Adding Mixed Numbers

Adding mixed numbers. Suppose we want to add two mixed numbers, like two and one third plus three and one fifth. We can do this by separating the whole number parts and the fractional parts and adding them separately. So we can take the two and the three and group them here. Then take one third and one fifth and group them here. Then we add them all up. Two plus three is five, so that part's easy. But now we have to deal with the fractions. For a moment, let's just look at the fractional part of the problem. If these fractions had the same denominators, then we could just add the numerators. But our luck isn't running that way today. We've got unlike denominators, a three and a five. So how can we make our denominators more likable? Let's start out by doing something unexpected. Let's multiply each of our fractions by one. Now, multiplying something by one doesn't change anything. And that's the point here. Let's look at the denominators. We need these denominators to be the same so that we are able to add the numerators. We need to make the change without changing the actual values of our two original fractions. To do this, we're going to take those ones that we're multiplying our original fractions by and convert them to fractions that are still equal to one. That way, when we multiply them by our original fractions, we get new fractions that have the same denominator. Since we are just multiplying our original fractions by ones and disguise, we won't be changing the values of the original fractions. The easiest way to do this is to multiply the denominators of the original fractions. Three times five is 15. 15 is going to be our new common denominator. Okay, we know where we're going, but how do we get there? We'll take those ones and give them a secret identity. Any number over itself, as long as it's not zero is equal to one. If I cut a pie into five pieces and then give someone five of those slices, I've actually given them the whole pie or one pie. So 5/5 is equal to one, and just like that, we can make the other one into 3/3. So while the ones are being transformed into fractions, we aren't changing their values. We can use these new transformed ones to transform our denominators. Now let's look at our original fractions. We're multiplying one third by 5/5 and one fifth over 3/3. When we multiply the denominators, we get 15. Yay, common denominators accomplished, but we can't stop there. Since we multiply the denominators, we need to multiply the numerators, as well. To get 15 in the denominator, we need to multiply three by five. If we multiply the denominator by five, we need to also multiply the numerator by five. This gives us 5/15. If we do this for our other original fraction, we multiply the numerator by three, and this gives us 3/15, 5/15 plus 3/15. Our problem has now gotten much simpler with our new and much more likable denominators. Adding these is a piece of cake. We add our numerators five and three. Now we have 8/15. If we go back to our original problem, substitute our new and improved fractions, separate the whole number parts from the fractions, add everything up, and we get five plus 8/15, which is 5 8/15. Here's a general template for how to add mixed numbers. Be aware that sometimes the fractional part that you get may not be in the simplest form.