Simplifying Fractions

In this video, we will learn how to simplify fractions by finding the creates common factor. Let's say you're making a cake and your recipe calls for some butter. Butter comes in sticks, but the recipe might specify the butter in tablespoons. Most butter is wrapped with paper that marks off the equivalent of a tablespoon. There are usually eight tablespoons marked off on a stick of butter. So a tablespoon is one eighth of a stick of butter. If your recipe says you need four tablespoons, that's four eighths of a stick of butter. Four is half of eight, so it's reasonable to say that the recipe calls for half a stick of butter. We can write that as a fraction, 1/2 or one half. Saying one half of a stick of butter is simpler and easier to understand than saying four eighths of a stick of butter. Lots of fractions can be simplified in the same way. Here, we have a stick of butter and half a stick of butter. We can slice the whole stick into 18 slices and the half stick into nine slices, making them 18 18th and 9/18, but it's still the same amount of butter. That is pretty easy to see when we're talking about a whole something or a half of something. But what if we're dealing with a fraction that is less easy to wrap our head around? Say seven 13th or 6/24, how can we find a simpler way of expressing these fractions? Or is there even a simpler way to say these fractions? To do this, we're going to have to look at the fraction and find the greatest common factor of our fractions top number and bottom number. Great, you say, but what the heck is a factor? That's a very important question. Factors, it turns out, are super useful in working with fractions. Let us digress for a moment and talk about factors. To start off our digression, let's look for some factors out in the wild. Here's a simple multiplication problem two times three equals six. In the language of math, the six, which we might think of as the answer is called the product. In this case, the product isn't something you buy at a store. Think of it as something more like the result of the multiplication we just did. And the two and the three, those, my friend, are a couple of factors of six. One way to think of factors is that they are the numbers that when multiplied, give a particular product. Two and three are factors of six because when you multiply them, you get six. And one and six when multiplied, also gives a product of six. So they are factors of six, too. Any number can be its own factor if multiplied by one. Okay. So now we know that one, two, three, and six are factors of six. Numbers that are bigger than six can't divide evenly into it. Let's look at another number, S eight. Let's go on an expedition looking for factors of eight. We'll shove our table here over to make some room and look at all the numbers 1-8. We'll fire up our calculators, and if any of those numbers can divide evenly into eight with no remainder, then we've caught ourselves a factor. So eight is the dividend, the number being divided. The potential factors are the divisors, the numbers being divided by. The quotients, the results of the division, we'll put underneath each potential factor. And below that, we'll put the remainders. Here are our results. One, two, four, and eight have remainders of zero and are in. The rest are out. Now we know what our factors are for eight. You can use this method for manually finding factors, and for small numbers, it can be very practical. But for larger numbers, well, I don't know about you, but I ain't got that kind of time. Calculators often have a function that will give you the factors of a number, as do most spreadsheets. Personally, I just ask Google, which can sometimes be very helpful that way. Your results may vary. So factors. Yeah. Where were we? Oh, simplifying fractions. Okay. Let's go back to our butter example. How do we get from four eighths to one half using the greatest common factor? We need to find the greatest common factor of the numbers in the numerator, the top number, and the denominator, the bottom number. In this case, that's four and eight. For four, the factors are one, two, and four. For eight, they are one, two, four, and eight. The greatest common factor of the numerator and the denominator is four. Now we divide the numerator and the denominator by four. Remember, if we do the same operation to both the top and the bottom of the fraction, the amount represented by the fraction remains the same. 4/4 is one. That's our numerator. 8/4 is two. That's our denominator. This shows that four eighths is the same as one half. So our procedure is one, find the largest factor of both the numerator and the denominator. Two, divide both the numerator and the denominator by this common factor. Three, replace the numerator and denominator with the answers you got in the fraction.