Reapplying Adding Fractions

March 19th, 2017

S: I forgot how to add fractions.

J: 1/2 + 2/3

S: I’m going to see if I can solve that without any help…  1 is the greatest common factor of 2 and 3…  I have no idea what to do next.

J: It’s the least common multiple, actually.

S: So, 6.

J: Yes, but you didn’t necessarily have to find the least common multiple.

S: Oh yeah?

J: Just recall that anything can be multiplied by 1 and it’s the same number.  So, we can multiply the 1/2 term by 3/3, and it’s still 1/2.  And we can multiply the second term, 2/3, by 2/2, where the 2 is coming from the denominator of the first term.

S: Now I’ll try again.  1/2 + 2/3     1/2 · 3/3 = 3/6     2/3 · 2/2 = 4/6     There.  I knew the denominator would be 6!

J: Cool.  So, now add the two terms together.

S: 3/6 + 4/6 = 7/6, or 1 1/6.

J: Yes, that’s correct.

S: Now, tell me what the story problem behind that was.

J: Well, George eats 1/2 of a peanut butter sandwich before lunch and 2/3 of John’s leftover egg sandwich after lunch.  So, how much sandwich has George eaten?

S: I know!  1 and 1/6 sandwiches!  This time, we can start with a story problem, and I’ll try to solve it without help.  And without watching Help!.

J: Paul has written a melody in 3/4 time.  And John insists that it be scored with his melody that is in 7/3 time.  What will be the time signature of the resulting score?

S: Poor George Martin has to do the math.  If we weren’t specifically adding fractions, I would have no idea what the equation would be.

J: Well, guess what?  It’s not necessarily adding fractions.

S: But that’s the title!

J: I know.

S: I was just thinking that the problem does not make sense with adding.

J: You were?  Great!  How do you think it does make sense, then?

S: Well, first I want to say that time signatures can change in the middle of a song, so it doesn’t matter.  The problem has no meaning.

J: You just want to argue?  Or do you want to figure out what the answer is?

S: Well, you know that I love arguing, but, fine: THE ANSWER.  THE MEANINGLESS ANSWER.

J: Okay.

S: I shall solve it…as soon as I figure out what the equation is.  Multiplication?

J: The trick is to do the same conversion that you do when you’re adding fractions.

S: But, is the full equation multiplication?

J: Yes.  Yes it is.  Just multiply the two fractions and you’ll get your answer.

S: Good.  Multiplication is easy.  3/4 · 7/3 = 21/12, or 1 9/12, or 1 3/4.

J: Mm-hmm.  But the time signature would just be the fractional representation.

S: Wait!  Because this is Beatles math, I must say that “Revolution” just randomly came on the neighbors’ speaker system a few second ago!  Beautiful coincidence.  Beatles math + Beatles song that was not played on purpose.  Anyway, the time signature would be 21/12, if that was how music scoring worked, which it isn’t.

J: So, you could reduce that.

S: To 1 3/4.

J: Let’s do it in terms of a fraction with two integers.

S: So, 7/4.  Hey!  That uses the denominator of 3/4 and the numerator of 7/3!

J: It’s just an accident that it worked out that way.

S: Oh.  Anyway, imagine counting like 1-2-3-4-5-6-7, 1-2-3-4-5-6-7, 1-2-3-4-5-6-7!  Next problem!

J: So, 2/5 of the time, Ringo is practicing with all the Beatles, and 3/7 of the time, Ringo is practicing with either himself or only one or two of the other Beatles.

S: Hang on a minute!  Ringo never practiced by himself because he didn’t need practice because he was the best drummer in the world!

J: But he wasn’t even the best drummer in the Beatles!

S: The point is, he didn’t practice by himself, so the other half of the story problem should just be with one or two other Beatles, not just with himself.

J: But the point of this is to get you to add those two fractions together to find out how much of the time he is practicing.

S: Yes.  Okay.       2/5 + 3/7     2/5 · 7/7 = 14/35     3/7 · 5/5 = 15/35     So, 14/35 + 15/35 = 29/35.

J: Yes.  Correct!

S: So, this means that I can now add fractions without help.  Give me one more problem.

J: This is contentious, but just go with it, alright?

S: Okay.

J: John is 7/5 as good as Elvis, Paul is 8/3 as good as Elvis, George is 3/2 as good as Elvis, and Ringo is 5/2 as good as Elvis.

S: Where is this data coming from?

J: I said it was contentious, so you don’t get an answer.  So, how much better than Elvis is the average Beatle?

S: So, add them together and take the mean of the sum…  Ooh!  George and Ringo have common denominators, so I’ll start with them…  3/2 + 5/2 = 8/2, or 4.  Now for John and Paul.     7/5 + 8/3     7/5 · 3/3 = 21/15     8/3 · 5/5 = 40/155     21/15 + 40/15 = 61/15     4 + 61/15, or 4/1 + 61/15…     41 · 15/15 = 60/15     61/15 · 1/1 = 61/15     60/15 + 61/15 = 181/15     181/15 ÷ 4 (We divide by four because there were originally four fractions.) CALCULATOR…  On average, each Beatle is 3.016 times better than Elvis.

J: Yes.  Very good.

S: The end!!!

Adding Fractions

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!