MARCH 4TH, 2017
J: The Beatles will have pie. The next task is adding up all the pie.
S: Wait! We need to add it in a special, more Beatle-y way than just adding.
J: John wants a 1/3 slice of pie.
S: Wow. That’s a lot of pie. But George was the hungry one who would ACTUALLY want 1/3 of a pie.
J: George wants all the rest of the pie after the other three have their slices. Paul wants a 1/5 slice of pie.
S: George is going to get, like, nothing!
J: We’ll see. Ringo wants a 1/7 slice of pie.
S: For fractions, isn’t there Euclid’s algorithm?
J: Yes. You’re thinking of the least common multiple, and that’s exactly what you want here. So, how much pie does George get?
S: Hmm… Well, I don’t remember Euclid’s algorithm!
J: That’s okay! We don’t have to use it because I cleverly picked prime numbers for the denominators.
S: I still don’t know what to do.
J: Well, George gets the leftover pie, right?
S: Yes.
J: And that would be one pie minus John’s slice, Paul’s slice, and Ringo’s slice.
S: Yes. But you aren’t supposed to draw a picture when you add fractions.
J: You can if you want to, but we don’t have to. Let’s write how much John, Paul, and Ringo have all together.
S: That’s what I’ve been trying to do!
J: So, do it!
S: 1/3 + 1/5 + 1/7
J: These have different denominators. We need to give them-
S: COMMON DENOMINATOR! COMMON DENOMINATOR! I remember how to do that! Wait. Too bad their only common factor is 1. But, multiples! So, 7 · 2 = 14; no; 7 · 3 = 21; no; oh! I remembered 35, which is 5 · 7, and if I multiply 35 by 3… 35 · 3 = 105. I got the least common multiple!
J: Yes! Correct! So, how many 105ths of a pie does John get?
S: However many times 3 goes into 105?
J: Correct.
S: Then it’s 35/105, or 1/3. We’re back to where we started.
J: No, no, no! Keep the 35/105!
S: This is not Beatle-y enough. We have to find a Beatle-y way to actually DO the math.
J: Can’t we sneak that in at the end?
S: Oh, fine. I just want this to be over as soon as possible… BECAUSE I’M NOT HAVING FUN!!!
J: So, how much pie does Paul get?
S: 3 · 7 · 1 = 21, so 21/105.
J: Yes. And-
S: How much pie does Ringo get? 3 · 5 · 1 = 15, so 15/105. And then we add up the numerators! 15 + 21 + 35 = 71, so, together, John, Paul, and Ringo get 71/105 of the pie. What flavor is it?
J: It’s pumpkin.
S: If I knew the Beatles’ pie preferences, I could tell you if they’d actually eat it or not, but I don’t because detailed information like that is SO HARD TO FIND just searching Google!!!
J: It’s for a photo shoot.
S: ANOTHER one?
J: Well, how much pie does George get?
S: George gets 34/105. Less than John by 1/105! There was no “we’ll see”!!! I WAS RIGHT!!!
J: But wait! John gives 1/11 slice of pie to George!
S: Awwwwww…
J: Do you think George has more now?
S: Yes.
J: Because…?
S: 1/105 and 1/11 have the same numerator, and 11 is a smaller number than 105, meaning that it is a bigger fraction. Wait! I don’t understand if the 1/11 is 1/11 of the whole pie or 1/11 of John’s slice.
J: The whole pie.
S: Well, then what’s wrong with my reasoning?
J: So, now, this looks fine. Now, what if John gave George 1/11 of John’s slice of the pie?
S: Ringo is most likely to do that, not John.
J: What if John steps out of the room, and Ringo gives George 1/11 of John’s slice of pie?
S: The point is that Ringo is too nice to do that! Let’s just have George steal the 1/11.
J: Okay! 1/11 of John’s.
S: So, I should divide 35/105 by 11? I know how to divide fractions.
J: Well, we could write John’s share in terms of 35/105 · 11/11.
S: That’s what I was going to do! Except, what does multiplying by 1 do?
J: We’re writing it in terms of the least common multiple!
S: Which isn’t 11! 11 was just a number pulled from your head!
J: So, can you multiply that and find out what John’s share is in terms of those numbers that have 11 as a factor?
S: I understood none of that sentence.
J: Can you perform the mathematical operation indicated seven lines ago?
S: Sure! 35/105 · 11/11 = CALCULATOR. *gets calculator and types in equation* Okay. It equals 385/1155.
J: Ah! So, what’s 1/11 of that?
S: I know what to do! On my calculator: whatever fraction is equivalent to 0.030303.
J: I would prefer the answer as a fraction of two integers.
S: Well, I forgot how to convert decimals to fractions, so poo-poo!
J: Well, then let’s go back to this step [five lines ago]. So, in the numerator, we have 35 · 11. What’s 1/11 of 35 · 11?
S: Oh. Two minutes left. And you haven’t done the Beatle-y twist!
J: Well-
S: Ooo! “Twist and Shout”! The Beatle-y twist!
J: Well, what’s 1/11 of 385 · 11?
S: Sorry. Bad reference.
J: Yes, at the end, they will dance to “Twist and Shout” while eating pie.
S: Was that your original plan?
J: No. But, can you write 35 · 11 / 11?
S: Yes. It is 35.
J: Yes. And so, you just divided by 11, but you made it so easy.
S: Now tell me what your original plan was.
J: To solve an algebraic equation of four variables: John, Paul, George, and Ringo.
S: No! I mean for the Beatle-y twist at the end!
J: I thought it was Beatle-y enough and that it would be done on one page!
Beatles Math 