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Post date: May 19, 2013 12:54:41 PM
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Slope: Slope here, to talk to everyone about dividing fractions.
Maud: Oh! I was going to put together a video on that...
Slope: Too late.
Cosine: Um, why bother dressing that way? Everyone knows your other identity now, it's...
Slope: Sis, I love you, but hush.
Lyn: Can I join you, like last time we talked fractions?
Slope: Yes, that would make sense.
UNCOMMON DENOMINATORS
Slope: Fractions are a lot like percents. They make NO SENSE unless you have the context of the WHOLE.
Lyn: Give us a "for instance".
Slope: Hey, Circe! What's bigger? Three quarters of a dollar, or half a hundred dollars?
Circe: Obviously three quarters is bigger than a half!
Slope: Ok, thanks, have fun with that then!
Lyn: But hold on. In mathematics, we're basing fractions off of the numerical value 1, in order to be consistent.
Slope: You're suggesting we reject real life examples? I think that's heresy or something.
Lyn: Actually I'm suggesting that for the purposes of this article we assume we're talking about the same "whole".
Slope: Aw, Linear, it's like you can read my mind.
CASE 1: ADDING FRACTIONS
Slope: I know. I said I'd talk dividing. We're building suspense.
Lyn: Okay. Then to add fractions, you need a common denominator.
Slope: No, you don't.
Lyn: Now THAT is heresy.
Slope: What's three quarters plus one half? I draw a picture. It's "two and one-half" halves, which is one and a half a half, also known as one and a quarter.
Lyn: That's... wait, what? I'm confused.
Slope: THAT'S why we have the common denominator. So that when I give an answer, it's not fractions on top of fractions, or mixed numbers, or some other standard.
Lyn: Ah! We want a simplest form.
Slope: More than that. If we have a common denominator, THE WHOLE DOESN'T MATTER ANYMORE. You're no longer comparing dollars to donuts, you're comparing cents to cents.
Lyn: Makes sense.
Slope: Yes. I should know better than to give you straight lines.
Lyn: My lines are already straight.
Slope: So, let's say we add 3/8 to 4/7. Different pieces of one. Messy. But at the same time, that's just 21/56 and 32/56.
Lyn: I see! Now that they're the same parts of one, we ignore the whole. 21 plus 32 is 53.
Slope: In fifty-sixths.
Lyn: Yes. Or whatever whole denominator is worked out earlier.
Slope: Precisely. No confusion in thinking we add denominators there.
CASE 2: MULTIPLYING FRACTIONS
Lyn: So, multiplying is where we're all "multiply the top and the bottom".
Slope: Actually, we can use a common denominator.
Lyn: We can? I mean, of course we can.
Slope: Just remember that what we're doing is taking a fraction OF a fraction. Here, multiply 1/2 by 1/3.
Lyn: Common denominator means 3/6 by 2/6. Multiply the top, keep the denominator and we get... six sixths? The heck?
Slope: We get six sixths where SIXTHS is the new whole. Thus 1/6.
Lyn: You lost me.
Slope: Then forget the common denominator, we'll come back to it. Given 1/2 times 1/3, even though both are with respect to one, the ANSWER would be with respect to ONE HALF, the first fraction.
Lyn: Ah, now I get it.
Slope: No you don't.
Lyn: No I don't.
Slope: 1/2 X 1/3 means HALVES is now my whole, and I carved out a third. Yet that third is a sixth... with respect to 1.
Lyn: Okay... so the answer is 1/3 with respect to halves, or 1/6 with respect to 1.
Slope: Alternatively, consider 1/3 times 1/2. The ANSWER is 1/2 with respect to thirds, or again 1/6 with respect to 1.
Lyn: True, but... sorry. I feel like I should be getting this.
Slope: No worries. Let's go again. Do 1/3 times 3/4.
Lyn: Okay. The answer is 3/4 - with respect to 1/3. But I want a "whole", so carve up said third into quarters. A third in quarters means I've got twelfths. Oh, but you said we've got three quarters, so I should have multiplied by three... it's 3/12. Or 1/4.
Slope: Exactly. Now consider 3/4 times 1/3.
Lyn: This time I'll start by multiplying numerators. So the answer is 3/3 - with respect to 1/4. I carve up my quarter into, uh, a whole. Still 1/4. I guess it makes sense I get the same answer.
Slope: Precisely. Now do 3/8 times 4/7.
Lyn: Oh bloody...
Reci: Lyn! Be good. No swearing.
Slope: Hey, Reciprocals! No eavesdropping!
Cotangent: ...Sorry, we're going...
Lyn: Ahem. Well, the answer is 12/7 with respect to EIGHTHS. I carve up my eighths into sevenths, and 12/7-th of an eighth is 12/56 of the whole. Or 3/14.
Slope: Now try it with a common denominator.
Lyn: Then as before, we use 21/56 and 32/56. The answer is 672/56-ths... but that's not of a WHOLE, it's of a 56th. So 672/3136 of the whole. Or 3/14. Huh.
Slope: Which I grant is the same as if you didn't bother with the common denominator.
Lyn: Or if you simply multiply the top and the bottom. But now it's more clear.
Slope: In one sense we didn't need this, but it brings us back to...
CASE 3: DIVIDING FRACTIONS
Slope: Here the common denominator is a lot more helpful than Case 2.
Lyn: I've got this. 3/8 divided by 4/7. Same as 21/56 divided by 32/56. The answer is 21/32 fifty-sixths, where... wait, is fifty-sixths the whole now? Because the answer IS 21/32.
Slope: That's one way to see it, yes. Alternatively, 1/3 divided by 1/2... you're giving to a "whole", and doing it in "halves". 1/3 + 1/3 = 2/3, which is the answer.
Lyn: Or 1/3 divided by 1/5... you're giving to a "whole", and doing it in "fifths". [1/3 + 1/3 + 1/3 + 1/3 + 1/3] = 5/3
Slope: Now, with 1/3 divided by 2/5, you'd have to cut your answer in half, because you've given to TWO "wholes". 5/3 halved is 5/6.
Lyn: Thus my 3/8 divided by 4/7, you're giving to a "whole", and doing it in 7ths, which gives [3/8 + 3/8 + 3/8 + 3/8 + 3/8 + 3/8 + 3/8] = 21/8 but that's four wholes, so [21/8][1/4] = 21/32.
Slope: Four wholes causing a quarter, which explains a bit of why the reciprocal is used in dividing.
Reci: I heard reciprocal. Did you call us back?
Circe: Hey, Slope! Did you scam me earlier?!
Lyn: Come to think, we can see fraction division as simply dividing the numerators and dividing the denominators. Which you can even do when dividing two rational functions, with variables... just track restrictions.
Slope: Maybe common denominators were even the norm, until someone with a multiplication fixation noticed that taking the reciprocal of the second helped the common denominator to "cancel out"! Which in truth means there's a multiplication of one.
ArcTan: Did someone call for inverses?
Cotangent: No, reciprocals.
Root: Hey hey, is this a party??
ArcSin: For SCIENCE!
Hyper: Damn it Nisano, stop using my schtick!
Lyn: Aaand we're off track enough to be done here. Except what about the issue of common numerators? Where are they helpful?
Slope: People can try that at home!
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With thanks to @nik_d_maths whose tweet inspired me to finally write this.
For more reading:
"Ask Dr. Math: Fractions and Common Denominators"
"Models for the Division of Fractions"
"Common numerator fraction division" by @trianglemancsd
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