Why are denomination explanations so complicated?

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paulbranon

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Why are all of the explanations as to why we need a common denominator when adding and subtracting fractions but not when multiplying them so complicated?

I've looked at quite a few explanations (one used algebra to explain it) I only got half way through that one! Isn't it because none of the ones I can find on the Internet are asking what does the question actually mean. Isn't it because when we ask what is 1/2 x 1/3 what we are actually asking is what is half of a third. When we ask what is 4/5 x 2/3 we are asking what is nearly all of two thirds


Is this not equally true of wholes multiplied by fractions? The demonstration of 2/3 x 12 takes quite a bit of space. But the answer to the question what is two thirds of twelve does not.
 
I'm not sure what your question is. I can't answer "why are the explanations so complicated" because I don't find them complicated! I also don't know whether you are asking about arithmetic (purely numerical fractions) or algebra. In either case the explanation is the same- we don't need a "common" denominator to multiply fractions because \(\displaystyle \left(\frac{a}{b}\right)\left(\frac{c}{d}\right)= \frac{ab}{cd}\) whatever the denominators are.

Adding fractions, however, the basic rule is \(\displaystyle \frac{a}{b}+ \frac{c}{b}= \frac{a+ c}{b}\) which requires that we have the same denominator for both fractions.

One way of thinking about this is to think of the denominator of a fraction as being like the "units" for a measurement. If I have a rectangle with length "3 inches" and width "2 feet" it is perfectly valid to say that the area is "6 feet-inches". That's an "unusual" unit for area but valid- we can multiply units like that just as we can measure energy in "pound feet". But we can't do that if we are adding. 3 inches plus two feet is not 5 anything! Instead we need to get the same units: 3 inches plus 24 inches is 27 inches.
 
I like the example of three inches plus two feet is not five anything.

I think there's an inherent problem when it comes to multiplying fractions which is shown most clearly with wholes. I'm sure that a lot of people consider multiplying to be the act of increasing a quantity. However 2/3 x 12 = 8 , the result is a smaller whole. The problem is not unrelated in the case of fractions multiplied together. But it's less dramatic. The problem centres around what the act of multiply by a fraction actually means.


(Adding and subtracting fractions is an entirely different proposition.)
 
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Perhaps it's also worth mentioning that it wouldn't be impossible to add fractions without a common denominator. The fact that we use one is merely a convention. Other things are manipulated unequal denominations without any problem, pounds, shillings and pence, pounds and ounces

A real problem would occur if we tried to add ounces to kilos. That wouldn't work.


So, we could talk about three quarters and three eighths, if we wanted to. We just don't.


I suspect the real problem would occur with dispensing with the convention in the case of subtraction of some unequal fractions. Without first converting them into the same denominations I suspect it would soon become impossible.
 
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paulbranon said:
Perhaps it's also worth mentioning that it wouldn't be impossible to add fractions without a common denominator. The fact that we use one is merely a convention. Other things are manipulated unequal denominations without any problem, pounds, shillings and pence, pounds and ounces

A real problem would occur if we tried to add ounces to kilos. That wouldn't work.


So, we could talk about three quarters and three eighths, if we wanted to. We just don't.

Actually, you can...

If you call 3/4 + 3/8 = 3/4 + 3/8 or 3 nickels + 1 dime = 3 nickels + 1 dime no body will object... as long as you don't want to simplify and call those with same "last-name" (dimension) like "eighths", or "fourths", or "nickel" or dimes.


I suspect the real problem would occur with dispensing with the convention in the case of subtraction of some unequal fractions. Without first converting them into the same denominations I suspect it would soon become impossible.
Click to expand...

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You could do it that way and nobody would object! I'm going to write that down.

Thanks for the considerations, guys. I just wanted to pin the whole thing down squarely.
 
So long as the feet are all of one denomination, or another, it rather looks as though it doesn't matter where they're planted.
 
paulbranon said:
So long as the feet are all of one denomination, or another, it rather looks as though it doesn't matter where they're planted.
Click to expand...

Cannot let those feet inch up the wall - those need to be in mid-air..... so said Denis...
 
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