Help me understand fractions

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 Tada! 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.>>>

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.

http://www.mathsisfun.com/numbers/ratio.html 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 ;)

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).

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.

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.

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.

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!

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.