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!!!
Beatles Math 