Finding a fraction of a number

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In summary, the reasoning behind multiplying a fraction of a number is that it represents dividing the quantity into equal parts and taking a certain number of those parts. This concept also applies when multiplying fractions, as it represents finding a fraction of a fraction. However, this explanation may not be applicable in all cases, as the philosophical question of the physical meaning of mathematical operations arises. Some suggest that multiplication can simply be thought of as "lots of" or repeated addition. The lack of understanding of these basic properties of fractions is alarming in senior high-school students.
  • #1
mtanti
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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
 
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  • #2
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
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
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 quantity 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.
 
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  • #5
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
mtanti said:
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?
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
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
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
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. :wink:
 
  • #10
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 weird 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
mtanti said:
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?

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 :(
 
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  • #12
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
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
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 explanation 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
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
 
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  • #16
Hmm... Nice approach, but highly theoretical...

I don't understand why you must find x or xy. However this is not a practicle explanation. Can anyone explain what is being done to the boxes above?
 
  • #17
mtanti said:
Can anyone explain what is being done to the boxes above?

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
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
mtanti said:
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

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 quantity 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.
 
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  • #20
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
verty said:
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.
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
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 don't 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...

the answer would be

[tex]40*\frac{3} {4}=30[/tex]

Edit: I realized 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
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
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)
 
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  • #25
mtanti said:
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.

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 [tex]\frac{1} {2} * 40[/tex].

It's like doing antiderivatives. First time I saw a [tex]\int{\frac{1} {x}} dx[/tex] i immediatelly wrote [tex]x^{0}[/tex] convinced all I have to do now is find its coefficient, and immediatelly I realized it's wrong. With quite a lot of surprise I might add.
 
  • #26
Robokapp said:
...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 [tex]\frac{1} {2} * 40[/tex].

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

Logic does 'process' all this. There's just the question if the brain does.
 
  • #27
Math doesn't have to stem from physical meaning. But let's say it does.

If I have half a pie, and I divide that half of a pie by 2.

[tex] \frac{\frac{1\,\,pie}{2}}{2} = \frac{1\,\,pie}{4} [/tex]I get a quarter of a pie.

I'll give a physical proof (which proves the math (since the math stemmed from a physical meaning)).

Now I just took a whole pie out of my fridge and divided it in half with a knife, thus giving me half a pie. I divided that half of a pie by 2 and I got 1/4 of my original pie. It worked! So that's why you multiply. Now use a lot of pies and induction to prove that multiplication yields the correct portion of a pie (I recommend strawberry).

There's this law that exists in civilized society. It says, don't shoot people. Why does this law work? Well it works because if people don't shoot each other, we don't die, and we pay taxes, and we all live happily. It just works. Multiplication and division just work. Isn't that a good enough explanation?

I'm kind of hungry, so I'm going to go multiply that quarter of a pie by 2 so I have more to eat.
 
  • #28
So what you're saying is that multiplication is said to work by inductive proof? Someone just tried if it worked and found that it does and that's it?? There is no actual proof or reason why multiplication has that property with fractions?
 
  • #29
Maybe studying the history of fractions will help you understand why and how they were developed and why multiplication is used. You may also want to look at what Binary Operations are.

It may seem confusing, but keep looking at it from as many perspectives as possible and eventually it will snap into place and youll go "Oh my God, I am an idiot!".
 
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  • #30
mtanti said:
So what you're saying is that multiplication is said to work by inductive proof? Someone just tried if it worked and found that it does and that's it?? There is no actual proof or reason why multiplication has that property with fractions?
Of course it has!
First, let us define, for any non-zero number a its reciprocal "1/a" that has the property a*1/a=1

We want to prove that (1/a)*(1/b)=1/(a*b)
By definition then, we have:
[tex](a*b)*\frac{1}{(a*b)}=1[/tex]
We multiply each side of this identity as follows:
[tex]\frac{1}{a}*((a*b)*\frac{1}{(a*b)})=\frac{1}{a}*1[/tex]
By using the property of associativity of multiplication and the property of 1, the left-hand side may be rewritten as:
[tex]\frac{1}{a}*((a*b)*\frac{1}{(a*b)})=(\frac{1}{a}*(a*b))*\frac{1}{(a*b)}=((\frac{1}{a}*a)*b)*\frac{1}{(a*b)}=(1*b)*\frac{1}{(a*b)}=b*\frac{1}{(a*b)}[/tex]
so that our entire identity now reads:
[tex]b*\frac{1}{(a*b)}=\frac{1}{a}[/tex]
Multiplying in a similar manner with 1/b, and using the commutativity of multiplication yields the desired result:
[tex]\frac{1}{(a*b)}=\frac{1}{a}*\frac{1}{b}[/tex]


When we construct the rationals as equivalence classes over the product set of integers, we can show that these new numbers constructed this way fulfill the set of axioms that we normally say "defines" the rationals (for example, the properties of multiplication I've used).
 
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  • #31
Dude, the question isn't about 1/a * 1/b = 1/ab... It's about why multiplication of a quantity by a fraction gives you the physical fraction of that quantity. Like 4 * 1/2 = 2 (half of 4)

Thanks for the proof though :)
 
  • #32
mtanti said:
Dude, the question isn't about 1/a * 1/b = 1/ab... It's about why multiplication of a quantity by a fraction gives you the physical fraction of that quantity. Like 4 * 1/2 = 2 (half of 4)

Thanks for the proof though :)

What is equivalent to [itex] \frac{4}{1} [/itex] ?
 
  • #33
mtanti said:
Dude, the question isn't about 1/a * 1/b = 1/ab... It's about why multiplication of a quantity by a fraction gives you the physical fraction of that quantity. Like 4 * 1/2 = 2 (half of 4)

Thanks for the proof though :)
Speculative posts regarding why the logic of maths is observed to hold in the "real" world belongs in the philosophy forum, not the math section
 
  • #34
But doesn't mathematics originate from applied observations? Especially such basics! It is a mathematical post to ask from where this operation originated.
 
  • #35
mtanti said:
But doesn't mathematics originate from applied observations? Especially such basics! It is a mathematical post to ask from where this operation originated.

No, that's philosophical. You're getting into epistimology here: how do we know math works? Why does it work? etc.
 
<h2>1. How do I find a fraction of a number?</h2><p>To find a fraction of a number, divide the number by the denominator of the fraction and then multiply by the numerator. For example, to find 1/4 of 20, divide 20 by 4 and then multiply by 1. The answer is 5.</p><h2>2. Can I use a calculator to find a fraction of a number?</h2><p>Yes, you can use a calculator to find a fraction of a number. Simply enter the number, press the division button, enter the denominator of the fraction, and then press the multiplication button followed by the numerator. The calculator will give you the answer.</p><h2>3. What if the fraction is a mixed number?</h2><p>If the fraction is a mixed number, convert it to an improper fraction first. To do this, multiply the whole number by the denominator, add the result to the numerator, and then use the sum as the numerator of the improper fraction. The denominator stays the same. Then, follow the same steps as finding a fraction of a number.</p><h2>4. How do I find a fraction of a decimal number?</h2><p>To find a fraction of a decimal number, first convert the decimal to a fraction. Then, follow the same steps as finding a fraction of a whole number. Finally, if needed, convert the answer back to a decimal.</p><h2>5. Is it possible to find a fraction of a number without using division and multiplication?</h2><p>Yes, you can find a fraction of a number without using division and multiplication by using equivalent fractions. For example, to find 3/4 of 20, you can first find 1/4 of 20 (which is 5) and then multiply by 3 to get the answer 15. This method is useful when working with fractions that have easy-to-calculate equivalent fractions.</p>

1. How do I find a fraction of a number?

To find a fraction of a number, divide the number by the denominator of the fraction and then multiply by the numerator. For example, to find 1/4 of 20, divide 20 by 4 and then multiply by 1. The answer is 5.

2. Can I use a calculator to find a fraction of a number?

Yes, you can use a calculator to find a fraction of a number. Simply enter the number, press the division button, enter the denominator of the fraction, and then press the multiplication button followed by the numerator. The calculator will give you the answer.

3. What if the fraction is a mixed number?

If the fraction is a mixed number, convert it to an improper fraction first. To do this, multiply the whole number by the denominator, add the result to the numerator, and then use the sum as the numerator of the improper fraction. The denominator stays the same. Then, follow the same steps as finding a fraction of a number.

4. How do I find a fraction of a decimal number?

To find a fraction of a decimal number, first convert the decimal to a fraction. Then, follow the same steps as finding a fraction of a whole number. Finally, if needed, convert the answer back to a decimal.

5. Is it possible to find a fraction of a number without using division and multiplication?

Yes, you can find a fraction of a number without using division and multiplication by using equivalent fractions. For example, to find 3/4 of 20, you can first find 1/4 of 20 (which is 5) and then multiply by 3 to get the answer 15. This method is useful when working with fractions that have easy-to-calculate equivalent fractions.

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