Why Do We Multiply Numerator & Denominator When Multiplying Fractions?

This is kind of a "dumb" question, but why do we multiply the numerators and denominators when multiplying fractions? For example:

1/5 x 2/3 = 2/15

Intuitively, I know why we need a common denominator when adding and subtracting fractions. We need to add apples to apples and oranges to oranges for it to logically make sense. But why do we suddenly not need a common denominator when multiplying fractions??? :confused: Wouldn't the same analogy apply here? Don't we need to do an apples to apples kind of operation?
 
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This is kind of a "dumb" question, but why do we multiply the numerators and denominators when multiplying fractions? For example:

1/5 x 2/3 = 2/15

Intuitively, I know why we need a common denominator when adding and subtracting fractions. We need to add apples to apples and oranges to oranges for it to logically make sense. But why do we suddenly not need a common denominator when multiplying fractions??? :confused: Wouldn't the same analogy apply here? Don't we need to do an apples to apples kind of operation?
Let's make a slightly easier problem, namely 1/5 X 1/3. Your problem is only a bit harder.

Imagine a pie cut into three large pieces, two of which have already been eaten, so that we have 1/3 of the pie. If we want to divide this piece of pie among five people equally, what fraction of the pie will each get? I'm using the idea that dividing by 5 is the same as multiplying by 1/5.

There are many different forms for the fraction 1/3, such as 2/6, 3/9, and so on. A form with a 5 in the numerator would be helpful - 5/15 would be a good choice. So one-fifth of 5/15 would be 1/15. After doing many such problems, you might get the idea that the answer could have been calculated more quickly simply by multiplying the numerators (getting 1) and the denominators (getting 15).
 
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Matterwave

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There are many different forms for the fraction 1/3, such as 2/6, 3/92, and so on.
I think you typo'd an extra "2" there.
 
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It's fixed now. I originally wrote 3/12, and neglected to get rid of the 2 when I changed the fraction to 3/9. Thanks for the correction!
 

NascentOxygen

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This is kind of a "dumb" question, but why do we multiply the numerators and denominators when multiplying fractions?
The most persuasive reason for doing it that way: so that you get the right answer! Do it any other way, and the answer won't agree with physical reality. No, I'm not kidding!
 

HallsofIvy

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One way to look at it is that the denominator of a fraction is a "unit". Just as saying the length of a line segment is "3 meters" means that I am using "meter" as my unit of length and the line segment is three of them, so the fraction "3/4" means that we are dealing with units of "one fourth" and we have three of them.

So just as a rectangle with sides "3 meters" and "5 meters" has area "15 square meters" or "15 m^2" so the fraction "3/4= 3 fourths" multiplied by the fraction "3/5= 3 fifths" is 9 (fourths x fifths)= 9/20.
 

Fredrik

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This is kind of a "dumb" question, but why do we multiply the numerators and denominators when multiplying fractions? For example:

1/5 x 2/3 = 2/15

Intuitively, I know why we need a common denominator when adding and subtracting fractions. We need to add apples to apples and oranges to oranges for it to logically make sense. But why do we suddenly not need a common denominator when multiplying fractions??? :confused: Wouldn't the same analogy apply here? Don't we need to do an apples to apples kind of operation?
It's not a dumb question at all. This is one way of looking at it: 1/5 is by definition the real number x such that 5x=1. 1/3 is defined similarly. 2/3 should be interpreted as ##2\cdot\frac{1}{3}##. To multiply the fractions 1/5 and 2/3 is to solve the equation ##\frac 1 5 \cdot \frac 2 3 =x##. If you multiply both sides by 5, you get ##\frac{2}{3}=5x##. If you multiply both sides of that by 3, you get 2=3·5·x. If you multiply both sides of that by ##\frac{1}{3\cdot 5}##, you get ##\frac{2\cdot 1}{3\cdot 5}=x##.

You also asked about the common denominator when we're doing addition. I would say that the reason is that we would like to use the distributive law: a(b+c)=ab+ac. If we see a sum ab+ac with a common factor (in this case a) in both terms, the distributive law tells us that we can rewrite the sum as a(b+c). The point of rewriting ##\frac 2 3+\frac 4 5## with a common denominator is that it enables us to identify a common factor in each term:
\begin{align}
&\frac 2 3+\frac 4 5 =\frac 2 3\cdot 1+\frac 4 5\cdot 1 =\frac 2 3\cdot \frac 5 5 +\frac 4 5\cdot\frac 3 3 =\frac{2\cdot 5}{3\cdot 5}+\frac{4\cdot 3}{5\cdot 3}\\
& =\frac{1}{15}\cdot 10+\frac{1}{15}\cdot 12 =\frac{1}{15}(10+12)=\frac{1}{15}{24}=\frac{24}{15}.
\end{align} The first calculation I did shows that we don't need a common denominator when we multiply fractions. The second calculation should explain why: The distributive law is used only when both addition and multiplication are involved.
 
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Thanks guys for the examples. I do have something to ask and follow-up on, but will write back this weekend. Need to cram for a quiz today (not in math though)!

It's odd, though, how a simple math concept can be so confusing at times. I'll write back more once I get the chance. Thanks for now everyone!
 

symbolipoint

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Put my post back!
 

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