Why is it that you can multiply two fractions with each other without converting them to equivalent denominators?
Because that’s the way fractional multiplication is defined, based on what works in real life.
I don’t think you can give a much more detailed answer to this one.
Because there’s no reason to convert to equivalent denominators. The reason to do this for addition and subtraction has to do with what you’re doing when you add or subtract.
Let me try to explain why we use a common denominator for addition (I won’t explain it for subtraction, but since subtraction is just the opposite of addition, it makes sense that we’d do it there too). I’ll use an analogy to adding fruit.
Say you want to add 2 apples and 3 apples.
You can think of this as combining one group of 2 apples and one group of 3 apples. So how do you do it? Obviously, you just combine the groups the groups together and count the total. 5 apples.
Now, let’s say you want to add 2/7 to 3/7.
You can think of this as having an object called “a seventh”. You have one group of two sevenths, and one group of three sevenths, and you want to add them just like you did with apples a minute ago. So now you get 5/7.
OK, so far so good. But now, say you want to add two apples and three oranges.
You can’t just add them, because they’re to different kinds of things. So you need to convert them into a single kind of thing. For instance, we can say 1 apple equals 1 fruit, and 1 orange equals 1 fruit. So now we’re combining 2 fruits and 3 2 fruits, which we can do as above and get five fruits.
So now let’s say we want to add 2/7 and 3/5.
We can’t just do it, because we have a group of 2 sevenths, and a group of 3 fifths. But sevenths and fifths aren’t the same thing. So we convert them. We say, 1 seventh equals 5 thirtyfifths, and 1 fifth equals 7 thirtyfifths. So now we have one group of 10 thirtyfifths, and one group of 21 thirtyfifths. We add them and get 31 thirtyfifths, in other words 31/35. Just like adding fruit (except the conversion wasn’t a one-to-one ratio.)
OK, so that’s why we use a common denominator for addition. Now that that’s clear (I hope), let’s look at why we don’t use a common denominator for multiplication. Again, let’s start with fruit.
Say we want to multiply 3 apples by 2 oranges. You can look at this and say 6, but 6 what? It’s not 6 apples, and it’s not 6 oranges. Actually, 3 apples times 2 oranges gives us 6 appleoranges, where appleoranges is a new unit representing the product of apples and oranges. If you’ve studied physics, for example, you’re used to composite units like that. Otherwise, maybe that seems pretty weird to you. You might think we can get out of this by going back to fruit. But we can’t. Even if we say we now have 3 fruit times 2 fruit, that won’t give us 6 fruit. In fact it gives us 6 fruit squared, where “fruit squared” is a new unit. It’s just like multiplying lengths. 5 feet times 2 feet doesn’t equal 10 feet, it equals 10 square feet.
So now lets multiply 2/7 times 3/5. Just as with fruit, we can multiply 2 times 3 and get 6. But 6 of what? Well, it’s 6 seventhsfifths. But what does "seventhsfifths" mean? It’s the unit for something that’s a product of sevenths and fifths. But we know what that is – a seventh times a fifth is a thirtyfifth, since 7 times 5 is 35. Thus, 2/7 times 3/5 is 6 thirtyfifths. In other words, 6/35.
As you can see, it was never necessary to find a common denominator, just as it was unnecessary for multiplying fruit. (We could have expressed the fractions in terms of a common denominator if we wanted to, but it would have been a waste of time because eventually everything would simplify back down to the same answer.)
That’s the best answer I can come up with. I hope it made sense. Sometimes, the “simple” things are the hardest to explain. 
Sorry, I must have been worn out by that long reply, because I forgot to preview.
Should read:
Obviously, you just combine the groups together and count the total. 5 apples.
Should read:
So now we’re combining 2 fruits and 3 fruits, which we can do as above and get 5 fruits.
Maybe this will help:
Adding fractions with a common denominator is sort of like “adding like terms” (or “adding similar terms”). 2x + 3x = 5x because of the distributive law: 2x + 3x = (2+3)x. (Or, 2 apples + 3 apples = 5 apples.) But you can’t combine 2x + 3y, or 2x + 3x[sup]2[/sup].
Similarly, 2/7 + 3/7 = 2(1/7) + 3(1/7) = (2+3)*(1/7) = 5/7. But 2/7 + 3/11 can’t be combined this way: in 2(1/7) + 3(1/11) you’re mixing up addition and multiplication (or division, since division = multiplication by the reciprocal).
However, with multiplication, (2/7)(3/11) = 2*(1/7)3(1/11) = 23(1/7)*(1/11) = 6 / 77. It’s all just multiplication, and you can multiply in any order.
Why settle for two answers when you can get three?
I’ll start by apologizing for offering a very visual example with no pictures. I’ll try to be as descriptive as I can.
The conversion becomes necessary so that the numbers mimic real life.
Let’s say you’ve ordered two pizzas, and eaten different amonts of each of them, so that when you’re ready to pack them away in the fridge, you have 3/4 of one of the pizzas, and 1/2 of the other left over (let’s pretend you’ve got some weird pizza place that delivers them uncut, but you order from them anyway 'cuz the crust and cheese are really good from there, and you have a pizza slicer anyway). You want to combine the pizzas to see how many boxes you’re going to have to stuff into the fridge. A quick look at the situation will tell you that you’ve got 1-1/4 pizzas here, and you’re either going to have to cram both boxes in the fridge, lay some extra pizza on top of the other one, or get out some tupperware.
But how do we represent this numerically? If we just add the fractions together, top and bottom, you get 3/4 + 1/2 = 4/6, which is the same as 2/3. That’s obviously not correct.
Fortunately, what we need to do with the numbers exactly matches what we ned to do with the pizzas. To get as much pizza into one box as possible without layering, we need to take our uncut half a pizza and cut it into two pieces, or 2/4 of a pizza. Now we can put one of the quarters of a pizza into the box with the 3/4 of the other pizza (3/4 + 1/4 = 1!), and do whatever we need to do with the leftover 1/4. (3/4 + 2/4 = 1-1/4, after all.)
Now for multiplication. Let’s say you and a friend order a pizza from this place, but you’re a little short of cash, so your friend pays for two thirds of the cost. “No problem”, they say, “eat what you want, I just want a fair share of what’s leftover.” Turns out neither of you is that hungry after all (or it’ts a REALLY big pizza), and after cutting the whole thing in quarters, you split one of the quarters and you’re both full.
So now you’ve got 3/4 of a pizza leftover, and your friend is due 2/3 of it. No problem: you already cut the thing in quarters and only ate one of them. You’ve got three pieces, and your friend gets two of them. (3/4 * 2/3 = (32)/(43) = 6/12 = 1/2 = 2 of your quarter-pizza-sized slices). No conversion necessary.
You COULD do the conversion if it makes you feel better, it’s just unnecessary work. The lowest common denominator is 12, so you get:
3/4 * 3/3 = 9/12,
and
2/3 * 4/4 = 8/12.
9/12* 8/12 = 72/144 = 1/2.
OR
9/12 * 8/12
= (3/4 * 3/3) * (2/3 * 4/4)
= (3/4 * 3/3 * 2/3 * 4/4)
= (3/4 * 2/3 * 3/3 * 4/4)
=(3/4 * 2/3) * (3/3 * 4/4)
=(6/12) * (1)
= 6/12
=1/2
So the conversion in multiplication is optional, that is, unnecessary unless you;re just in the mood for math.
Wow, you either are, or would have been, the world’s greatest teacher.
Actually, there’s nothing wrong in converting two fractions to equivalent denominators before multiplying them. You’ll get exactly the same answer. So I think that the simple answer to your question is “Why bother?”. From the excellent explanations above, I think you already know why you need common denonimators for adding or subtracting fractions.
There is something here that some people get that seperates them from others.
Because converting to common denominators is done by multiplication. If multiplication was required to do multipication, it would take forever to pass 3rd grade math.
Muad’Dib, the other day I searched for “adult math” because I wanted to see if there was anything about teaching oneself math and I saw your old thread on the same question. Now that I see you are asking this question, does that mean you are doing it? Do you mind if I ask how you are doing it and what your plan is? Even if you want to email me about it, I would appreciate it very much, if it’s not an imposition.
I don;t know. I am a college student that has passed first year calculus. Yet, while reviewing my math skills, I realize that I do not really understand why some of these basic math problems work. I learned the rules without ever learning the why.
Heh, well I think the thread was 2 years old. I guess going to college and taking calculus is a good way to do it.
Let’s go simpler. It might help to recall that multiplying by 1/n is the same thing as dividing by n. You can take pizza, divide it in 2 (multiply by 1/2), then chop one of those halves into thirds (multiply by 1/3), and one of those thirds into sevenths (multiply by 1/7), and so on, without doing any strange mathematical trickery.
1 times 1/2 times 1/3 times 1/7 is the same as 1x1x1x1, divided by 2, divided by 3, divided by 7.
I imagine that what confuses the person asking the question is that you do have to convert to equivalent denominators when adding fractions. Perhaps the best way around the confusion is to realize that you don’t have to convert to like denominators when multiplying because once the multiplication is finished ytou basically have done so, i.e. you’ve devised a new denominator that both the original fractions can be expressed in.
I found the same thing when I hit college.
The problem is that elementary math education is designed to give you a lot of random skills that dovetail into each other if you keep with it and take higher math in High School and College. So suddenly in sophomore year you find yourself needing something they taught you (and you ignored) in fourth grade and expected you to retain all through adolescence.
Bad plan. Something needs to be done, IMO.