Hi everybody! I need presentation help

Hi everyone

I'm a math novice that needs help in presenting my answers.

How do you use the implies, therefore and because when presenting your answers.

In a questions like:

"When the water is turned on, a bathtub will be filled in 12 minutes. When the drain is opened it takes 20 minutes for the tub to drain. If the water is turned on and the drain is left open, how long until the tub is filled?"

I can solve the problem but don't not how to lay things out properly. Do I write words, sentences, use symbols such as imples? Help?

Re: Hi everybody! I need presentation help

If you really can solve the problem, it is unlikely that you need help with presentation. The problem is nontrivial because, unlike the rate of incoming water, which is constant, the outgoing rate slows as the water level goes down.

A typical mathematical proof is a piece of grammatically correct English prose. Words like "implies," "therefore" and "because" have their usual meaning. Some symbols are used as abbreviations; for example, instead of "x equals y" one writes "x = y." A proof often introduces names for mathematical objects, which are used later. One rule that is frequently violated by beginning proof writers is using names that have not been properly introduced.

Feel free to post your attempt at the solution to get feedback.

Re: Hi everybody! I need presentation help

Hi emakarov

Thank you for response. I never considered, that the outgoing rate was not constant. The question quoted was from a book teaching problem solving techniques so I assume that they meant for it to be constant. Plus the answer given is 30 minutes.

How would you write out the answer?

My brain freezes when I have to describe the problem elements. It's something to do with writing English sentences and mathematically expressions at the same time - I feel I should be able to show the workings without having to write words. Assuming that's the correct way.

I know Mathematicians don't approve if workings are not presented correctly. I just don't know what the rules are and how to get info on this topic. How to practice getting it right. Are you suggesting I need to learn how to write proofs?

Thanks

Re: Hi everybody! I need presentation help

no need to be so formal [will you wear a tuxedo presenting this? ;) ]

assuming the fill and drain rates are constant ...

combined rate in tubs per minute = fill rate - drain rate

$\displaystyle R = \frac{1 \, tub}{12 \, min} - \frac{1 \, tub}{20 \, min}$

$\displaystyle R = \frac{5 \, tubs}{60 \, min} - \frac{3 \, tubs}{60 \min} = \frac{2 \, tubs}{60 \, min} = \frac{1 \, tub}{30 \, min}$

Re: Hi everybody! I need presentation help

Hello, MtahsNoviceO1!

I wouldn't worry about the *format* of your solution.

If you have a clear explanation, you're golden!

Quote:

When the water is turned on, a bathtub will be filled in 12 minutes.

When the drain is opened, it takes 20 minutes for the tub to drain.

If the water is turned on and the drain is left open, how long until the tub is filled?"

I can solve the problem but don't know how to lay things out properly.

Here's my explanation . . .

I assume that the water is flowing in at a constant rate.

I assume that the water is draining out at a constant rate.

. . Otherwise, the problem requires Differential Equations.

The tub can be filled in 12 minutes.

. . In one minute, $\displaystyle \tfrac{1}{12}$ of the tub is filled.

The tub can be drained in 20 minutes.

. . In one minute, $\displaystyle \tfrac{1}{20}$ of the tub is emptied.

With the water turned on **and** the drain opened,

. . in one minute, $\displaystyle \tfrac{1}{12} - \tfrac{1}{20} \:=\:\tfrac{1}{30}$ of the tub is filled.

Therefore, it will take 30 minutes to fill the tub.

Re: Hi everybody! I need presentation help

Speaking about problems where water leaks out of a container, almost 100 years ago (in 1916), the outstanding Russian educationalist and the author of many popular science books Yakov Perelman wrote in "Physics Can Be Fun" (or, "Entertaining Physics"), vol. 2, that "problems about reservoirs (with water leaking out) have no place in books of arithmetic problems." This book was translated into English, but I found only the first volume online. He also wrote "Mathematics Can Be Fun." Both books are highly recommended.

Re: Hi everybody! I need presentation help

Hi all

Thanks, however I was more interested in finding out how to present any solution, rather than an answer to this particular problem.

Just for information my attempt was as follows:

*12m x 5b = 60m (a)*

*20m x 3b = 60m (b)*

*2b = 60 implies_symbol 1b = 30m*

*1/12m - 1/20 = 5/60m - 3/60m = 2/60m = 1/30m*

Therefore_symbol 1/30b in a minute implies_symbol 1/30b x 30 = 1b Therefore_symbol 30 minutes.

m stands for minutes

b stands for bath tubs

Apologies, don't know how to insert symbols into post.

Any links which talk about how to present equations, working, formulae, proofs or whatever they maybe called will be appreciated.

Re: Hi everybody! I need presentation help

I meant it when I said that a proof is a gramatically correct English prose, not a sequence of formulas. There are two exceptions to this rule. In mathematical logic, there is a formal definition of a proof as a sequence of formulas, which can be encoded as a number. It is necessary to represent proof as precise mathematical objects in order to study what can and cannot be proved. Similarly, automated theorem proving is concerned with developing programs, called proof assistants, that help people construct proofs. The program builds a proof object and verifies that it uses valid proof steps.

However, these formal proofs are not intended for human consumption. Proofs are made formal in order to make them the subject of mathematical investigation or computer processing. It may seem like avoiding potentially ambiguous natural language text would also make proofs that are intended to be read by humans more concise and precise, but in practice it is almost impossible to understand a stream of formal symbols, just as it is almost impossible to understand a program written in some programming language without comments.

Quote:

Originally Posted by

**MtahsNoviceO1** *12m x 5b = 60m (a)*

*20m x 3b = 60m (b)*

*2b = 60 implies_symbol 1b = 30m*

*1/12m - 1/20 = 5/60m - 3/60m = 2/60m = 1/30m*

Therefore_symbol 1/30b in a minute implies_symbol 1/30b x 30 = 1b Therefore_symbol 30 minutes.
m stands for minutes

b stands for bath tubs

This does not make much sense. Minutes times bathtubs give minute-bathtubs, not minutes. You can't write "Therefore_symbol 30 minutes" because "therefore" must be followed by a proposition (something that is either true or false), and "30 minutes" is neither true nor false. It is not clear how the first two lines imply the third one, and so on.

My advice is not to try to be too formal because it often makes a proof hard to follow. Soroban gave a nice explanation.

If you want to learn how to write proofs, enroll in a course that deals with proofs and read the textbook. If you want to insert formulas in posts on this forum, refer to the LaTeX Help subforum. You can also click on the "Reply With Quote" button to see how a post was typed. Many mathematical symbols can be approximated in plain text. For example, you can type => instead of $\displaystyle \Rightarrow$ to denote "therefore."

Re: Hi everybody! I need presentation help

Hi emakarov

I'm not sure I understand what you mean by 'not to try to be too formal'. Your comments would seems to indicate that I wasn't formal enough as what I've written 'doesn't make sense'. C'mon guys don't you think I know what I've written is probably not correct. It's obvious that I am looking for help/assistance to determine what's appropriate. It would be more helpful to refer me to a reference source which explains points likes the ones you have mentioned e.g. ' "therefore" must be followed by a proposition'. Rather than pick holes in the rather arbitrary example provided. A reference would allow me to look at the issue(s) holistically, rather that the specific issues relating to a single example. I was hoping that someone could recommend a website or book that would explain what I should and should not do when showing my workings.

Please appreciate that I do not consider myself a mathematician. I am a adult learner (with limited time) trying to gain a better understanding of some fundamental aspects of maths in preparation for an MSc (which will have involve discrete maths, algebra etc).

Re: Hi everybody! I need presentation help

These are part of differential equations...one finds relations between:

$\displaystyle \frac{dQ}{dt}=\text{rate} \text{of} \text{filling}$

$\displaystyle \frac{dR}{dt}=\text{rate} \text{of} \text{drain}$

Re: Hi everybody! I need presentation help

Quote:

Originally Posted by

**MtahsNoviceO1** I'm not sure I understand what you mean by 'not to try to be too formal'. Your comments would seems to indicate that I wasn't formal enough as what I've written 'doesn't make sense'.

By "formal" I meant writing exclusively with formulas and no plain text. In most contexts, writing with formulas only is not helpful. Of course, one can easily write formulas that are wrong or don't make sense.

Quote:

Originally Posted by

**MtahsNoviceO1** C'mon guys don't you think I know what I've written is probably not correct. It's obvious that I am looking for help/assistance to determine what's appropriate.

As I said, Soroban wrote an excellent solution to the problem in post #5.

Quote:

Originally Posted by

**MtahsNoviceO1** It would be more helpful to refer me to a reference source which explains points likes the ones you have mentioned e.g. ' "therefore" must be followed by a proposition'.

Actually, this follows from my advice to write grammatically correct English prose. One has to write a complete statement, not a number, after "therefore."

Quote:

Originally Posted by

**MtahsNoviceO1** A reference would allow me to look at the issue(s) holistically, rather that the specific issues relating to a single example. I was hoping that someone could recommend a website or book that would explain what I should and should not do when showing my workings.

Higher math, unlike high school math, consists of theorems and proofs rather than problems and solutions.I personally never studied how to write proofs per se; I just read proofs in a textbook or lecture notes on some particular subject, such as calculus or abstract algebra, and learned to write proofs in homework assignments. However, some schools have courses whose purpose is to teach proofs. They often use discrete mathematics as the subject matter. If you want to follow this road, see this thread on MathOverflow that has recommendations for textbooks that teach proof writing.

Quote:

Originally Posted by

**MtahsNoviceO1** Please appreciate that I do not consider myself a mathematician. I am a adult learner (with limited time) trying to gain a better understanding of some fundamental aspects of maths in preparation for an MSc (which will have involve discrete maths, algebra etc).

Especially since you have limited time, I would recommend finding out what textbooks you will be using and starting to read them now. I did this very little, but even when I did not understand much at first, the material stayed in my head and when I started the actual course, it was much easier to understand it.

Re: Hi everybody! I need presentation help

Thank you clarifying what you meant by formal

Quote:

Originally Posted by

**emakarov** By "formal" I meant writing exclusively with formulas and no plain text. In most contexts, writing with formulas only is not helpful.

You mentioned Soroban's solution, which I appreciate

Quote:

Originally Posted by

**emakarov** As I said, Soroban wrote an excellent solution to the problem in post #5

However, as I said in post #1

Quote:

Originally Posted by

**MtahsNoviceO1** I can solve the problem but don't not how to lay things out properly.

The post is about the methodology/process/rules and not about solving the **example** problem used for illustrative purposes.

You mentioned:

Quote:

Originally Posted by

**emakarov** Actually, this follows from my advice to write grammatically correct English prose.

.

But where did you learn this heuristic or hard and fast rule that you now apply? As I said in post #3

Quote:

Originally Posted by

**MtahsNoviceO1** I know Mathematicians don't approve if workings are not presented correctly. I just don't know what the rules are and how to get info on this topic.

.

Btw- I've seen the text books and the past exam paper and they don't cover what's required. Hence the original request for information.