 So let's do some arithmetic with fractions, starting with the addition of fractions. So to begin with, here's an idea to keep in mind. Suppose you have three apples and add two oranges. What do you have? You have three apples and two oranges. The important idea here is you can only combine terms when they are of the same type. So let's consider this addition of two fractions, and remember how you speak influences how you think. This is not three over twelve plus four over twelve. It's three-twelfths and four-twelfths. Since we're adding, we're going to put them together. So all together we have three plus four seven-twelfths, which we write as a fraction. And this leads to an important result. If two fractions have the same denominator, the sum will be a fraction with the same denominator and a numerator equal to the sum of the numerators. So let's take that addition again, and we see that the fractions have the same denominator, so by our theorem we can add the numerators. And this gives us our final answer, seven-twelfths. What happens if our denominators are different? Again, how you speak influences how you think. We have three-fourths and two-thirds. If you read them this way, that reminds you that we have three of one type of thing and two of a different type of thing. So all together we do have three plus two equals five things, but they're not apples, they're not oranges, they're pieces of fruit, which is something different entirely. So remember, since the fourths and thirds are different, we can't combine them meaningfully. Now we can't meaningfully add two things together if they're of different types. However, we can transform fractions into equivalent fractions. So remember that given any fraction a over b and any non-zero number n, a over b is the same as na over nb. So this fraction three-fourth, I can transform into an equivalent fraction by multiplying numerator into denominator by two or multiplying numerator into denominator by three or multiplying numerator into denominator by four and so on. And we can do the same thing to the fraction two-thirds, multiplying numerator into denominator by two, by three, by four, or by anything we want. So if we look at our possibilities, we see that three-fourths is the same as nine-twelfths and two-thirds is the same as eight-twelfths and because they have the same denominator we can add the numerators and get our sum, seventeen-twelfths. What made this work is that twelve was a common denominator. In our fraction three-fourths, we can multiply the original denominator four by three to get twelve and in the fraction two-thirds we can multiply the original denominator three by four to get twelve. And this suggests a useful idea, a common denominator of the fractions A over B and C over D is BD, the product of the denominators. So, for example, we'll find a common denominator for one-sixth and three-tenths, then add and, if possible, reduce the fractions. So our theorem says that a common denominator of the fractions can be found by multiplying the two denominators together. So a common denominator is six times ten and so, if we take our fraction one-sixth, because we want a denominator of six times ten, we have a missing factor. We have a six, so we need a ten, so we'll multiply the denominator by ten and, because we need to make sure that we have an equivalent fraction, we'll also multiply the numerator by ten. Now, here's a useful idea. Because we eventually want to reduce and we can only reduce by removing common factors in numerator and denominator, it's worth leaving the denominator in factored form. So we'll leave our denominator as six times ten. What about the numerator? Well, because we're adding the two fractions, the numerator is going to be a sum, it will not be a product, so we might as well multiply the product ten times one gives us ten. How about this other fraction, three-tenths? Well, again, we want our denominator to be six times ten and we have a ten, but we're missing a factor of six, so we'll supply it, but we also have to multiply our numerator by six. Again, we'll leave our denominator in the factored form six times ten, but multiply out the numerator. And now, our fractions have the same denominator, six times ten, so we can add the numerators ten plus eighteen. So we'll get a fraction of the numerator twenty-eight and denominator six times ten, which is the same denominator we started with. Our arithmetic is bookkeeping, and it's always good to keep track of what we've actually done. This fraction we got by adding one-sixth and three-tenths, so let's write down that fact. And we do want to reduce, so here's where. Having that denominator in factored form is useful. Whether six or ten is a factor of twenty-eight and the answer is no, but because the numbers in our denominator are smaller, they're easier to factor. So six is two times three, ten is two times five, and both of these two are also factors of twenty-eight. So we'll factor twenty-eight, we can remove our common factors, and now at the end of the problem, we can multiply the remaining factors together. So our numerator only has seven, our denominator three times five, fifteen. Now sometimes when adding fractions, we talk about the lowest common denominator, and that is the least common multiple of the denominators, and it's sometimes useful to find it, but it's never necessary. And since life is hard enough, we're not going to waste our time finding lowest common denominators because all we need are common denominators. And so remember that we can always find a common denominator by multiplying the denominators together.