the factors involve symbolic expressions), but there are caveats. $\endgroup$
    – Jyrki Lahtonen
    Commented Nov 15, 2021 at 8:49
  • $\begingroup$ @JyrkiLahtonen I disagree. Just because it has been drilled into us to simplify fractions or radicals automatically, doesn't mean that simplifying is more valid than not simplifying. Only if the question asks us to simplify, or when simplifying is part of the pedagogical purpose of the exercise should we penalise the answerer for not doing so. $\endgroup$
    – Adam Rubinson
    Commented Nov 15, 2021 at 13:10
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    $\begingroup$ I've seen a student in university linear algebra carry $3/6$ through a (correct) computation to the end. When asked why, he replied "you don't allow calculators". (This has occurred repeatedly.) There are sound pedagogical reasons for asking students to simplify fractions, the first of which is that many don't know how to do so, and that's evidence of a problem. Another, more subtle, is that not doing so leads to losing a "feel" for which of $3/4$ and $45/60$ is bigger ... $\endgroup$
    – Dan Fox
    Commented Dec 19, 2021 at 10:35

14 Answers 14

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The answer to your question depends on the pedagogical goal of the exercise, and what learning outcomes you have identified. It basically comes down to the following question:

Is manipulating fractions one of the skills which you are emphasizing in this class?

If one of the goals of your class is to get students to more handily work with fractions, then yes, you should require them to simplify fractions. On the other hand, if working with fractions is not one of the identified learning outcomes for the class, then you probably shouldn't insist upon it.

However, if you are not the one setting the marking guidelines, then the only possible answer is "Ask whoever is in charge."

As an example from my own teaching: in my precalculus classes, we spend some time talking about various equations for a line. Starting from very basic Greek geometry, we can conclude that if $(x_1,y_1)$ and $(x_2, y_2)$ are any two points in the plane, then an equation for the line is $$ (y-y_1)(x_1 - x_2) = (x - x_1)(y_1 - y_2). $$ After a little bit of manipulation, this can be written as $$ y - y_1 = \left( \frac{y_1 - y_2}{x_1 - x_2} \right) (x-x_1) $$ (assuming that $x_1 \ne x_2$) which is the point-slope equation for a line. For example, the line through the points $(2,5)$ and $(1, 3)$ is given by $$ y-3 = \left( \frac{3-5}{1-2} \right)(x-1). $$ I am perfectly happy to give full credit to this answer on an exam, as I expect my students to be capable of basic arithmetic, and the goal of the exercise is to demonstrate recall of this equation.

On the other hand, I would expect them to simplify this equation in the context of a question like "Find an equation for a line which is perpendicular to the line through $(2,5)$ and $(1,3)$ which passes through the point $(-4,5)$. Certainly, if they wrote $$ y - 5 = -\left( \frac{1-2}{3-5} \right)(x+4), $$ I would probably give them full marks (as it demonstrates understanding of the key concept), but I would encourage them to simplify the fraction, as this will (1) make it easier to see the relation between the two slopes ($2$ vs $-\frac{1}{2}$) and (2) simplify further computation.

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    $\begingroup$ This is so, in theory. May be my brain is frozen at a strange place (or at a point of time in a bygone era), but I cannot bring myself to give full marks, when a student writes (in response to a question that involves pluggingin values to trig identities). The answer is $$\sqrt{1+\frac{8^2}{15^2}}.$$ I did not know that we are not allowed to use symbolic calculators in the exam. $\endgroup$
    – Jyrki Lahtonen
    Commented Nov 15, 2021 at 9:01
  • $\begingroup$ If this answer is correct, it raises some important questions in my mind: when in school, was it drilled into us that we should simplify where possible, or was this just a habit that I have gotten into? Should I abandon this habit? And assuming it was drilled into us to simplify where possible, why? Why do we "need" to simplify (specifically fractions, radicals, etc)? Surely, so long as we know that we can simplify, or more generally, that we can change the form of an expression/equation etc to fit the purpose of our pedagogical/learning goal, this is the most important thing? $\endgroup$
    – Adam Rubinson
    Commented Nov 15, 2021 at 13:19
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    $\begingroup$ @AdamRubinson We simplify when it makes doing computation more simple. I will almost never bother to "simplify" $1/\sqrt{2}$, but if I end up needing to sum a bunch of fractions, I will write $1/\sqrt{2} = \sqrt{2}/2$, as this will make it easier to do work later on. If $40/50$ were part of an intermediate step (rather than a final answer) I might simplify it. On the other hand, I was working some optimization problems with students this week, and "simplifying" an expression like $80^2 + 4(16)5 = 6720$ actually made the problem harder (it was better to write $80(84)$). $\endgroup$
    – Xander Henderson ♦
    Commented Nov 15, 2021 at 13:50
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    $\begingroup$ It is also a matter of who the audience is, and what the goal of the exercise is. I am pretty insistent that "word problems" or "applications" should generally be answered with a decimal approximation (with the correct number of significant figures) with units given in such a way that a "layperson" will clearly and unambiguously understand the answer. For example, in the particular question asked in the original question, "40 out of 50 ought to pass" is a good example if you want to know how many certificates to print out, but "80% pass" is probably a better answer for institutional reporting. $\endgroup$
    – Xander Henderson ♦
    Commented Nov 15, 2021 at 13:54
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    $\begingroup$ I will note, however, that I am generally in favor of simplifying. I tend to discourage it on exams, because exams are a totally artificial form of assessment, and if I am not specifically testing for an ability to simplify, I assume that my students are capable of simplifying, so they should just move on. But this question was about how to mark an answer, which I think requires one to think about the pedagogical objective of the task. I will also note that I have deducted 0.01 points (out of 3) from students who simplify poorly, or make other stupid (but not pedagogically relevant) mistakes. $\endgroup$
    – Xander Henderson ♦
    Commented Nov 15, 2021 at 14:00
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The short answer to your question is: everyone is right.

I agree with people here that in many contexts, $0.75$ or $\frac{3}{4}$ would be a more desirable answer than $\frac{45}{60}$. I also agree with several here that when the context is "out of 50 people", an unreduced fraction like $\frac{40}{50}$ makes perfect sense. Certainly, when I write \$13.25 cents as "Thirteen Dollars and $\frac{25}{100}$" on my checks, the bank would be concerned about me if I wrote "Thirteen Dollars and $\frac{1}{4}$." What matters more than anything else is the context of the question and the audience for whom the audience is intended.

Consider simplifying radicals -- this is an important skill in Geometry class in US High Schools, especially when dealing with fractions with radical denominators. Leaving a fraction of the form $\frac{1}{\sqrt{8}}$ is unheard of, because dealing with radicals, and rationalizing denominators, is a skill taught in the course. Now consider the same situation in an AP Calculus class. Students in this course are not only allowed, but encouraged by the format of the test to leave this fraction alone in the free-response section. Why? Because verifying that the student performed the Calculus correctly is more important than seeing if they make a computational mistake in the simplifying algebra afterward.

Most multiple choice standardized tests, including the AP Calculus test, will show simplified fractions most of the time -- but not all of the time.

As a result, in my courses, I typically explicitly vary the context, and make it obvious when I do so. In one chapter, we might be emphasizing simplification, so I will do all practice problems with a simplified answer, and note in the test instructions in bold that answers must be simplified to receive full credit. In another chapter, I will state that simplification isn't as important, practice problems will be done with non-simplified answers, and the test instructions will either say that simplification is unnecessary or omit discussing simplification at all. (I do the same thing with calculator use, and the same thing with being able to look up or reference formulas.).

In my mind, this helps them be prepared for varying contexts they will encounter in the future.

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    $\begingroup$ The bank example is irrelevant because that's not a fraction in the first place. It's a quantity of 25 of the unit type cents, aka hundredth's of a dollar, with the unit type symbol of "/100". $\endgroup$
    – Loren Pechtel
    Commented Nov 13, 2021 at 22:05
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    $\begingroup$ @LorenPechtel My checks have “Dollars” pre-printed on them to the right of that field, which means I could be writing “Thirteen and one-quarter [dollars],” which certainly is accurate but is also rather odd. (Then again, I really doubt the bank would be “concerned” by it.) $\endgroup$
    – KRyan
    Commented Nov 14, 2021 at 3:35
  • $\begingroup$ @Kryan They can bounce it because the cashier didn't know what to do, but that's a completely separate problem. $\endgroup$
    – Nelson
    Commented Nov 14, 2021 at 16:35
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    $\begingroup$ I don't get rationalising denominators. In my mind, $\frac{1}{\sqrt{8}}$ is easier to parse than $\frac{\sqrt{8}}{8}$ $\endgroup$
    – masher
    Commented Nov 15, 2021 at 7:21
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    $\begingroup$ @masher Depends what you mean by "parse". Notice that $\sqrt{8} = 2 \sqrt{2}$, thus $\frac{1}{\sqrt{8}} = \frac{\sqrt{2}}{4}$. If you know that $\sqrt{2} \approx 1.4$, then you can immediately conclude $\frac{\sqrt{2}}{4} \approx 0.35$. It is much easier to divide $\sqrt{2}$ by an integer than divide an integer by $\sqrt{2}$. I wouldn't be able to approximate $\frac{1}{\sqrt{8}}$ to any reasonable precision without rationalizing the denominator. $\endgroup$
    – Stef
    Commented Nov 15, 2021 at 17:21
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I am a GCSE Maths examiner. For a question like this, any correct equivalent decimal, percentage or fraction, whether simplified or not, would receive full marks. It is only specifically if it says in the question that the answer should be simplified that a simplified fraction is required to obtain the final accuracy mark. The reason for this is that the question is not testing ability to simplify fractions unless this is mentioned in the question.

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    $\begingroup$ Whatever your GCSE rules say, this makes absolutely no sense. Would you still give full marks for 1256637061435916/1570796326794895?? $\endgroup$
    – user21820
    Commented Nov 14, 2021 at 4:31
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    $\begingroup$ @user21820 What do you mean by "makes absolutely no sense"? What if the answer to your hypothetical had been "yes, that answer would receive full marks"? Just trying to understand where you are going with this commentary. $\endgroup$
    – Steve
    Commented Nov 14, 2021 at 12:14
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    $\begingroup$ @user21820 The rule can still be applied with an answer of “yes” in your last example, implausible as it may be in the context of a question posed to a 9-year-old, so we are still in search of an example where the rule does not “make sense” or “fails” (= cannot be applied?). I also do not accept that any practical guideline must account for every conceivable situation up front to be successfully implemented. Before we go further off the rails or get moved to chat, are you suggesting any improvement to this answer or just rejecting it outright? $\endgroup$
    – Steve
    Commented Nov 14, 2021 at 12:47
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    $\begingroup$ @user21820 My answer addresses how answers such as 40/50 would be marked in a GCSE exam, since this is what OP is preparing a student for. As I have said, mark schemes typically contain the phrase 'or equivalent' to allow (in this case) answers such as 0.8 or 40/50 or 8/10 to receive credit. In practice, out of tens of thousands of GCSE scripts, answers with extremely large numbers or $(5+e^{i\pi})/5$ will not be given by any student to this 1 mark question, so how to mark such answers is not a practical concern. $\endgroup$
    – A. Goodier
    Commented Nov 14, 2021 at 13:26
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    $\begingroup$ A lot of flags have been raised on the above discussion. While it is true that the discussion was not particularly friendly, I would like to preserve the comments because I think this discussion could be useful for a reader of this question years in the future. $\endgroup$
    – Chris Cunningham
    Commented Nov 15, 2021 at 16:22
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I tell my students this story when this issue comes up:

Imagine you are answering the phone at the local pizza place. Someone on the other end says "Yes, I'd like to place an order. I'd like twelve thirds pizzas." What would you think?

They often come up with the following explanations:

  • It is a prank call.
  • Maybe I misheard them?
  • Maybe they don't know what thirds are?

When someone gives a final answer like "$\frac{40}{50}$," the possible explanations are the same.

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    $\begingroup$ I use a similar hypothetical, wherein the student, having been pulled over in their car and asked if they “have any idea how fast” they were driving, tells the officer “only about 413 miles per 7 hours”. Instant ticket. $\endgroup$
    – Nick C
    Commented Nov 13, 2021 at 1:16
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    $\begingroup$ The point is that OP should explain the student that if they say "40/50" as their final answer, people who see that will likely assume one of the three options I outlined. $\endgroup$
    – Chris Cunningham
    Commented Nov 13, 2021 at 4:42
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    $\begingroup$ Sure, but that doesn't apply to 40/50, which can be read as "forty out of fifty", a pretty normal way to say that when there were 50 people. $\endgroup$
    – trlkly
    Commented Nov 13, 2021 at 6:26
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    $\begingroup$ I don't think this example is relevant, because no one orders pizzas in fractions of any kind (half/half pizzas aside), but the question is all about fractions. "You wouldn't convert perfectly good whole numbers to improper fractions" is a different lesson than figuring out when not to simplify proper fractions $\endgroup$
    – TylerW
    Commented Nov 13, 2021 at 13:39
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    $\begingroup$ @TylerW Plenty of pizzerias will sell you pizza by the ⅛th or ⅙th! They just call it pizza by the slice. $\endgroup$
    – Hearth
    Commented Nov 13, 2021 at 19:34
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Í am a physics teacher, not a mathematics teacher, but I would reward the mark.

It was a multipart question and you did not show us the other parts. But when I formulate tests I try to have all parts independently of each other, so a mistake in a previous part does not impact the latter parts. Assuming that the parts are independent, the student does have to do a lot of work for 1 mark. First calculate the number of adults, the number of adults that fail, the number of adults that pass, the number of teens that pass and lastly the total number of persons that pass. And this is the shortest path (other solutions require more steps). So what is the mark for: for the students to show he has the insight to make all steps above, or to show he can simplify fractions?

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tl;dr: the answer should be given in the reduced form.

Say there are min. 40 students who are taking the exam. Would you want to find the reduced version of the fraction $394.784176044 / 493.480220054$ 40 times? (assuming every time the answer will be given differently but of same complexity)

Another issue is that the rational number has infinitely many fractional representation, hence the fact that student gave the correct answer doesn't mean that you can verify the correctness of the answer in a finite amount of time. That is why, to make the answer unique, you have to stick with a unique representation (or a small subset of possible representations) to make sure that a correct answer can easily be recognised!

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    $\begingroup$ Though these points are technically correct, don't you think in practice the answers received will be a small subset of possible representations? $\endgroup$
    – Steve
    Commented Nov 13, 2021 at 18:16
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    $\begingroup$ @Steve That is an assumption that doesn't necessarily hold in every single case. Even though we are talking only about fractions in there. The issue is of more general in nature. In another course's exam where the answer has many different representation, the students can also argue that their answer correct and should be accepted even if might take hours for the instructor the verify the correctness of their answer. $\endgroup$
    – Our
    Commented Nov 14, 2021 at 11:00
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Let me start with an analogous question on phil-SE:

Teacher: What is 2+2?
Student: 2+2 is 2+2

What is wrong with this answer?

Now it may seem to be an issue with using tautologies for communication. However as I point out law-of-identity/tautology is a red-herring to address this issue. To see that, let's change the exchange slightly:

Q : What is 2+3
A : 2+3 is 3+2

Most of us would admit this is correct but "uh-uh". Yet the answer is sufficiently different from the question that pure logic — aka tautologies/law-of-identity etc does not help us.

We need a math answer not a logic one

Note: Ironically some overzealous member there misleadingly edited the question-title adding the word "tautology"😝

As I commented there

What you are looking for is the idea behind the technical term ground term of Rewrite systems. Alternatively normal form in the lambda calculus.

Likewise here: sometimes 40/50 could be a better answer. Sometimes not.

If we back off from this specific question it's clearly a question about...

The definition of "Simplification"

Now much of school-math, at least arithmetic and algebra are just about simplification.

But even just going from school arithmetic to algebra the rules change: So in arithmetic $2+3$ unequivocally simplifies to $5$.

But in algebra between $x^2 - 5x +6$ and $(x-3) (x-2)$ there is no a priori reason why one is simpler than the other. (And calling the first "simplified" and the second "factorized" only adds to the basic problem viz that simplification is context defined)

In short therefore, at least for math teachers, if not the general math-literate public, it would be a good idea to go from a fuzzy notion of simplification to...

A reified concept of simplification

This question (and many such) suggests that it's increasingly imperative to teach basics of lambda-calculus (LC) and/or rewriting-systems (RS) as fundamental (and not "advanced") math. That is: put something like LC/RS on par with at least calculus, if not algebra/geometry, in the math curriculum.

Also this conversation is worth reproducing.

Now I show you a circle and ask: what is the ratio of the perimeter and the diameter? The answer is Pi. Now I ask: what is Pi? The answer is: it is the ratio of the perimeter and diameter of any circle. That's circular. You could also say that Pi is approximately 3.14, but it's not exactly 3.14, so there really is no better answer than the circular "Pi is the ratio of the perimeter and the diameter, and the ratio of the perimeter and the diameter is Pi."

My response :

Yes... Sometimes one writes $\pi$ as 3.14, sometimes as the more correct sophisticated formulae. And sometimes we leave $\pi$ as $\pi$. (not to mention sometimes making pi into $\pi$ as I've done or $\pi$ into pi as you've 😉). IOW people answering condescendingly are only succeeding in displaying their ignorance of the non-triviality hiding behind the issue of simplification.

Summary

The notion of simplification is not simple! Nor unique. It should at the least be made rigorous.

And preferably mechanically computable. Though thats a dispensable frill.

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    $\begingroup$ Often times simplification is unique, e.g. simplifying fractions so that numerator and denominator have no common factors than $1$, or writing a number to three significant figures (rounding up if the number ends in a $5$), which I think is also unique. But if the question doesn't specify how to simplify then yes, it can be a matter of interpretation as to what is an acceptable answer, which is a problem. This is probably why so many people have said to specify in the question what form the answer should be in. $\endgroup$
    – Adam Rubinson
    Commented Nov 16, 2021 at 12:38
  • $\begingroup$ @AdamRubinson Are ambiguities of simplification the norm or the exception?? I am conflicted... Here is a question re choice between $x$ and $1x$ With at least one commenter claiming the $1x$ helped an IEP student. I'm reasonably sure such could be multiplied many times over from this SE site itself. Maybe it would be a good idea to have a CW-answer collecting all these examples strewn across answers/comments/....?? $\endgroup$
    – Rushi
    Commented Nov 17, 2021 at 5:52
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I then stopped for pause and admitted, maybe they're right, but I'm not sure if they would get the mark if they do not fully simplify the fraction or give the answer as a decimal.

Life is not about getting points. A person who doesn't automatically simplify 40/50 to 4/5 (or 0.8) is being silly and annoying, or showing a lack of competence. This person may be unlucky enough to have a teacher who doesn't enforce this expectation. If so, then they're lucky to have you to provide the guidance that they should be getting.

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    $\begingroup$ "A person who doesn't automatically simplify 40/50 to 4/5 (or 0.8) is being silly and annoying, or showing a lack of competence." I disagree. There are numerous examples in which simplifying this fraction might be a case of premature optimization, or where simplification might obfuscate the problem. In this case, I think that 40/50 is actually a better response in some contexts, as it very clearly says that 40 of the 50 test-takers passed. $\endgroup$
    – Xander Henderson ♦
    Commented Nov 12, 2021 at 20:41
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    $\begingroup$ I disagree that giving an answer 40/50 makes any more sense when there happens to be forty passes from fifty people in the group. The question asks for a probability; at the very least a pragmatic answer would reduce it to 4/5 (repeat this random draw five times, expect 4 to be passes) or bump it to 80/100 (reporting percentages are much more common and straightforward for those with worse number sense to handle). It's not wrong to say 40/50, it is just, as said in the post, annoying. Premature optimization is not especially relevant when this is the end result being reported. $\endgroup$
    – Nij
    Commented Nov 13, 2021 at 4:34
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    $\begingroup$ -1 this is purely an opinion, and one which doesn't seem justified at that $\endgroup$
    – Drake P
    Commented Nov 13, 2021 at 4:35
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    $\begingroup$ I hope you've never used 50% or 80%, then. That would be silly and annoying, and show a lack of competence. $\endgroup$
    – Eric Duminil
    Commented Nov 13, 2021 at 9:51
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    $\begingroup$ Continuing that thought, I wonder if some of us are focusing on this particular example of 40/50 in this context (as I am), and others are thinking of more extreme examples like answering 1/5+3/5 with a complicated integral that equals 4/5. I think stuff like this has to be dealt with on a case-by-case basis and generalizing doesn't work here. $\endgroup$
    – Thierry
    Commented Nov 13, 2021 at 14:44
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(Adding this because I want to - the first version of this answer is near the end after a horizontal line.)

I think that the acceptable answers should succeed in communicating with whoever is reading the answer. And the context (who is reading, what the answer is supposed to tell to the reader,...) should be taken into account.

  • Here $40/50$ is ok, unsimplified, because the entire population is a nice round number $50$, and this answer conveys the desired piece of information in a useful manner.
  • Similarly, I don't care whether a student choose to write $1/\sqrt2$ or $\sqrt2/2$. Or $\dfrac{22}7$ instead of $3\dfrac17$. True, with more difficult fractions the latter form may give a better about the size of number. I mean, if I'm applying for a grant, and the formulas tell me that the budget total is $10345712/171$ euros, I still should not use that form of the number in my application.
  • To reiterate the point from my old answer. Simplification should perhaps be seen as a tool for converting an answer into a form that is better suited at communicating with the reader as opposed to as an end in itself. In addition to learning simplification tricks, the students could (should?) be tested (or at least trained) in exhibiting good judgement about which forms of an answer are useful for the reader.

Not really answering the title question, and possibly this should be a comment only.

But because your pain is my pain, may be we can work around this as follows.

Whenever the question has a numerical answer, and we really want the kids to simplify the end result, may be a way to achieve this is to turn the question into a multiple choice one? Give a list of alternatives, and make the students pick: A) 37/50, B) 4/5, C) 12/25, D) 3/4?

Of course, this may lead to different kinds of games. Like students complaining if the only numerically correct answer is not fully simplified.

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  • $\begingroup$ But the problem, apparently, is us wanting them to simplify the answer. For the question in the OP, 40/50 is just a good an answer as 4/5 or 0.8 because there is no good reason to simplify the fraction for this particular question. The fact that we have been drilled into us that 40/50 should automatically be simplified to 4/5 without much thought is the problem, and we should resist this urge. $\endgroup$
    – Adam Rubinson
    Commented Nov 15, 2021 at 13:07
  • $\begingroup$ @Adam Here I would be willing to accept an answer of 40/50. If only because the entire population is a nice round number of 50. In some other case I'm not sure. I'm not willing to resist the urge to simplify even though I won't necessarily insist that my students simplify (except in a problem where that is specifically the skill being tested). But, I definitely give partial credits according to a scheme not unlike that outlined by KvdLinden. There are occasions where automatic simplification is not unlike mathematical potty training. But sometimes simplification is very non-trivial. $\endgroup$
    – Jyrki Lahtonen
    Commented Nov 15, 2021 at 13:30
  • $\begingroup$ @AdamRubinson I understand that you're being hyperbolic, but there are good reasons to want to simplify 40/50: it scratches my itch, for one. Slightly less facetiously: it takes a microsecond shorter to process 4/5 than to process 400000/500000 (I haven't taken into account the tediousness of having to carefully count/match the number of zeros in the numerator & denominator) or even just 40/50. $\endgroup$
    – ryang
    Commented Nov 17, 2021 at 8:54
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    $\begingroup$ "Simplification should perhaps be seen as a tool for converting an answer into a form that is better suited at communicating with the reader." Spot on. Though one might also note that simplification is not just for the reader, but also for the person who is doing the work. It is often... simpler... to work with simplified expressions. $\endgroup$
    – Xander Henderson ♦
    Commented Nov 17, 2021 at 20:42
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The general purpose of simplification or writing an expression in some "standard form" is to be able to have a unique canonical answer, and thereby easily compare student work to an official answer for grading, to spot-check against an answer at the back of a book, to compare between two cooperating students, etc.

So generally the protocol in a classroom or testing situation will be "yes", it should be written in the canonical unique form to support that kind of efficient check.

I also point out that in the linked sample test, some questions are marked with a "calculator" symbol, while the present question is not. We can deduce the final answer is not meant to be given as a decimal (which would change this advice: admittedly no need in the middle of a work product), but rather to be given as a fraction. And given the above protocol, it should be simplified, as that's almost surely the form on the answer sheet being used for grading.

(That said, the answer from A. Goodier claims from experience that actual GCSE Maths examiners don't require that; whereas I agree with the commenter about that not being fully coherent.)

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  1. The test should indicate in what format the answer is expected to be given. This should be clear both to you and to the students. Otherwise grading becomes subjective. If you accept 40/50, would you accept another student giving 120/150, and another 468/585?

  2. The test covers basic math skills. Simplifying fractions seems to fit the overall material covered by the test. It seems appropriate to require students to simplify the fractions.

  3. If the student doesn't want to spend their time simplifying the fraction, you need to do it. How much time are you willing to spend? Consider you spent time writing a SE question about it. If the student knew they were asked to simplify the fractions and they leave it to you, they are disrespectful.

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  • $\begingroup$ "would you accept another student giving 120/150, and another 468/585?" Apparently, yes. See A. Goodier's answer. $\endgroup$
    – Adam Rubinson
    Commented Nov 14, 2021 at 21:02
  • $\begingroup$ @AdamRubinson You wrote before "I expect students to simplify fractions where possible unless there is a good reason not to do so". But if A.Goodier's answer convinced you to no longer to expect that, I guess that's fair as well. $\endgroup$
    – user1079505
    Commented Nov 14, 2021 at 21:29
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    $\begingroup$ @AdamRubinson it's fair if your workload is reasonable, and all students are treated in the same way. E.g. it might be unfair if you asked the students to simplify the fraction, but then disregarded the effort of those who did. It might be unfair if some students submit their answers in a format that is difficult for you to process (e.g. very poor handwriting), but perhaps that's never the case? $\endgroup$
    – user1079505
    Commented Nov 14, 2021 at 21:47
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The probability is 4/5’s or 80% not 40/50. The number of people that passed could be described as 40/50 (aka 40 out of 50), but that would require a different question.

Whether the answer should or should not be simplified is going to depend upon the question. It’s fundamentally no different from whether the answer should be 1/3, .3, .33, .3333, etc: the question determines the form and precision. The question should be worded in such a way that the form and precision is obvious.

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  • $\begingroup$ But in reality, questions about probability usually don't say, "give your answer as a percentage/decimal/fraction in simplest form". $\endgroup$
    – Adam Rubinson
    Commented Nov 13, 2021 at 15:30
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    $\begingroup$ I agree the acceptable answers will be defined by the question (and global expectations established at the start of the exam). One minor quibble: if the question was "how many people passed?", wouldn't the answer be 40, not 40/50? It seems 40/50 is just a different representation of 4/5, not a shorthand for "40" with a prepositional phrase appended. $\endgroup$
    – Steve
    Commented Nov 13, 2021 at 18:21
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    $\begingroup$ "In the early 1980s, this was called standard form: The final answer was expected to be expressed in "lowest terms," with no whole-number factors (other than 1) common to both the numerator and denominator. A fraction greater than one (e.g. 22/7), in standard form, had to be expressed as a combination of a whole number and a fraction less than one (e.g., 3 1/7)." @AdamRubinson - Every math problem I ever did that resulted in a fraction, it either said this, or it was assumed because that's day one of fractions. $\endgroup$
    – Mazura
    Commented Nov 14, 2021 at 0:40
  • $\begingroup$ @Steve I agree: "40, out of 50, students passed", "out of 50 students, 40 passed" and "40 students passed, out of the 50 of them", etc. are equivalent to one another, and all are claiming that the number of students who passed is 40. Those parenthetical commas are deliberate and important; had they been omitted from my first example, then its meaning would have been changed to "the number of students who passed is 0.8", haha. Similarly, for a car going at 70km/h, its total distance travelled in 1 hour is just 70km, rather than 70km/h. $\endgroup$
    – ryang
    Commented Nov 17, 2021 at 9:00
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I would say you might expect a meaningful answer.

In your case, there might be multiple meaningful answers:

  • $\frac{4}{5}$ : this is the answer in the most simplified form.
  • $\frac{40}{50}$ : this answer contains the original total number of people (50).
  • $\frac{24 + 16}{50}$ : this answer contains the number of adults and the number of teens and the total number of people.
  • even $\frac{80}{100}$ : from this answer you can easily read the probality in percent, which is 80.
  • An answer like $\frac{12}{15}$, however, would not be acceptable as this notation does not reveal any information.

... however, for the third I would make the remark "You have given as an answer that the probability equals $\frac{a+b}{c}$. You do realise that this only makes sense if $a+b \leq c$. How can I know from your answer that you have actually checked your answer?".

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  • $\begingroup$ I think in general if they haven't automatically simplified it to $4/5$ then that's something I should question them about, as they should automatically simplify it to $4/5.$ $\endgroup$
    – Adam Rubinson
    Commented Mar 20, 2023 at 14:45
  • $\begingroup$ @AdamRubinson: that's a matter of taste, which might be different from you, that one student or the entire class, but I give you the benefit of the doubt: it's a simple task, equal for everybody, while mine is an invitation for doubt :-) $\endgroup$
    – Dominique
    Commented Mar 20, 2023 at 14:51
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If the student knows how to simplify and is, on purpose, not simplifying in order to save time to give themselves more time for the rest of the test, then this is an OK reason not to simplify, so long as the student also knows that they will not lose marks by not simplifying. I do find this weird because for me, simplifying a fraction is something I do automatically - without thinking too much (and it doesn't take much time) - unless I spot a good reason to not simplify - which in the question in the OP, there was not a good reason to not simplify, and simplifying the fraction actually brings the question more to life to me, as it is easier to imagine $4/5$ of something rather than $40/50$ of something.

If the student is not simplifying because they are bad at simplifying, then this issue needs to be raised with the student I am tutoring and dealt with (worked on), as simplifying fractions is an essential skill in order to make fractions more meaningful and relatable (one can "see" $1/2$ in their minds eye very quickly, but not $93274384/186548768$ until they have simplified it). Simplifying fractions and ratios is arguably an important life skill.

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