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Addition and subtraction of fractions

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Regina rode her bike 2 and 1/4 miles from her house to school and then 1 and 5/8 miles to her friend's house. How many miles did Regina ride in total? So she first rode 2 and 1/4 miles, and then she rode 1 and 5/8 miles. So the sum is the total number of miles she rode. So to take this sum, we've seen that we can add the whole number parts, because this is really the same thing as 2 plus 1/4 plus 1 plus 5/8, so we can just switch the order, if you want to view it that way. So we can add the 2 plus the 1 first, and then we get-- let me do that here. So 2 plus the 1, you get 3, and then we need to add the 1/4 plus 5/8. And to add these two fractions, we have to find the least common multiple of 4 and 8. That'll be our new denominator. 8 is divisible by both 8 and 4, so that is the least common multiple of 4 and 8, so our common denominator will be 8. Obviously, 5/8 will still be 5/8. Now to go from a denominator of 4 to 8, you have to multiply the denominator by 2, so we also need to multiply the numerator by 2, so 1 times 2 is 2. And, of course, we still have this 3 out there. So 2 and 1/4 plus 1 and 5/8 is the same thing as this right here, and this is equal to-- we have our 3 plus, and then over 8 we add the 2 plus 5. We have 7/8. So this is going to be equal to 3 and 7/8 miles. She rode a total of 3 and 7/8 miles. Now, I want to make one thing very clear. So far when we've been adding these mixed numbers, the fraction part always ended up as a proper fraction. The numerator was smaller than the denominator. But I want to do a quick example to show you what you do when the numerator is not smaller than the denominator. So let's say we had 1 and 5/8 plus 2 and 4/8. So if you add just the whole number parts, 1 plus 2, you get 3. Plus 5/8 plus 4/8, 5/8 plus 4/8 is 9/8, so you get 3 plus 9/8. Now it would be really strange to just say, OK, that's the same thing as 3 and 9/8, because you have a mixed number with a whole number and an improper fraction. If you're going through the trouble of making it a mixed number, the fraction better be a proper fraction. So what you need to do is rewrite 9/8, and you know that 9/8 is the same thing as 1 and 1/8, right? 8 goes into 9 one time with 1 left over, so it's 1 and 1/8. So this is the same thing as 3 plus 1 and 1/8. So now we can add the whole number parts. 3 plus 1 is equal to 4, and then you have your 1/8 over there: 4 and 1/8. I just wanted to give you that special circumstance when your fraction part ends up improper.