# Math things I should have realized earlier

So just this morning I had this realization. To divide a fraction, you can just divide the numerator by the divisor and leave the denominator alone.

So, 8/15 divided by 2 is 4/15.

All my life, I’ve been saying, OK, 8/15 divided by 2, lessee, that’s the same as 8/15 x 1/2. So 8 x 1 is 8; 2 x 15 is 30. So that gives you 8/30. Almost done! I can reduce that to 4/15. Whew! I think I got it.

:smack:

PS: One I always liked is my grandfather taught me this trick. If you have a roll of wire, you can estimate the length by measuring the diameter, and multiplying by 3, and multiplying that by the number of coils you have. He wasn’t sure why it worked, but was happy it did!

That works, as long as you’re dividing by a number that goes into the numerator evenly, of course . . . otherwise, you have to do it the “long” way that you indicated.

A long time ago I realized that multiplying by five is the same as taking half the amount and moving the decimal over, you know, like multiplying by .5.

So 5 times, 360 becomes 1,800 and there’s almost no though to it. I estimate other calculations similarly. 3 times 175 is like a third and then move the decimal to get 600. It’s useful for estimating.

Somehow, in calc 101 I missed the fact that solving an integral is just calculating the area under the curve defined by the function. So I never understood why we would want to bother, or indeed that in principle, the way you calculate it is actually pretty intuitive: approximate it with a bunch of rectangles that fit under the curve and then keep making the width of those rectangles smaller and smaller until they are infinitesimal and thus giving you the true area under the curve.

I went through all of my courses just blindly calculating these things with memorized techniques instead of having this fundamental understanding. I think it really held me back from being better at math.

My wife doesn’t really understand that fractions and decimals are two ways of expressing the same number, i.e., that 1/4 is the same thing as .25, and that to get the decimal equivalent of a fraction, you just divide the bottom(right) number into the top(left) number (Don’t even bother using “numerator” and “denominator”, that just confuses the issue, okthxbai!)

My daughter used to be amazed at my ability to multiply by “9” in my head… until I told her the secrets of the ancients, that is. (multiply by 10, subtract the original value. 47*9=(470-47)=423, for example.)

“Per” means “divided by.”
“cent” is an abbreviation for “one hundred.”
“of” means “times.”

So 17 per cent of 875 is 17 divided by 100, times 875 which equals 148.75

This might be more intuitive if you think of denominators as being like denominations (as of money). For example, 8 five-dollar bills divided by 2 (people) is 4 five-dollar bills (for each person).

This might also help to explain why you need a common denominator when adding or subtracting.
What’s 8 five-dollar bills plus 3 five-dollar bills? 11 five-dollar bills.
What’s 9 ten-dollar bills minus 4 ten-dollar bills? 5 ten-dollar bills.
What’s 3 ten-dollar bills plus 7 five-dollar bills? 10 fifteen-dollar bills? Wrong! The 3 ten-dollar bills would be equivalent to 6 five-dollar bills; and 6 five-dollar bills plus 7 five-dollar bills equals 13 five-dollar bills.

Pie is really good to illustrate fractions.

a pie is precut into 5 slices for a party. the 5 people unexpectedly arrive with two friends.

how do you serve equal portions of pie? you cut each slice in half. creating 10 slices and you serve 7. 3 slices of the original 10 remains.

most people can relate to that better than just writing fractions on a paper. 1/4=2/8=4/16=8/32=16/64 you just keep cutting that slice of pie in half

Congratulations to him for discovering pi.

I hate to ruin the magic, but you can find circumference by Pi times diameter. That’s why it works.

*robert_columbia beat me to it.

NM

Actually that is a very clever way to estimate wire length. Geometry is fine but seeing a very easy used practical example is quite interesting.

I’ll give it a try with water hose.

In a similar vein, I didn’t understand until long after high school that this is related to the fact that if the sum of the digits is divisible by 9, then the entire number will be as well.

Or that if you consider that for any number, say 50761, the numeral is shorthand for a polynomial, with x=10; in this case,

`````` 5x[sup]5[/sup] + 7x[sup]3[/sup] + 6x[sup]2[/sup]+1
``````

Similarly, that you can generalize this principle to any value of x, such that if you show that the sum of the coefficients is divisible by (x - 1), then the entire expression will be.

I might have been able to learn this in HS if only I could have handled the basic algebraic abstractions needed.

This is probably trivial but dividing by 5 is kinda like multiplying by 2.

Quick. What is \$415 split five ways?

Take 415 and double it to 830.

Actually I think its a little more complicated than that. I think you need the area formula rather than the circumference formula
The area covered by the coil’s cross section = Pi * R^2 = (length of tape)*(width of tape)

but width of tape is equal to the radius divided n =number of coils, so you get

Pi * R^2 = (length of tape)* R/n, so with a little algebra

Length of tape = PiRn
I figured this out once on my own and used to ask this to my fellow math students worded in this way,

“If you have a piece of tape that was 1 km long and 0.1 mm thick how many wraps would you need to coil it up.” Most would try to do some calculation summing the series of wrap lengths, but the geometric calculation was much easier.

Dude, I said HE was unaware why it worked, not me. Your teacher must not have used Bikini Calculus. Somewhat NSFW Youtube video. Integral calculus starts at 1:40.Intro to Bikini Calculus - YouTube

I have tended to explain things like this with ‘banana algebra’ which is apparently easier than ‘letters algebra’.

8/15 is eight ‘fifteenths’. A fifteenth is a ‘banana’. What’s half of eight bananas? Four bananas. So, four fifteenths.

But start calling a banana ‘x’ and everybody totally loses their shit.

ETA I concur with aceplace, that pies are a great way to teach manipulation of fractions. Long live food algebra.