Why-do-the-denominators-need-to-be-the-same-when-adding-the-fractions

Saumyojit

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TOO LONG PLEASE READ. NEED HELP @Dr.Peterson @JeffM @Subhotosh Khan

SO I found a already given answer in this "https://www.quora.com/Why-do-the-denominators-need-to-be-the-same-when-adding-the-fractions"
I was going through Carol Hartman Shadwell & Alan Bustany answers
1: FROM ALAN BUSTANY ANSWER I QUOTE "
Numerators Equal
Suppose the numerator is a common value, n, then what is the value of n/b+n/d ? Suppose it is p/q then we can multiply through by qbd to get:
nqd/qbd+nqb/qbd=pbd/qbd
So our problem is reduced to a situation of denominators equal.

Denominators Equal
Suppose the denominator is a common value, m≠0, then what is the value of a/m+c/m ? Suppose it is p/m then multiplying through by m gives us:
a/m+c/m=p/m⇒a+c=p so--->a/m+c/m=(a+c)/m"

1:I did not understand what he is trying to tell in this particular quote other than "Denominators equal turns out to be much simpler "
2: he multiplied by qbd both sides num & denom to make denom equal ; does he found out the value of n/b+n/d or it is not neccesarry in this context ;he was trying to make denom same right?or he was giving us some instinct ? I have not got the logic why denom has to be equal after reading this.

2:
FROM Carol Hartman Shadwell answer I QUOTE "
Likewise, in fractions, we add pieces of the same size. We add halves to halves and 12ths to 12ths but not halves and 12ths.To add 1/3 and 1/2 of a pizza, we have to do something first: rewrite them as fractions with common sizes on the bottom. (common denominators) The halves need to be divided into 3 equal parts to create 6ths and the thirds need to be divided into 2 equal parts to create sixths. The result is a pizza that is made of 6 equal parts. 3 out of the 6 equals 1/2 and 2 out of the 6 equals 1/3. Now add the sixths. 1/3+1/2=2/6+3/6=5/6 "

THE THIRDS PART FIRST: i was drwaing tiles and picturing it ;he says there is only one pizza out of which to take 1/3


1/3 looks like this:
|____|_____|___| each box is 1/3 ---->then i pick 1 one third slice&divide into 2 equal parts to create sixths -->|____|_____| two box each 1/6 worth
a equivalent pic of that one 1/3 slice box = 2 slices of (1/6)box look like just imagine there is a arrow from this box to upper 1/6th box|_________|


HALVES PART: then i take 1/2 of the same whole pizza or different pizza?IF same, I have already cut the pizza into 3 slices and taken 1 slice(1/3) and then divided into two sixths as shown above ; REMEMBER i have still two 1/3 slices left which i havent done anything ;now 1/2 means" i need to slice the pizza into 2 parts and consider 1 part right"; which part of already sliced pizza will i have to halve ;whether it is one of the two 1/3 slices which i think it might be or the already sliced 1/6th pizza.

i cannot drw the next pic of the 1/2 part as there are many possbilities ringing in my mind.
 
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Saumyojit

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IHAVE EDITED due to spacing problemn
 

Dr.Peterson

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First, as you said, this is far too long, and I resisted answering. I've already said that it's much easier for a helper to just answer your personal questions, rather than trying to explain what someone else has said. If you don't understand what you read, then there's a good chance that it is not written in a way that you can understand, so you should find a better source. And it would be best if that source were one well-written textbook or site that takes you through the topics in an orderly way, from one consistent perspective. Looking at random help sites is a horrible way to learn.

SO I found a already given answer in this "https://www.quora.com/Why-do-the-denominators-need-to-be-the-same-when-adding-the-fractions"
I was going through Carol Hartman Shadwell & Alan Bustany answers
1: FROM ALAN BUSTANY ANSWER I QUOTE "
Numerators Equal
Suppose the numerator is a common value, n, then what is the value of n/b+n/d ? Suppose it is p/q then we can multiply through by qbd to get:
nqd/qbd+nqb/qbd=pbd/qbd
So our problem is reduced to a situation of denominators equal.

Denominators Equal
Suppose the denominator is a common value, m≠0, then what is the value of a/m+c/m ? Suppose it is p/m then multiplying through by m gives us:
a/m+c/m=p/m⇒a+c=p so--->a/m+c/m=(a+c)/m"

1:I did not understand what he is trying to tell in this particular quote other than "Denominators equal turns out to be much simpler "
2: he multiplied by qbd both sides num & denom to make denom equal ; does he found out the value of n/b+n/d or it is not neccesarry in this context ;he was trying to make denom same right?or he was giving us some instinct ? I have not got the logic why denom has to be equal after reading this.
Yes, he's just saying, with far too many words, that if the denominators are not equal, you need to make them equal in order to carry out the addition, so why not make them equal as soon as possible. Or something like that.

But he did not multiply the top and bottom of anything by qbd. He multiplied the top and bottom of n/b by qd, and multiplied the top and bottom of n/d by qb, so that both denominators become qbd. That one fact, which he skipped over, is probably the main thing you need to understand!

2: FROM Carol Hartman Shadwell answer I QUOTE "
Likewise, in fractions, we add pieces of the same size. We add halves to halves and 12ths to 12ths but not halves and 12ths.To add 1/3 and 1/2 of a pizza, we have to do something first: rewrite them as fractions with common sizes on the bottom. (common denominators) The halves need to be divided into 3 equal parts to create 6ths and the thirds need to be divided into 2 equal parts to create sixths. The result is a pizza that is made of 6 equal parts. 3 out of the 6 equals 1/2 and 2 out of the 6 equals 1/3. Now add the sixths. 1/3+1/2=2/6+3/6=5/6 "

THE THIRDS PART FIRST: i was drwaing tiles and picturing it ;he says there is only one pizza out of which to take 1/3

1/3 looks like this:
|____|_____|___| each box is 1/3 ---->then i pick 1 one third slice&divide into 2 equal parts to create sixths -->|____|_____| two box each 1/6 worth
a equivalent pic of that one 1/3 slice box = 2 slices of (1/6)box look like just imagine there is a arrow from this box to upper 1/6th box|_________|

HALVES PART: then i take 1/2 of the same whole pizza or different pizza?IF same, I have already cut the pizza into 3 slices and taken 1 slice(1/3) and then divided into two sixths as shown above ; REMEMBER i have still two 1/3 slices left which i havent done anything ;now 1/2 means" i need to slice the pizza into 2 parts and consider 1 part right"; which part of already sliced pizza will i have to halve ;whether it is one of the two 1/3 slices which i think it might be or the already sliced 1/6th pizza.

i cannot drw the next pic of the 1/2 part as there are many possbilities ringing in my mind.
So find a page that does better, including appropriate pictures! Try this: https://www.mathsisfun.com/fractions_addition.html

That shows how to add 1/3 + 1/6 of a pizza, then 1/3 + 1/5, all in pictures.

Now tell me what part of that you are having trouble with.

But I'll show you my way to demonstrate the problem you're doing (keeping in mind that we don't need pictures once we've grown past these initial ideas):

Code:
Here is 1/3: +---+---+---+  Here is 1/2: +-----+-----+
             |XXX|   |   |               |XXXXX|     |
             +---+---+---+               +-----+-----+

We can split the third into two equal sixths, and the half into three equal sixths:
             +-+-+-+-+-+-+               +-+-+-+-+-+-+
             |X|X|       |               |X|X|X|     |
             +-+-+-+-+-+-+               +-+-+-+-+-+-+

Putting these together, we have 5/6:
             +-+-+-+-+-+-+
             |X|X|X|X|X| |
             +-+-+-+-+-+-+
I don't need to take both pieces from one pizza in my initial thinking! But I ended up taking 5 of the 6 pieces of the one pizza.
 

Jomo

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Whenever you add things they have to be of the same type and size. to have mathematical meaning. You need common size for comparison. For example if you eat two regular sized slices of pizza and I eat a HUGE size, you can't really say that you ate 2 slices and I ate one slice. There really is no comparison since my slice could have equaled 5 of your slices.

When you add 3 and 7 you are really adding 3 ones and 7 ones. So the units are ones.
Now you can add 3/11 and 2/11 since 3/11 = 3*1/11 and 2/11 equals 2*1/11. Here you are adding 1/11s!. You have 3eleventh and you are adding 2 eleventh getting a total of 5- eleventh or 5/11.

Now if you try to add 2/3 and 3/4 what are the common units which you are adding? This is why you can't add these numbers before writing them with common denominator.
 

pka

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Out of perverse curiosity what would say about:
1) \(\dfrac{\sqrt 7}{9}+\dfrac{\sqrt[3]{13}}{5}\) OR

2) \(\dfrac{9}{\sqrt 7}+\dfrac{\sqrt{13}}{5}\)
 

Jomo

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Out of perverse curiosity what would say about:
1) \(\dfrac{\sqrt 7}{9}+\dfrac{\sqrt[3]{13}}{5}\) OR

2) \(\dfrac{9}{\sqrt 7}+\dfrac{\sqrt{13}}{5}\)
\(\dfrac{\sqrt 7}{9}+\dfrac{\sqrt[3]{13}}{5}\) = \(\dfrac{5\sqrt 7+9\sqrt[3]{13}}{45}\)
 

Dr.Peterson

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Out of perverse curiosity what would say about:
1) \(\dfrac{\sqrt 7}{9}+\dfrac{\sqrt[3]{13}}{5}\) OR

2) \(\dfrac{9}{\sqrt 7}+\dfrac{\sqrt{13}}{5}\)
I'd say, since this question is posted under Arithmetic, those can wait. I certainly wouldn't use pictures!

But, yes, students need to get pretty far away from the concrete, and get very used to more or less rote manipulations, before handling those. The OP has some distance to go.
 

pka

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I'd say, since this question is posted under Arithmetic, those can wait. I certainly wouldn't use pictures!
Are you saying that is not part of Arithmetic? If so what have we come to?
 

Saumyojit

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Here is 1/3: +---+---+---+ Here is 1/2: +-----+-----+

We can split the third into two equal sixths, and the half into three equal sixths:

+-+-+-+-+-+-+[/CODE]I don't need to take both pieces from one pizza in my initial thinking! But I ended up taking 5 of the 6 pieces of the one pizza.
@Dr.Peterson PLEASE READ I HAVE STATED MY CONCERN

From mathisfun.com, I saw that from the eg : 1/3+ 1/5 there are 2 pizzas not one !!... one of which is divided into 3 slices considering one & second pizza divided into 5 considering one. They are showing it like they have sliced two different pizzas and glued into one . THAT WAS EXACTLY MY DOUBT. So How many pizza 2 or 1 are there?

MY WAY: If it was 1 pizza and if it was given from beginning the original fractions are 5/15 & 3/15; I would have sliced it into 15 slices first and taken 5 to meet the demand of first fraction (5/15) and of the 10 slices left i would have taken 3 slices of the remaining 10 i.e 3/15 , then it can be said as 8/15 of one whole pizza not two.This is not possible as the original fractions are given 1/2 , 1/3 ; considering one pizza, then i slice that one pizza into 3 slices and take one & then 1/2 is there so MY DOUBT IS i take 1/2 of which slice coz the pizza has already been sliced ???


this is not a doubt-->
: Another thing from this eg 1/3+ 1/5 --> I discovered that when i am splitting one 1/3rd slice into 5equal fifteenths; the other two 1/3rd slices need to be splitted into 5 equal fifteenths; all the thirds need to be divide into 15ths . I thought that the other two 1/3rd slices will be untouched.

Yes, he's just saying, with far too many words, that if the denominators are not equal, you need to make them equal in order to carry out the addition
SO i have to accept this thing as a rule that FRACTIONS have to have the same size or denom to do addition or sub. There is no logic here right or any logic present?
 
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Saumyojit

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@Dr.Peterson @Jomo

CONCLUSION: THERE ARE TWO PIZZA INTIALLY. IF IT IS:

i think now that 1/3 + 1/5 means there are two pizzas initially it makes more sense ...then we divide each 1/3 and each 1/5 into fifteenths ;then 5 slice out of 15 i consider of the first pizza and 3 slices out of 15th i consider of the sec pizza and then join them 8 slices(8/15) formed and there are 22 ( 1/15th) slices remaining. I know that 8 slices have been considered and the remaining 7 unconsidered slices shall come from which of the 22 unconsidered slices to make it a complete pizza(8/15) that i am holding after lcm. this is my doubt if and only if initially there are two pizzas.If not then go back to my post before this.

THE CONFUSION
started from Carol Hartman Shadwell answer I QUOTE "To add 1/3 and 1/2 of a pizza "---> i though there is only one pizza!!!!
 

Saumyojit

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@Dr.Peterson please reply i want to end the post by today and thank u for seeing the post
 

HallsofIvy

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I think of the denominators of fractions as the "units". "one third plus two fifths" is like "one mile plus two furlongs. In order to add those we have to get the same unit. One mile is 1760 yards and one furlong is 220 yards so "one mile plus two furlongs" is "1760 yards plus 440 yards" or 2200 yards.
 

Saumyojit

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I think of the denominators of fractions as the "units". "one third plus two fifths" is like "one mile plus two furlongs. In order to add those we have to get the same unit. One mile is 1760 yards and one furlong is 220 yards so "one mile plus two furlongs" is "1760 yards plus 440 yards" or 2200 yards.
see my recent doubt and new two posts
 

Saumyojit

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Dr.Peterson

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Please stop nagging. You make me think of a child pulling at my clothes to get me to pay attention when I am busy with something else. I am not your personal servant. All this does is to make me want to avoid answering your long, complex "questions". (Note that others are generally avoiding them already. I will eventually join them.)

So How many pizza 2 or 1 are there?

SO i have to accept this thing as a rule that FRACTIONS have to have the same size or denom to do addition or sub. There is no logic here right or any logic present?
You can add pieces from one pizza, or from several. No glue is needed, just a plate. The quantity is the same. And in any case, we're adding fractions in the abstract, and it doesn't matter what they mean in an application. You're getting upset about a mere illustration.

There is plenty of logic; it is not just an arbitrary rule. Most simply stated, you can't add 1 apple and 2 oranges; you have to use the same unit, such as 1 cup of apple and 2 cups of orange to make 3 cups of fruit.

But if you can't understand it, just take it as a rule.

THE CONFUSION started from Carol Hartman Shadwell answer I QUOTE "To add 1/3 and 1/2 of a pizza "---> i though there is only one pizza!!!!
The phrase "1/3 and 1/2 of a pizza" doesn't specify whether they are "of the same pizza" or not; it just describes two quantities, and "pizza" is the unit. Do you honestly think you'll be eating different amounts if they come from two (equal sized) pizzas or one?
 

Saumyojit

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You can add pieces from one pizza, or from several. No glue is needed, just a plate. The quantity is the same. And in any case, we're adding fractions in the abstract, and it doesn't matter what they mean in an application. You're getting upset about a mere illustration.
@Dr.Peterson @Jomo @HallsofIvy

I am now afraid when u told that u will not be answering my future questions. I dont know whats the problemn.. i have doubts in my mind which i cannot understand myself thats why i say everything that i have in my doubts and thats why it will be long . If u read it when u are not busy ; it will be helpful of what i have written in post #9 & post#10 .
You can add pieces from one pizza, or from several
Suppose 1/2 & 1/3 are original fractions; considering one pizza first i slice the whole pizza into half and then there are two halves out of which i consider one half , the rest half is unconsidered so do i slice that rest half into three slices as the other fraction is (1/3)

Code:
  Here is 1/2: +-----+-----+
               |XXXXX|     |
               +-----+-----+
slcing the whole piza into half.Then taking the other unconsidered half |1/3|1/3| 1/3| breaking the other half into 3 slices each is 1/3

Am i right till this diagram?
 
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Dr.Peterson

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@Dr.Peterson @Jomo @HallsofIvy

I am now afraid when u told that u will not be answering my future questions. I dont know whats the problemn.. i have doubts in my mind which i cannot understand myself thats why i say everything that i have in my doubts and thats why it will be long . If u read it when u are not busy ; it will be helpful of what i have written in post #9 & post#10 .
Suppose 1/2 & 1/3 are original fractions; considering one pizza first i slice the whole pizza into half and then there are two halves out of which i consider one half , the rest half is unconsidered so do i slice that rest half into three slices as the other fraction is (1/3)

Code:
  Here is 1/2: +-----+-----+
               |XXXXX|     |
               +-----+-----+
slcing the whole piza into half.Then taking the other unconsidered half |1/3|1/3| 1/3| breaking the other half into 3 slices each is 1/3

Am i right till this diagram?
If you have one pizza from which you want to cut 1/2 and 1/3, divide it into 6 equal parts and take 3 of them for the half, then 2 more for the third. What is hard about that?
 

Saumyojit

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If you have one pizza from which you want to cut 1/2 and 1/3, divide it into 6 equal parts and take 3 of them for the half, then 2 more for the third. What is hard about that?
@Dr.Peterson
AS far as i have understood the First two operations are :
1: I cut half of one whole pizza and then consider one half slice ok?
Code:
Here is 1/2: +-----+-----+

             |XXXXX|     |

             +-----+-----+
2: Then i think i need to cut any one of the already cut 2 slices into 3 equal slices for 1/3rd part. Which one should i cut into 3 equal slices the considered half slice (the crossed one) or the uncrossed half slice? Please make me clear till this . After i understand this ,then we will go to dividing in to 6 equal parts operation.
 
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Dr.Peterson

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If you want to add two slices, then the second one should not be part of the first one, should it? Of course the 1/3 comes from the second half!

But (when I explain this to children) I think in terms of cutting the entire pizza into 6 parts, not cutting only the second half into 3 parts. Take it like this:

Code:
  1/2
+-----+-----+
|XXXXX|     |
+-----+-----+

  3/6
+-+-+-+-+-+-+
|X:X:X| : : |
+-+-+-+-+-+-+

  3/6  2/6 (= 1/3)
+-+-+-+-+-+-+
|X:X:X|X:X: |
+-+-+-+-+-+-+

    5/6
+-+-+-+-+-+-+
|X:X:X:X:X: |
+-+-+-+-+-+-+
 

Saumyojit

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Now you can add 3/11 and 2/11 since 3/11 = 3*1/11 and 2/11 equals 2*1/11. Here you are adding 1/11s!. You have 3eleventh and you are adding 2 eleventh getting a total of 5- eleventh or 5/11.
@Jomo @Dr.Peterson My intution suggests that if i am eating 3 slices out of 11 of 1st pizza and 2 slices out of 11 of 2nd pizza; then eating 5 slices out of 22 . But in mathemtical notation we write 5/11. why so.. i know the lcm logic
 
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