Help me understand fractions
June 19, 2013 4:34 PM   Subscribe

Dear math people: please help me understand fractions and ratios intuitively.

I feel really embarrassed about this, but I must have missed a fundamental period when I was learning fractions as a child because I have a really hard time with understanding them intuitively. I am hoping someone can help me to see where my understanding breaks down and how I can make it more intuitive.

I know what fractions are and I fully understand addition and subtraction of fractions. I can also convert fractions (thought I don't remember the term for this). I think this is because I like to bake so when I think of adding or subtracting fractions, I always imagine cups of flour or something. So 2 1/2 + 1/3 makes perfect sense to me because I can picture it. I know that I need to find a common measurement (denominator) and the easiest way (but not the only way) to do that is to multiply both denominators. So, I would convert so I have this:

5/2 + 1/3

make them have common denominators:

15/6 + 2/6 = 17/6 = 2 5/6

That part is easy for me! Makes total logical, intuitive sense.

Multiplying and dividing fractions make no sense (maybe because I never have to do it in real life??). So I have to rely on remembering the "rules" (which I can never remember). I guess (because I looked it up) that to multiply you just multiply both numerators and then multiply both denominators. But to divide you have to flip one of the fractions and then multiply across. Anyway, I get that the rules are simple but this doesn't make logical sense to me (like why you would flip the fraction).

So to bring this into real life matters where it is really bothering me is something like the P/E ratio. Here are the "rules" I know about it:

- market value of share divided by earnings per share
- a higher P/E means people think the company has better potential for future earnings growth
- PE ratios are only comparable to similar companies

What I don't understand is the logic. So if someone asks me - "what happens to the PE ratio if a company's earnings per share falls?" I see this as what happens to the fraction itself if either the numerator or denominator or both rises or falls. The ONLY way I can figure this out is literally by trial and error using normal numbers. This is literally what I do:

20/5 = 4
20/4 = 5

So if the numerator falls, the fraction solution rises. So if the above were a PE ratio, the PE ratio would go up, which means the market thinks the company has better future growth prospects than it did before (extrapolating, if the EPS has gone down but the share price stays the same, the market thinks the lower EPS is temporary, but still has faith in the company because the share price hasn't changed and the EPS doesn't fully reflect the value of the company or whatever. THIS PART DOESN'T REALLY MATTER FOR THE PURPOSES OF MY QUESTION).

Another example would be financial/balance sheet ratios where understanding intuitively how fractions work and what it means when the numerator or denominator changes is super important to understand the balance sheet and the company. So if I wanted to look at net profit margin, the ratio is:

Net income/revenue

And if someone asked what decreased revenue means for a company's net profit margin, I would like a way to understand this intuitively through my understanding of fractions (and not my understanding of how companies and economies work) without having to use my manual trial and error divisions of twenty that I used above, Does that make sense? To make it perfectly clear:

Variable A/Variable B = Bigger Picture XYZ

What are the implications for Bigger Picture XYZ if Variable A changes?

PLEASE HELP. Where is my understanding of fractions breaking down???
posted by anonymous to Education (29 answers total) 9 users marked this as a favorite
What about if you thought about baking for multiplying/dividing fractions?

So, you have a recipe that calls for 2 1/2 cups flour. You want to make a half recipe - so you need to divide in half.

So, you need 1 1/4 cups of flour - half of the two cups is one cup, and half of the half cup is a quarter cup.

I'm not sure if this helps at all.
posted by insectosaurus at 4:45 PM on June 19, 2013

Math is fun is geared towards children. It has nice graphics and easy to understand explanations. Don't laugh, but I will use this website when I can't remember how to figure out something mathematically. Class of '79 ;)
posted by JujuB at 4:49 PM on June 19, 2013 [2 favorites]

So if the numerator falls, the fraction solution rises

Denominator. The numerator is on top.

Studying the basic rules of algebra might help by giving names to the rules you've memorized but don't fully understand. I.e., you're not really "flipping the fraction", you're multiplying both sides of the equation by the fraction's multiplicative inverse and then distributing it. Another link to Math is Fun, this time for laws of algebra, and for the multiplicative inverse.

But really I'd just advocate converting to decimal notation most of the time. Especially for the baking. It's hard for anyone to intuitively process mixed-denominator fractional expressions; that's why we spend so much time trying to convert everything to the same denominator, but why bother with all of that if you have a calculator to do it for you? (Sure you lose some precision, but in your real applications it's probably not significant)
posted by qxntpqbbbqxl at 5:00 PM on June 19, 2013

Yeah, I agree that you can use baking examples for multiplying fractions. Let's say you're making pizza. The recipe calls for 2 1/2 c. flour. but you want to make 3 pizzas, because you have friends coming over. And you want to substitute half the flour with whole wheat flour, to make it healthier. So how much white flour do you need? It's the same as multiplying by 3/2 (or multiply by 3, divide by 2).

5/2 * 3/2 = 15/4 or 3 3/4 cups of white flour.

Your friend likes the pizza so much, you give her the recipe. But she doesn't have any whole wheat flour, so she's just going to use white flour. And she only wants one pizza, because it's just for her. How much flour should she use? She needs to reconstruct your original recipe. So, she needs to multiply by 2 and divide by 3 (= multiply by 2/3 = divide by 3/2...which is what you multiplied by).

15/4 / 3/2 = 5/2


The other general concept I would think about is direct and inverse relationships. I used to teach middle school science, and this comes up a lot. For example, force = mass * acceleration. This is Newton's second law, and it makes intuitive sense. If you push a ball twice as hard, it will accelerate twice as much. If you push two balls the same, but one is twice as big as the other, the big one will accelerate half as much as the small one.

So, force and acceleration have a direct relationship. They change together. If the mass is the same, then as force goes up, acceleration goes up. As force goes down, acceleration goes down.

But mass and acceleration have an inverse relationship. They change in opposite directions. If the force stays the same, then as the mass goes up, the acceleration goes down (the bigger the mass, the less it accelerates with the same push). As mass goes down, the acceleration goes up (the smaller the mass, the more it accelerates with the same push).

You can think of these relationships with fractions.
f = ma <> f/1 = ma/1
m = f/a <> m/1 = f/a
a = f/m <> a/1 = f/n

These are three ways of writing the same thing. If you pick any one of the letters f, m, or a, and hold it constant, then you can ask how the other two change in response to each other.

If two letters are both on the top of fractions, then they have a direct relationship. As one goes up, the other goes up. Force has a direct relationship both to mass and to acceleration.

If one letter is on the top of a fraction and the other is one the bottom, they have an inverse relationship. As one goes up, the other goes down. Mass is inversely related to acceleration.
posted by pompelmo at 5:05 PM on June 19, 2013 [3 favorites]

Variable A/Variable B = Bigger Picture XYZ
What are the implications for Bigger Picture XYZ if Variable A changes?

If the numerator increases, the value increases. However, this not what you are explaining by "Net income/revenue" and then saying that the revenue increases because revenue is the denominator and not the numerator. On preview, this is a problem with your first example as well. It can help to think "denominator = down".

It doesn't matter what the name is though. When the top number increases, the value increases. When the bottom number increases, the value decreases. You illustrated that well with 20/4 vs 20/5. Now think 40/4 and 40/5.
posted by soelo at 5:05 PM on June 19, 2013 [1 favorite]

I'm not actually sure you're doing anything wrong beyond not trusting yourself.

If you asked me the net profit margin question, I'd think "The denominator's smaller, so the whole thing gets bigger." But there's a tiny part of my brain going "Gets smaller is like going from dividing by 2 to dividing by 1." right where the comma was in that sentence.

You quite possibly learned about fractions by analogy with pie or pizza. All these numerator or denominator gets bigger or smaller questions are doing are changing either how many slices of pizza there are or how many people are eating pizza. More pizza for same number of pizza has to mean more pizza per person, and same amount of pizza for more people means less pizza per person.

You totally know how to do that computation and do it all the time when baking (well maybe, my example is really contrived). Say you have a recipe that makes 9 muffins (yeah WTF?) and, of course, you have a muffin pan that makes 12 muffins. You have to decide whether to adjust the recipe (and multiply by 4/3--which I'm sure you can do) or decide that it'll be okay to stretch the batter between 12 spaces instead of 9. And you'll think "Well, they'll be kind of small muffins and that might mess up the baking time." because you know that the bigger denominator (the muffin spaces) means smaller muffins because batter/muffin decreased.

Anyway, I get that the rules are simple but this doesn't make logical sense to me (like why you would flip the fraction).

Maybe it makes more sense if we break up the fractions, at least for multiplication. Say you're doing 2/3*1/4. Well, we know that 2/3 is twice 1/3, so we could multiple by 1/3 and then by 2 and get the same thing. You kind of have to just believe me that it makes sense that 1/3*1/4=1/12, but then multiply by 2 gets you 2/12, which is what you'd get if multiplied them using the rules and not thinking about it.

For division, you want to think about division as multiplication by the multiplicative inverse. The multiplicative inverse of some number x is the thing (call it a) that you multiply x by to get 1 (so xa=1=ax). And what's this? It's 1/x. This is really clear if you think about integers--if I have eight widgets and need to get one, I need to take 1/8th of my widgets.

But let's try 7/8. We want (7/8)*(8/7)=1 and that's what'll happen if you follow the rules. If we multiply 7/8 by 8, we're going to get 7 (you can think of it as having 1/8+1/8+1/8+1/8+1/8+1/8+1/8 (seven times) and taking that eight times, so you'd have 1/8 added together 56 times, which we can, of course, regroup into 7 copies of 1/8 added to together 8 times (i.e. 1)), but then we've got 7 and not one, so we have to divide by 7, i.e. multiply by 1/7.

So I'm trying to convince you that if we have (1/4)/(3/7), it makes sense to think of it as (1/4)*(7/3), which (if you believed the multiplication paragraph) is 'naturally' 7/12.
posted by hoyland at 5:08 PM on June 19, 2013

When we learn fractions in elementary school, it's raw mechanics. They are squiggles on a page. We learn this rule of flip to divide but we're not taught why. Now in the real world, you're seeing ratios, which are mechanically the same as fractions, but conceptually they represent real physical things like revenue. Math (mostly) doesn't exist for its own sake; it's a tool for understanding the world. It's a shame that math education often fails to instill these connections.

With ratios we are usually interested in understanding how one number compares against another. Let's take price and earnings as an example. To understand how well a company is doing, you have to simultaneously understand the price of their stock, and their earnings per share, or however that works. These are two numbers, let's say price is 20 and earnings is 2. Let's say another company has a price of 20 and an earnings of 10. We know a good price is beneficial, but what's a "good" price for a stock? The number 20 on its own doesn't give us enough information to tell us whether the stock is valuable or not. So we have to think about price and earnings at the same time.

When we take the price/earnings ratio, we're comparing two numbers. You can read this as "price is how many times bigger than earnings"? So for 20/2 = 10, price is 10 times bigger, and the P/E ratio is 10. For 20/10 = 2, price is twice as big, and the P/E ratio is 2. By dividing one by the other, we get an answer that expresses the relative size of the two numbers. This is useful, because often in the real world we care about the relationship between two numbers rather than the values of each of them.

If you read "X = A/B" as "A is X times bigger than B" then it answers questions about how X changes with respect to changes in A or B. If B gets larger, and A stays the same size, then X has to shrink -- A is less bigger than B than it used to be, and X tells us "how much bigger is A relative to B".

By the way that stuff you did with the 20/5 and 20/4 is how I built an intuitive understanding of fractions when I was learning them as a kid and it's how I out more complicated things today. I test them and try to figure out the rules of how they operate, over and over again, until they make sense. The best way to demystify is to practice.

By the way, when you're talking about ratios, a ratio of one is the magic number. Calculus professors call it "unity", because it means the two numbers are equal to each other. If it's smaller than one, it means the number on the bottom is bigger. If it's bigger than one, it means the number on the top is bigger. Even basic information like "which number is bigger than the other" is incredibly valuable when measuring the real world. Say, for example, you took the ratio of two currency prices, and plotted how that ratio changed over time -- if you identify the times when the ratio dips above or below one, you know when one currency became more valuable than the other.
posted by PercussivePaul at 5:11 PM on June 19, 2013 [3 favorites]

Have you tried, "The bottom number is how many pieces a whole thing is broken down into, and the top number is how many of those pieces you have?"
posted by La Cieca at 5:12 PM on June 19, 2013 [4 favorites]

So it actually looks like you do pretty much get fractions. Your only issue is that you're not quite comfortable yet with generalizing what you know.

A fraction is just a division problem. The fraction A/B just says "take A and divide it by B." You can convert from a fraction to a decimal by simply doing the long division (I'm assuming you remember how to do long division), or just using a calculator.

Your examples of 20/5 and 20/4 are actually perfect. You totally get that a fraction is a division problem! You just need more practice to prove to yourself that the general principle holds, regardless of what the specific numbers in the numerator or denominator are.

Making graphs might also help you -- it will visually show how Bigger Picture XYZ changes as A and B change.

Take the fraction A/B. Set a value for A -- say, A = 5. Write down a list of values for B -- say, B = 1, B = 2, ..., B = 10. Then for each B value, you can do the division problem A/B = C and figure out what C is. Write the C values down next to their corresponding B values. Then graph C vs. B (B on the horizontal axis, C on the vertical axis) however you're comfortable -- you can use Excel, for example. (I'm assuming you're comfortable with graphing and understand the concept. If not, here's a decent explanation.)

Now, repeat for a different value of A, say A = 3, using the same list of B values. Call this fraction A2/B = C2, and graph C2 vs. B in a different color.

Do this as many times as you like with different values of A. You'll start to see the pattern in what happens when B changes while A stays the same, or when A changes while B stays the same.
posted by snowmentality at 5:13 PM on June 19, 2013

You can think of a fraction like 3/4 as three apples divided among four people. Each gets 3/4 of an apple.

Now if you the increase the number of people from 4 to 5, you expect each person's share to get smaller, 3/5.

If you decrease the number of people from 4 to 3, you expect each person's share to get bigger, 3/3 = 1.

In other words, if you divide by a smaller number, each share gets bigger. If you divide by a larger number, each share get smaller.
posted by JackFlash at 5:14 PM on June 19, 2013

But really I'd just advocate converting to decimal notation most of the time.

I'd vote the exact opposite. Being dependent on decimals is basically being dependent on a calculator. It's easy to remember decimals for things like 1/8 and 1/3, but that's about it. But people heavily reliant on calculators tend to not have a good grasp of the calculation they're doing, in the sense that it's much harder for them to see where they are. And then there are the typos (and the messing up parentheses on the calculator).
posted by hoyland at 5:18 PM on June 19, 2013 [2 favorites]

I actually have a little trouble with this, and my way out was learning geometry and algebra. It is really hard to visualize weird fractions. But when you internalize the rules of the road as far as how to handle them, you start to understand them without having to visualize.

We know a fraction is just another way to express division. 1/2 is 1 divided by 2, which is 1 half, or 0.5. Further, we know that whole numbers are just fractions with 1 as the denominator. 4 halves is 2 wholes, right? So 4/2 can be 2/1. So when you multiply 4 times three, it is also 4/1 times 3/1. Which is 4 times 3 over 1 times one, which is 12/1.

Now, you can do the same thing with any denominator. 4 divided by 3 is 4/3. Or, saying it out loud, four thirds. Which is the same as four times one third.

Another way to think about it is by thinking of the denominator as the types of unit being used to count the thing. It is, actually, right there in the name. It is a whole, denominated (or referred to) in parts of a whole. One tenth is the same thing as ten hundredths. One mile is 5280 feet. So one mile, denominated in feet, is 5280 5280ths.

Lastly, I would advise against converting to decimal unless it is super straightforward. The best kind of measuring is the kind you don't have to do at all. Learn the conversions. A tablespoon is 3 teaspoons. So instead of trying to figure out how to measure out 1.6666666 tablespoons, you will know that you just need 5 teaspoons.
posted by gjc at 5:24 PM on June 19, 2013

But to divide you have to flip one of the fractions and then multiply across.

Gonna make a stab at explaining this with some ASCII art!

Here's four out of four, or four fourths, or 4/4, or 1: [ x x x x ]
Here's three out of four, or three fourths, or 3/4: [ x x x ] x
Here's two out of four, or two fourths, or 2/4: [ x x ] x x

"Out of 4" means that 4 parts makes up 1 whole, so 4/4 = 1 whole (which we just write as "1").

Okay, so now say we want to take 1/2 out of 3/4. What does it mean to take something "out of" 3/4? Well, it's actually really similar to what we just did!

So here's three out of four, or 3/4: [ x x x ] x
And here was two out of four, or 2/4 = 1/2: ( x x ) x x
So if we select "two out of four" out of these "three out of four", it'll look like this: [( x x ) x ] x

Does this make sense? All this is saying is, you start with three out of four, and then out of that, you select two out of four.

So far, so good. But now, remember, we have a fraction that's "out of 3/4". So now, "3/4", not 4, makes up 1 whole.

So we're going to "zoom in" to that 3/4, so that it makes up our entire whole.

That just looks like this: ( x x ) x

And you might notice that now, we can actually write this differently. There are 3 x's, and we've selected 2 of them. That means what we're looking at is the exact same beast as 2/3.

And indeed, 1/2 ÷ 3/4 is 2/3! Bingo.


Here's another approach. Take a big piece of paper and make the left end 0 and the right end 1.
0          1
[          ]
Then mark a line for the fraction on top (numerator) and the fraction it's "out of" (on the bottom, denominator).
0     N  D  1
[     |  |  ]
Then just fold over the piece of paper that's to the right of the D line and look at where the numerator is.
0     N  D
[     |  ]
And now we pretend that D is our new 1.
0     N  1
[     |  ]
Voila, a line 2/3 down the middle of the page!

(Note: if N were bigger than D, then it would be beyond the fold, meaning it would be greater than 1.)


And finally, here's another way to treat this symbolically. First, make everything have the same denominator. So we'd be thinking about 2/4 ÷ 3/4.

Now just put it in simplest form. So you might remember that if you have (for example) 4 / 8, that's not simplest form, because that's (1 × 4) / (2 × 4) and the 4s cancel, giving you 1/2. If that still seems weird, look up simplifying fractions and you'll see what I mean here.

In the case of 2/4 ÷ 3/4, you can rewrite this as (2 × (1/4)) ÷ (3 × (1/4)). (If this seems weird, remember, 2 = (2 / 1), so 2/1 × 1/4 = (2 × 1) / (1 × 4) = 2 / 4. Same deal for the denominator.)

Now the (1/4)s cancel out, and you're left with 2/3.


Feel free to MeMail me if this doesn't make sense!
posted by en forme de poire at 5:55 PM on June 19, 2013

But to divide you have to flip one of the fractions and then multiply across. Anyway, I get that the rules are simple but this doesn't make logical sense to me (like why you would flip the fraction).

The easiest way to "get" this, at least from my perspective, is to use a half as an example.

If you want to halve something, there are two ways to think about it:

I need to divide this by two (÷2)
I need to multiply this by a half (×½)

So these two things are the same.

You can also think of it in these terms:

1/2 is the opposite of 2/1
÷ is the opposite of ×

Changing one of them would change the equation. Changing both allows you put the equation in a different form, without changing the actual meaning.

And once you've got that, the main reason why you would choose to do this is because it's much easier to apply the multiplication rules because you don't get tangled up with diagonal operations - numerator x numerator and denominator x denominator is much easier to keep organised.
posted by robcorr at 6:21 PM on June 19, 2013

This is the clear and explicit answer to your question of why:
posted by TestamentToGrace at 6:55 PM on June 19, 2013

Net income/revenue = XYZ

Think of it as :
(Net income) x (1/revenue) = XYZ

Two separate quantities. XYZ has a geometric relationship with net income and XYZ has an inverse relationship with revenue.
posted by cmcmcm at 7:01 PM on June 19, 2013

What I don't understand is the logic. So if someone asks me - "what happens to the PE ratio if a company's earnings per share falls?" I see this as what happens to the fraction itself if either the numerator or denominator or both rises or falls.

Another way to think about this is maybe just ignore the formal notions of fractions and ratios for a second.

If the earnings fall and the price stays the same, the ratio is higher. This is like if the number of chips in the bag dropped but the price stayed the same, the price/chips ratio is higher. You are paying the same amount for less stuff.

I bet in your normal life you think of stuff like this all the time without worrying about which side is the denominator and come up with the right answer. Huh if you buy a pizza before six you get the same pie but it's two dollars less. The price-pizza ratio is lower.

I think other people in this thread have a lot of good advice about this topic more generally but I think on some level you already know this and the words are throwing you off:

Variable A/Variable B = Bigger Picture XYZ

What are the implications for Bigger Picture XYZ if Variable A changes?

If you are the grocery store and it says six eggs for three dollars or 12 eggs for $5.75, you know which one is cheaper per egg without doing fractions right? What you are doing is still ratios. You are calculating the PE ratio except this time it's the Price-Egg ratio.

Also, there's nothing wrong with doing trials in your head, or trying to come to the same calculation through another route. It's a great way to check yourself and make sure you haven't made a mistake. Everyone makes mistakes all the time. If anything, even if you knew this backwards and forwards I'd say developing the habit of doing trial calculations is just a good idea.

One last thing:

- PE ratios are only comparable to similar companies

This has nothing to do with math. Don't let this trip you up. I mean, PE ratios are strictly speaking *not* only comparable between similar companies, but this is something people say for a variety of reasons. The short reason for this rule is someone will say like, "Well, this is a web startup who's only cost is headcount and it's growing by leaps and bounds" vs. "This is a railroad company that has huge amounts of capital sunk into railroad junk and they make a profit carrying freight for people". These companies are going to trade at very different multiples. It's not necessarily reasonable to say that the train company is cheaper than the startup by comparing their P/E ratios because their whole capital structure and everything will be totally different.
posted by jeb at 7:21 PM on June 19, 2013

Your question seems kind of muddled to me but a couple of things come to mind that helped my math-challenged son and hopefully will help you:

When he was little and struggling, I happened to have a pie at home. I pulled it out and handed him a butter knife. I told him to divide it into two halves. And divide it again, into four pieces. And divide it again into eight pieces. Seeing the pie divided into multiple smaller pieces helped him understand why the number gets "smaller" as the denominator gets bigger. (In other words, why 1/8 is less than 1/2.) That relationship suddenly made sense. Before that, he just did not get it.

Recently, we were talking about how women make 2/3 as much money on average as men, thus men make 1.5x as much. His brother said "Just flip it when you reverse the relationship." It suddenly made sense to him.

In other words: Women equal 2 berries. Men equal 3 berries. Women to men is 2/3. Men to women is 3/2. You are just making a comparison. You can make the comparison in either direction. Flipping it keeps the comparison valid. It just shows a relationship.

Women=2 compared to 3=Men

Flip the direction of comparison (women to men or men to women) flips the numbers in your ratio (2/3 or 3/2).

I hope that helps.
posted by Michele in California at 7:59 PM on June 19, 2013

I like this question for some reason!

A simple thing to remember would be this:
A fraction tells you how many times the denominator can fit inside a numerator.

So, an example:
For the P/E ratio, imagine that the Numerator = Market Price = a Car, and the Denominator = EPS = a Clown.

What happens to the P/E ratio if Market price increases? Well if the Car increases in size, MORE Clowns can fit inside. The Fraction increases (P/E ratio increases)

What happens to the P/E ratio if EPS increases? Well if the Car stays the same size, but the Clowns get fatter, FEWER Clowns can fit inside. So the Fraction decreases (P/E ratio decreases).

What happens to P/E if Market price decreases? If the Car gets smaller, FEWER Clowns can fit inside. So the Fraction decreases (P/E ratio decreases).

What happens to P/E ratio if EPS decreases? If the Car stays the same size, but the Clowns are getting skinnier, MORE clowns can fit inside. So the fraction increases (P/E increases).

Always remember the denominator trying to fit inside the numerator.
posted by watrlily at 8:02 PM on June 19, 2013 [1 favorite]

Not sure if this will help, but it all became clearer to me when I stopped saying "divided by" and started saying "fitted to". Like instead of twenty divided by five (mysterious to me) I would say 20 fitted to 5. Oh, it's four times bigger.

So you're fitting (comparing) the sizes of two things in a fraction. The top is "fitted to" the bottom.
posted by telstar at 8:02 PM on June 19, 2013 [1 favorite]

This might not help, but when I'm trying to use fraction intuition, I replace the "/" with the mathematically-equivalent "÷". In other words, don't think of 3/5 as a fraction, but as an operation "3 divided by 5". If you divide by a big number, you end up with a small number and vice-versa. If you multiply "3 divided by 5" by "7 divided by 4", you may be able to more-easily see that, when you split it up, it's simply (((3 * 7) divided by 5) divided by 4).

Like I said, it may not help, but that's the way I do it.
posted by originalname37 at 8:08 PM on June 19, 2013

I would suggest thinking about fractions and ratios as totally separate concepts that just happen to use the same math. Fractions make intuitive sense when you're thinking about one number: 3/4 cups of flour is just a quantity of flour. Ratios make intuitive sense when you're thinking about the relative sizes of two numbers: when you're baking a cake, you need a certain ratio of flour to sugar.

PE ratio is the ratio of price to earnings. What happens to the PE ratio when earnings go down? Well, now you're comparing the price to a smaller number, so the price is larger relative to that.
posted by Vampire Cat at 10:05 PM on June 19, 2013

Based on the two examples you gave as your goal:

First, don't worry about fractions. The mechanics of manipulating things that look like "p/q" is sometimes useful, but not here. You want to know about division. And it won't help you to worry about whether the number in the bottom (denominator) looks like "2", "5.92", or "8¾".

Additionly, thinking about fractions leads you to think about one single static math problem at a time. That's not what you want. You want to get a feeling for variables that are related to each other, and what happens when they change dynamically. [And those variables represent the quantities you are interested in.]

Echoing what pompelmo said, you want to be able to recognize when
  • two variable vary directly with each other
  • and when
  • two variable vary inversely with each other.
Varying directly means that when one goes up, the other goes up along with it.
Varying inversely means that when one goes up, the other goes down, always in the opposite direction, like the two ends of a teeter-totter.

To pick off whether it's a direct or inverse relationship, you math the equation around (imagine me waving my hands around here), and get the variable you're interested in on the left side by itself. Then, variables in the top (numerator) vary directly, and variables in the bottom (denominator) vary inversely.

Since you've thought about P/E Ratios a bunch already, let's try it on a new example: Gases. (This is great example, as people usually get a handle on this in chemistry or physics classes, not in math classes).

A sample of gas has a pressure (P), volume (V), and temperature (T). For the sample, think of a bicycle pump filled with air, and with the end plugged up so it can't escape. These three variables are related by the equation
PV = kT
where k is just some fixed number. You can just pretend it's equal to 6, or 4, or some convenient value. All we care is that it never changes, no matter what happens to the gas.

Now, suppose we care about the volume. We re-arrange the equation to say
V = kT/P
which tells us that (1) volume and temperature vary directly, and (2) volume and pressure vary inversely. Let's check it against our intuition.
  • If the temperature goes up, the gas expands and the volume goes up - that checks out.
  • If the pressure goes up, we smoosh it, and the volume goes down - that checks out too
There is a third possibility: suppose both temperature and pressure change? Well, maybe they both pull the same way. If T goes up, and P goes down, those both make V go up, so V really goes up.

But if they pull opposite ways, then it depends on which pulls harder. It's even possible that they perfectly counter-balance each other. Suppose we are told that the volume is fixed so it can't change, and the temperature goes up. Well, T increasing normally makes V increase (they vary directly with each other), so P has to counteract that by pulling V to decrease. Since P and V vary inversely, V decreasing goes along with P increasing --> the pressure goes up. Final story: T increases pulling V up, but P also increases, pulling V down by the same amount, leaving V unchanged.

Anyway, hope that gives some help. If I was tutoring you I'd have you play around with a graphing calculator a bit to get more of a feel for of it, but the key is to think of the variables as continuously varying quantities, and understand how they affect each other to always maintain the relationship that your equation expresses.
posted by benito.strauss at 10:29 PM on June 19, 2013

Some of the stuff on this website might help. Full disclosure: my website, but not my content. I suggest the activities page.
posted by b33j at 10:43 PM on June 19, 2013

Here's a way to think about multiplying fractions.

You have a 1/2 cup of flour (which means you have divided a cup into two sections, the top and the bottom, and one section, the bottom, is full of flour).

Now let's compare doubling that (x2) vs. halving that (x1/2).

If you double it, you now have two halves (2/2) instead of one half (1/2). So you doubled the number of halves you have. Two halves make one whole (2/2 = 1). The cup is full.

If instead you halve that 1/2 cup of flour, you are taking half of half. So you take that 1/2 cup of flour on the bottom of the cup and remove the top half of it. Now you have 1/2 of 1/2, which is 1/4.

So you can think of multiplying by a number greater than 1 as increasing the amount and multiplying by a number between 0 and 1 as decreasing the amount (since a number less than 1, like 1/2, is reducing something rather than increasing it).

Maybe the word "times" is not the best word for understanding multiplication. Instead maybe thinking of it as "how many" would be better. How many half cups?

1 half cup: 1 x 1/2 = 1/2
2 half cups: 2 x 1/2 = 2/2 = 1
A 1/2 half cup (or 1/2 of a half cup): 1/2 x 1/2 = 1/4
posted by Dansaman at 11:18 PM on June 19, 2013

Here's another way:

5/2 * 3/4 = ?

This is 5 times one-half of the stuff to the right.

So ... find one-half of the stuff to the right:

1/2 * 3/4 = 3/8.

Now we want 5 of that. 5 copies of three-eigths.

5 * ( 1/2 * 3/4 ) = 5 * (3/8) = 15/8.
posted by sebastienbailard at 11:29 PM on June 19, 2013

This is the way to understand division by fractions: how many episodes of Ren and Stimpy can you watch in 3 hours?

Well, Ren and Stimpy is a half-hour show, so we're trying to figure out what is 3/(1/2). You can watch 2 episodes per hour, so if you watch for 3 hours, that is 3x2=6 episodes.

But lo and behold! That's the same thing you would get by inverting and multiplying. i.e.

3/(1/2) = 3 x (2/1) = (3/1) x (2/1) = 6/1 = 6.

Now how many episodes can you watch in an hour and 45 minutes? Three episodes makes an hour and a half, and an extra 15 minutes is another half of an episode, so the total is 3+1/2=7/2 episodes.

If we invert and multiply, we get the same thing. Remember that 1 hour and 45 minutes, converted to fractions, is 1+3/4 = 7/4 hours. So, dividing by 1/2, we get

(7/4)/(1/2) = (7/4) x (2/1) = 14/4 = 7/2.

See? Easy!
posted by number9dream at 12:39 AM on June 20, 2013

There is a saying in mathematics that says, "math is not a spectator sport." Thus, I would encourage you to not only read these responses, but work through them and other fraction problems. I think it's easiest to start in concrete terms. Once you understand fraction multiplication and division in terms of say cookies and area, you will have the intuition to understand fractions and proportions when they describe more abstract situations. I also think it's easier to write out fractions like this: instead of like this: 1/2.

A very simple cookie example. You have 12 home made cookies. You want to give half of these cookies to your neighbor. How many cookies will you get? On your sheet of paper draw 12 cookies all in one row. Shade in 1/2 of them. Now there are 6 shaded and 6 unshaded cookies, right? In other words, you get 6 cookies.

One way to write this is is 12/2=6. You can see from your drawing that this is true. You also get 6 when you reduce or simplify 12/2.

Let's look at this problem another way. Instead of cookies we will be counting squares, or brownies if you will. We'll be using the area model as described in the above link. On a sheet of (graph) paper, draw a rectangle that is 1 square tall and 12 squares wide. Verify (by counting) that this rectangle has an area of 12. We also know that the area of this rectangle is 12 since 1*12=12. Now I want you to draw a horizontal line to divide the rectangle into two equal pieces. Shade in one of these two pieces. What is the area of the shaded region? If you count you should find that the shaded region is 6 units. This should make sense since we learned from the cookie diagram that 12/2=6. Now I want you to label your big rectangle with its dimensions. The width is 12, and the height is 1. I also want to label the shaded rectangle with its dimensions. The shaded rectangle is still 12 squares wide, but it is only 1/2 square tall. We know that the area of a rectangle is the width times the height, so twelve times one half.

One way to write this is 12 x .5=6. But 1/2 is the same as .5, right? In addition, any number divided by one, is the original number itself. (Think about this for a moment). All I'm saying is that 12/1 =12.

12 .... 1
--- x --- =6.
1 ..... 2

(The ... in the equation aboveis just keeping everything lined up for this post). Do you see how this is the same equation as earlier?

--- =6


12 x 1
-------- =6.
1 x 2

Now what would happen if you wanted to divide your 12 cookies (and brownies) among 3, 4, 6, and 12 people?
posted by oceano at 8:17 AM on June 20, 2013

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