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

Cubist

Full Member
Joined
Oct 29, 2019
Messages
645
@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
In this example every pizza in the shop is divided into 11 slices, not 22 slices. Therefore you are buying 5/11 of a single, whole, pizza. It doesn't matter that the pizza restaurant had, say, 10 whole pizzas in stock before your arrival (a total of 110 slices). All that matters is that you have purchased 5 slices. The remaining 105 slices left in the shop after your visit are still the property of the restaurant. Do you really think that you purchased 5/110 (less than 5%) of a whole pizza?
 

Saumyojit

Full Member
Joined
Jan 21, 2020
Messages
293
In this example every pizza in the shop is divided into 11 slices, not 22 slices. Therefore you are buying 5/11 of a single, whole, pizza.
I know i buy 3 out of 11 of 1st pizza and 2 out of 11 slices of 2nd pizza. Therefore I am buying 5/11 from 2 whole pizzas not single!
 

Dr.Peterson

Elite Member
Joined
Nov 12, 2017
Messages
9,133
I know i buy 3 out of 11 of 1st pizza and 2 out of 11 slices of 2nd pizza. Therefore I am buying 5/11 from 2 whole pizzas not single!
We have several times mentioned the importance of a unit in discussing fractions.

The fractions 3/11 and 2/11 refer to 3/11 of a pizza, and 2/11 of a pizza. The unit here is one pizza. Each fraction represents an absolute quantity, and when we add them, we have 5/11 of a pizza -- that is, 5 slices, each of which is 1/11 of a pizza.

Your fraction 5/22 is a relative quantity, a ratio of part to whole: 5/22 of the two pizzas from which you took them. Each has a meaning; but only the former (using a consistent unit) is addition of fractions.
 

Saumyojit

Full Member
Joined
Jan 21, 2020
Messages
293
The fractions 3/11 and 2/11 refer to 3/11 of a pizza, and 2/11 of a pizza. The unit here is one pizza. Each fraction represents an absolute quantity, and when we add them, we have 5/11 of a pizza -- that is, 5 slices, each of which is 1/11 of a pizza.Your fraction 5/22 is a relative quantity, a ratio of part to whole: 5/22 of the two pizzas from which you took them. Each has a meaning; but only the former (using a consistent unit) is addition of fractions.
5/11 of a pizza how .. those 5 slices belong to 2 diff pizzas so 5/11 of two pizzas ? i am hearing the first time the usage of absolute & relative quantity in fractions. Relative to what 2 pizzas or whole ..did not get the meaning sir.
U said "5/22 is a relative quantity, a ratio of part to whole" 5/11 is also a part to whole ratio
@Dr.Peterson
 

Jomo

Elite Member
Joined
Dec 30, 2014
Messages
8,290
@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
5/(22/2) for 1 pie. Yes, you ate 5/22 of 2 pies which equals 5/11 of one pie. So you are right! But you need to say that you ate 5/11 of a pie or 5/22 of two pies. The choice is up to you!
 

Saumyojit

Full Member
Joined
Jan 21, 2020
Messages
293
5/(22/2) for 1 pie. Yes, you ate 5/22 of 2 pies which equals 5/11 of one pie. So you are right! But you need to say that you ate 5/11 of a pie or 5/22 of two pies. The choice is up to you!
@Jomo @Dr.Peterson 5/11 of one pie means the considered slices belong to only one kind of pie from which i ate ; where in the question 5 slices that i ate comes from both of 2 pies. Not fitting in my mind..

Doc said "Your fraction 5/22 is a relative quantity, a ratio of part to whole" relative to what? 2 pizzas or 22 slices total?
If it was 5/11 then its a absolute quantity nothing relative as said in post23 as there is one unit. "My intution suggest that in 5/11 ; 5 slices i consider relative to 11 slices . 5/11 is also a relative quantity acc to my unsure intution.

Why sir said part to whole ? 5/11 or 5/22 are both part to whole ratio. Did he mean something else significant by part to whole?
 

Dr.Peterson

Elite Member
Joined
Nov 12, 2017
Messages
9,133
Do you know what a unit is? It is something like a meter or a kilogram or a pound or a day, used for describing a quantity (in an absolute sense, relative to a fixed standard, not a specific situation).

Here we are thinking of ONE PIE as the unit of measure for a quantity of pizza. That is a natural unit.

Similarly, I might use 1/4 of a cup of flour as a measurement. If there were 4 cups of flour in the bag I took it from, that wouldn't affect the quantity I measured out. I would not call it "1/16" -- unless I wanted to know how much of the bag I used, in which I would explicitly call it "1/16 of the bag" or "1/16 of the flour I started with". That is what I referred to as a relative amount -- relative to a certain total amount.
 

Saumyojit

Full Member
Joined
Jan 21, 2020
Messages
293
Similarly, I might use 1/4 of a cup of flour as a measurement. If there were 4 cups of flour in the bag I took it from, that wouldn't affect the quantity I measured out. I would not call it "1/16" -- unless I wanted to know how much of the bag I used, in which I would explicitly call it "1/16 of the bag" or "1/16 of the flour I started with".
Code:
+-------+ +-----+  +-----+  +-----+
| | | |   | | | |  | | | |  | | | | 
-------    ------  -------  -------
suppose 4 cups of each 20 gm;total 80gm of flour.Each box is worth 1/4th of 20gm=5gms.Each box can hold 5gms of flour.
Now if i take 1gm of flour from first cup,2gm frm second cup ,1gm each from last two cups . Then i use 5gms from all the 4 cups of flour and conclude that i used 1/4 of a cup of flour. I dont know am i right or not?
1/4 of a cup of flour ; numerator means 5 gms of flour(1 part) i am using from a cup of 20 gms (4 parts)of flour.
 

Jomo

Elite Member
Joined
Dec 30, 2014
Messages
8,290
@Jomo @Dr.Peterson 5/11 of one pie means the considered slices belong to only one kind of pie from which i ate ; where in the question 5 slices that i ate comes from both of 2 pies.
where in the question 5 slices that i ate comes from both of 2 pies. It doesn't matter! Do you really think that if you ate all 5 slices from pie A or all 5 slices from pie B or some from each totaling 5 slices that you ate different amounts? Do you really think that your stomach will know the difference? It must if you are eating different amounts of pizza.
 

Saumyojit

Full Member
Joined
Jan 21, 2020
Messages
293
Do you really think that your stomach will know the difference?
I was purely talking from a mathemtical point of view. It seemed very untuitive to me. But as doc told there is a rule in fractions "only the former (using a consistent unit) is addition of fractions " so i need to accept it and move on.

See my post no28
 
Last edited:

Jomo

Elite Member
Joined
Dec 30, 2014
Messages
8,290
5 slices i consider relative to 11 slices of one pie.

To say you ate 5/11 has no meaning at all! You ate 5/11 of something and you must state that something. I ate 5/11 of one pie or I ate 5/22 of two pies. Note that i ate 5/22 of two pies is the same as eating 5/11 of 1 pie.
 

Saumyojit

Full Member
Joined
Jan 21, 2020
Messages
293
@Jomo I ate 5/22 of 2 pizzas of each size 1/11 . Which one is the quantity and amount? 1/11 that is the size can i say it is the qunatity , 5 is the amount ? and the whole 2 pizzas are ...what? amt or qunatity? Or it does not matter too much in math the english terms.
@JeffM see my post 28
 

firemath

Full Member
Joined
Oct 29, 2019
Messages
587
This is making me laugh.

Amount, quantity, it doesn't really matter what you call it. Those are English terms for a mathematical idea. Just stay consistent with one of them to avoid as much confusion as possible.
 

Saumyojit

Full Member
Joined
Jan 21, 2020
Messages
293
Code:
+-------+ +-----+  +-----+  +-----+
| | | |   | | | |  | | | |  | | | |
-------    ------  -------  -------[/QUOTE] srry there will be one extra part in each of the cups then it will be 4 parts
 

Dr.Peterson

Elite Member
Joined
Nov 12, 2017
Messages
9,133
suppose 4 cups of each 20 gm;total 80gm of flour.Each box is worth 1/4th of 20gm=5gms.Each box can hold 5gms of flour.
Now if i take 1gm of flour from first cup,2gm frm second cup ,1gm each from last two cups . Then i use 5gms from all the 4 cups of flour and conclude that i used 1/4 of a cup of flour. I dont know am i right or not?
1/4 of a cup of flour ; numerator means 5 gms of flour(1 part) i am using from a cup of 20 gms (4 parts)of flour.
Of course you're right. Why do you doubt it? You've added 1/20 + 2/20 + 1/20 + 1/20 = 1/4 cup

I probably shouldn't have used "cup", as you are unaware that this is a standard unit of volume (equal to about 250 ml).
 

Saumyojit

Full Member
Joined
Jan 21, 2020
Messages
293
Similarly, I might use 1/4 of a cup of flour as a measurement. If there were 4 cups of flour in the bag I took it from, that wouldn't affect the quantity I measured out. I would not call it "1/16"
but in ur eg u said i would not call it "1/16" that means the original fractions were 1/4+1/4+1/4+1/4 thats the only way i would have mistakenly got 16 in the denom; if the fractions were those then acc to my 20gms eg i have taken 5gms from each 4cups so i conclude that i have used 1 full of a cup of fluor not 1/4 of a cup of flour.
The eg that u have given in post 27 how can i depict in fractions before addition. 1/20 +1/20+1/20+1/20 ;yes it is giving me 1/4 of a cup but it is not mistakenly given me 1/16 if i add denom then it is 1/80...
 

Cubist

Full Member
Joined
Oct 29, 2019
Messages
645
It's time for some music in this thread:- youtube link ( you might need to skip, or watch, an advert before the song plays)

This video animates the division of pizzas into an equal number of slices before adding/ subtracting slices. Initially it clearly shows that each slice has very different area, so it doesn't make sense to add/ subtract these slices. So each pizza is then divided into the LCM number of slices and you can see that the shaded areas can remain EXACTLY the same - but now both pizzas are carved into slices of equal area. This means that adding/ subtracting the equal sized slices makes sense!

And even better, I swear that one of the girls singing is Beyoncé. I guess it makes sense that she'd be great at math, so that she can keep track of her record sales ;)

Watch out for the damning pizza assessment lyric, "It smelled so good we had to give it a sniff" 🤣
 

Saumyojit

Full Member
Joined
Jan 21, 2020
Messages
293
It's time for some music in this thread:- youtube link ( you might need to skip, or watch, an advert before the song plays)
Still the way here it is explained in this forum is much satisfying and also intutive .
 

Dr.Peterson

Elite Member
Joined
Nov 12, 2017
Messages
9,133
but in ur eg u said i would not call it "1/16" that means the original fractions were 1/4+1/4+1/4+1/4 thats the only way i would have mistakenly got 16 in the denom; if the fractions were those then acc to my 20gms eg i have taken 5gms from each 4cups so i conclude that i have used 1 full of a cup of fluor not 1/4 of a cup of flour.
The eg that u have given in post 27 how can i depict in fractions before addition. 1/20 +1/20+1/20+1/20 ;yes it is giving me 1/4 of a cup but it is not mistakenly given me 1/16 if i add denom then it is 1/80...
I was saying that 1/4 of a cup is 1/16 of the 4-cup bag. And this has nothing to do with your 20-gram cups. I said it in post #27, before your #28.
 
Top