# Finding a fraction of a number

1. Aug 28, 2006

### mtanti

This sounds like first grade mathematics but I need to know the reasoning.

Why is it that to find a fraction of a number you multiply the number by that fraction?

ie to find (a/b)th of X you do X*(a/b). What operation is the multiplication doing in reality? This is used in precentages where you find a fraction of 100 by multiplying the fraction by 100.

10x

2. Aug 28, 2006

### Staff: Mentor

Percentages are different. "Per cent" means literally per 100. That's why you multiply a number by 100 to get its percent value. Like 2 x 100 = 200%.

3. Aug 29, 2006

### mtanti

yes but when it's a fraction you are not actually finding a 'per cent' as you won't even have one 'cent'. You are finding a fraction of hundred but the operation is still multiplication. Is there some complex mathematical reasoning behind this?

4. Aug 29, 2006

### uart

Well think of the fundamental meaning of a fraction like p/q, it essentually means to divide a quantity into q equal pieces and then take p of those pieces. Say I had a pizza and I wanted to eat 3/7th of it, I could achieve this by cutting the pizza into 7 equal pieces and then eating 3 of them.

Now the above is for the case where the quanity under consideratoin is unity, that is one pizza. What if I had 2 pizza's and I wanted to take 3/7 of the total. Clearly one way I could do it would be to cut each pizza into 7 equal pieces as above, and then to take 3 of these pieces from each pizza. So the process was to divide a unit by 7, then multiply by 3 and then multiple by 2. I hope that simple example helps you visualize why we multiply fractions the way we do.

Last edited: Aug 29, 2006
5. Aug 29, 2006

### mtanti

Hmm maybe it was a little to simple to apply it to numbers. So what you said is that:

Since a/b is of 1 and cannot be of any other number, you must first break the number X into a series of 1s, find a/b of each one, and then add them all up.

The background process is:
a/b + a/b + a/b + a/b + ... for X times which basically is a/b * X.
Good reasoning?

Another important point is that you are both finding the multiplication of a/b for X times and finding an [a/b]th of X.

However this cannot be applied when multiplying 2 fractions because the natural way of multiplication (addition for X times) doesn't make sense with 2 fractions. What does multiplication do then?

6. Aug 29, 2006

### Staff: Mentor

I don't understand your confusion. Multiplication and division and fractions are very straightforward. What exactly is your question? Can you give a concrete example of a situation that confuses you?

7. Aug 29, 2006

### mtanti

You see, mathematics is a subject which it taught terribly. You only start to learn it properly at university level. I want to know what happens when you multiply 2 fractions together. The bare facts, not just the rules to multiply, what are you finding? A fraction of a fraction, but why is that so?

8. Aug 29, 2006

### Staff: Mentor

Yeah, a fraction of a fraction. Think of the example of a pie chart (or the pizza example earlier). First you take half of the whole by multiplying 1/1 by 1/2. Then you can take half of what remains by multiplying 1/2 by 1/2 to get 1/4. Then you could get back to the whole again by multiplying 1/4 by 4/1 (4). Don't get too hung up on this. What other math subjects are you studying now?

9. Aug 29, 2006

### Swapnil

It seems to me that you are asking very philosophical questions. The question you are asking alludes to the fact that do all the mathematical operations need to have a "physical meaning" in one way or the other.

This is a very deep philosophical question. I think a college-level course in Philosophy of Mathematics would be the best for a person of such an intellect like yourself (that is if you are interested and your college offers such a course :tongue: ).

Personally, if I go from one mathematical equation to another using rules of aritmetic, algebra, trignometry, calculus, etc to another form, I don't worry about if the operations that I am doing have a physical meaning at EACH step. As long as the starting step and the ending step have a physical meaning, I am satisfied.

10. Aug 29, 2006

### mtanti

Thanks for the compliment Swapnil, if it was a compliment that is... But yeah I believe that mathematics must be taught that way, knowing exactly what you are doing before doing it if you are to understand the subject. I didn't understand what LCM was until recently and I've been using it for years!

The only mathematics which is purely practicle is the natural number mathematics... As soon as you get negatives it starts getting wierd and when you use real numbers it gets worse. When you learn complex numbers you're just plain lost!

OK, but that still doesn't explain what is happening during the 1/2 * 1/2 process... You are adding a 1/2 by itself for 1/2 a time. Wierd statement I say. Can anyone help out?

11. Aug 30, 2006

### uart

What's wrong with simply considering multiplication as meaning "lots of", that's how I teach it to under 10 year olds. 4 x 2 means 4 lots of 2, 1/2 x 1/2 means one half a lot of one half. Come on, its as easy as falling off a log.

Actually I do see many senior high-school students who have an alarmingly poor understanding of these most basic properties of fractions. It's something that really annoys me when I give a senior high-school student a question like "solve 13x =7" and they reach for their calculator. I say "no calculator please, just give me the answer as a fraction" and honestly they often just stare in disbelief with some kind of "how do you expect me to do that without a calculator?" look on their faces. Grrrrrrr, this seriously irks me :(

Last edited: Aug 30, 2006
12. Aug 30, 2006

### mtanti

It's not that I have a poor understanding of multiplication uart, it's just curiosity. I enjoy understanding mathematics more than using it.

You still can't explain what 1/2 a lot of 1/2 means though... What you are saying is that you are adding 1/2 for 1/2 a time. That's what 1/2 a lot of 1/2 means to me... So what does 1/2 a lot of 1/2 mean?

I know this is a question that bugs most of you since you probably never thought about this since it's so basic, but think about it... Does it really make sense?

13. Aug 31, 2006

### verty

Mtanti, I think your choice of words is limiting. Why you speak of "adding 1/2 for 1/2 a time", you assume that a 'time' is atomic.

Think of 1/2 * 1/2 as "one half of one half", which is "one part of the result of chopping (one half) into 2 equal parts".

Or in terms of multiplication, 1/2 * 1 might be "that entity X that one must double to form (a whole)" and 1/2 * 1/2 might be "that entity X_1 that one must double to form (that entity X_2 that one must double to form (a whole))". Is this a better formulation?

14. Aug 31, 2006

### mtanti

What is multiplication in practicle terms? It is the addition of equal groups.

[][][]
[][][]
[][][]
= 3*3 or 3 for 3 times

[
[
[
=1/2 * 3 or 1/2 for 3 times

┌┐┌┐┌┐
=3 * 1/2 or 3 for 1/2 times

=1/2 * 1/2 or 1/2 for 1/2 times

Now who can give a clear explaination of what this 'breaking in half' means in logical terms? I know it's hard to understand the question but is there a philosophical reason for multiplication to work that way?

15. Aug 31, 2006

### verty

Hmm. Does this help?

a/b is new notation which denotes k such that bk = a.

1/2 = (x : 2x = 1)
1/2 * 1/2 = (x : 2x = 1) * (x : 2x = 1)
= (x^2 : 4x^2 = 1)
= 1/4

Another example:
2/3 = (x : 3x = 2)
4/5 = (y : 5y = 4)
2/3 * 4/5 = (xy : 15xy = 8)
= 8/15

Last edited: Aug 31, 2006
16. Sep 1, 2006

### mtanti

Hmm... Nice approach, but highly theoretical...

I don't understand why you must find x or xy. However this is not a practicle explaination. Can anyone explain what is being done to the boxes above?

17. Sep 1, 2006

### Moo Of Doom

Don't you see that you have a quarter of a box when you multiplied 1/2*1/2?

An obvious interpretation of multiplication of a and b is finding the area of the rectangle with sides a and b. That's the interpretation you've shown us with your

[][][]
[][][]
[][][]

boxes.

18. Sep 1, 2006

### verty

Mtanti, why must we accept negative numbers? Why not limit mathematics to natural numbers only? Is it intuitive to say that I have -1 apples?

Negative numbers are simply (x : x + b = a). Well, if my bank account has an overdraft facility and I withdraw more than I have put in, it has a negative balance. The negative amount reflects that I owe that money instead of being owed it. Negative numbers certainly seem useful.

19. Sep 1, 2006

### Werg22

This is something actually few people can awnser. Axioms are taken for granted when they're actually not.

Ok, I'll try to make it simple. By definition, a/b*x represents 1/b of x repeated a times (not a/b or x by themselves, but their product). Say we have the 1/b of 1 object. Now if we multiply this 1/b by a, and then multiply it b we are left with a objects, simple logic. Thus the quanity we have is a/b. Now we have proved that 1/b*a = a/b. Now we prove that a/b of x object is ax/b. We start with x. We start with x objects. We divide them into b equal and additive parts, thus x/b. If we multiply these parts by a, we get x/b *a parts. Now if each of those parts are multiplied by b, we get x*a objects - here again simple logic. Thus the previous expression is equivalent to the bth division of a*x which is a*x/b.

Last edited: Sep 1, 2006
20. Sep 7, 2006

### mtanti

So this is what is happening:
The definition of X * a being X added by itself for 'a' times is only true when 'a' is integer. When it is a fraction what you do is you do the previous definition to the numerator of 'a' and then divide the result into the denomenator equal parts and state the quantity of one of those parts obtained. Or else you can first divide X into the denomenator equal parts and then multiply the result for the numerator times.

Now as for multiplying a number by a fraction to find that fraction of the number, you are finding one (a/b)th of the number (c/d) because:

as stated earlier, like finding fractions of integers, you need to add a (c/d)th of each '1' in (a/b) since c/d is only a fraction of a '1'.

therefore
a/b for c/d times
problem is that there isn't even one '1' in a/b as it's only a fraction of a '1'.

So what we have to do is make the number, which we are finding a fraction of, an integer. How do we do that?

Lets say that we have to find 1/2 of 1/2
Then to find that we have to add the 1/2 to itself for 1/2 a time. A more practicle approach would be to 'send the denomenator' to the other fraction by multiplying one fraction by 2 and the other by 1/2. As so:
(1/2)*1/2 * (1/2)*2
and thus we are adding 1/4 by itself for 1 time = 1/4

But of coarse we are doing a recursive problem here, we are still multiplying 2 fractions together, so here is what is happening in practice:

You have 2 circles divided into 2 parts each. You want to multiply each part together but you can't because you need to multiply by wholes. So what you do is make one circle a whole by adding it another half, thus multiplication by 2. However you still need to balance the two circles out. So if you multiplied one circle by two you need to divide the other circle by 2, thus breaking it into a further 2 pieces. And thus you have 1/4 * 1.

OK now who can tweak this up for me?

21. Sep 7, 2006

### HallsofIvy

If you believe that mathematics has anything to do with apples or bank accounts, then you do not know what mathematics is. You may have worked with applications of mathematics, but not with mathematics itself.

22. Sep 8, 2006

### Robokapp

Ok I thought and thought and this is what my explanation would be:

it's one of those questions that you don't spend any time thinknig about...but I think I have an acceptable expalnation. Multiplication was thought many years ago...

Let's say you got 40 apples and you only want to take home 3/4 of them. that's three quarters

so you divide them in quarters. 4 groups of 10. the operation you did was 40/4=10. Now you dont want 1 group but 3 groups...so you add 2 groups of 10 to your group. 10+10+10=30. the groups are identical since you divided them in identical groups so you might as well write it as 10*3=30.

so after you divided by 4 you multiplied by 3. Due to math properties, (40 * 3) / 4 is the same as (40 / 4) *3 so...

$$40*\frac{3} {4}=30$$

Edit: I realised I wrote all that and did not actually answer the question.

The answer is because it's the quickest correct way to obtain the wanted answer.

23. Sep 10, 2006

### mtanti

Aha! Now there's some progress. It's the quickest way, but which is the purest way? The way which is most obvious to logic.

24. Sep 10, 2006

### arildno

It is relatively simple to understand this, I'll take the 1/2*1/2 example (and keep it as informal as I want to):
Assume you have 1CAKE. Let a 1SLICE OF CAKE=1/2*1CAKE (that is, one-half cake, since 1/2*1=1/2)
Let a 1 PIECE OF CAKE=1/2*1SLICE OF CAKE.
Now, substitute 1 SLICE OF CAKE in the second equation by aid of the first equation:
1 PIECE OF CAKE=1/2*1/2*1CAKE=1/4*1CAKE
That is 1 PIECE is one fourth of the whole cake, whereas the same PIECE remains one half of the SLICE.

This is EXACTLY the same reasoning as the following:
Let 1 BOX OF CAKE=6*1CAKE, and 1 STORE OF CAKE=55*1BOX OF CAKE.
Thus, we have:
1 STORE OF CAKE=55*6*1CAKE=330*1CAKE

(By "definition", we may say that the statement 1 SLICE OF CAKE=1/2*1CAKE "really" means that 1CAKE=2*1SLICE OF CAKE)

Last edited: Sep 10, 2006
25. Sep 11, 2006

### Robokapp

Absolutely correct but probably what most of us have the problem in answering this question...(I know i did, it took me 10 minutes to come up with my answer and i still needed an edit) is that logic no longer processes all this. It is something that just becomes second nature. When you hear "half of 40" you picture the number 20, not the $$\frac{1} {2} * 40$$.

It's like doing antiderivatives. First time I saw a $$\int{\frac{1} {x}} dx$$ i immediatelly wrote $$x^{0}$$ convinced all I have to do now is find its coefficient, and immediatelly I realised it's wrong. With quite a lot of surprise I might add.