Krdytkyu

As I was looking through Spotify, I noticed that I listened to 2040 minutes of music this year. I did as follows: 2040=204*10, and 60=6*10. Thus 2040/60= 204/6. Intuitively, I'm not able to see why both of these actually equal each other. I mean, the closest I'm able to get to an answer is to assume that this division problem is a ratio, and that I'm trying to find how many minutes listened to 1 "real" minute. However, this makes no physical sense. Can someone please provide some intuition?

Thanks

Here is a little background before a rather involved answer.

The best question to ask in approaching this problem is "What is division, really?" Not very intuitive, but the short answer is that its the opposite (or, inverse operation) of multiplication. I ask, "What is $2040$ diviced by $60$ equal to?" And in mathematical notation, the answer to that question can be given a name: $x$. This translates to the familiar

$dfrac{2040}{60} = x = 34$.

The rules of math give us an equivalent question to ask, "What number do I multiply by $60$ to obtain $2040$?" The answer to this one is exactly the number of piles of $60$ objects required to have $2040$ total objects. That number is $x$!

If you like to organize your piles into rows, say, rows of length $60$ all stacked like pancakes, then $x$ is the number of rows, or the height of the piles. Your question is why (in a physical kind of way) can I first cancel factors from the numerator and denominator?

Here is my answer. I intentionally did not state "piles of what objects," because what if I had $204$ dimes, and I broke them into piles of $6$ dimes each? I would have $x$ piles of dimes when I'm finished. But what if I replace the dimes with sets of $10$ pennies? there would still be $x$ piles, I started with the same amount of money, but each pile now has $60$ coins in it!

The more abstract answer to your question is that the physical equivalent to canceling factors in both the numerator and denominator of a ratio is like viewing the numerator as a smaller collection of bigger objects, and breaking it into smaller piles of these bigger objects. In this case, "Bigger" means "each object contains the common factor number of the original objects."

In your example, viewing $dfrac{2040}{60}$ as $dfrac{204}{6}$ is like taking $2040$ minutes and breaking them into $34$ piles of $60$ minutes each, or taking $204$ ten minute blocks, and breaking all $204$ of them into $34$ piles of $6$ ten minute blocks. This is the same as taking $34$ sixty minute blocks and breaking them into piles of one hour, which gives one block per pile as desired.

How many sets of 6 go into 204? Ten times as many go into 2040. You need 10 groups of 6 to have one group of 60. We calculated the group's of 6 from the previous step so just divide by 10 to get the group's of 60.

To get how many times 60 goes into 2040 requires finding how many times 6 goes into 204, times 10, then divided by 10.

More generally:

Let us have ratio a/b=n, or how many sets of b go into sets of a. Then c times n is how many sets of b go into sets of ca. We know how many times a set of size b goes into ac, nc. How many times does a set of size c times bigger than b go into nc? It has to go in c times fewer. So we divide nc by c to get n.