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Adding and Subtracting Fractions

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[MUSIC PLAYING] Let's look at our objectives for today. We'll start by reviewing PEMDAS, the order of operations. We'll then look at adding and subtracting fractions with common or the same denominators. We'll see how to add and subtract fractions with uncommon or different denominators. We'll look at how to find the least common denominator. And finally, how to use order of operations with fractions.

Let's start by reviewing order of operations. PEMDAS is an acronym that we use to remember the order of operations, which is the order that we use operations when simplifying or solving problems, including problems involving fractions. PEMDAS stands for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction. It's important to remember that multiplication and division are done together from left to right, and addition and subtraction are also done together from left to right.

Here's an example of how we add or subtract fractions with common or the same denominators. The problem is 2/5 plus 1/5 is equal to 3/5. The 2, 1, and 3 are the numerators of the fractions, and the 5's are the denominators of the fractions. When we add or subtract fractions, we simply add or subtract the numerators together. Here we have 2 plus 1, which equals 3. The denominators stay the same.

Looking at a picture representation, our bar is split into five pieces, which matches the 5 in our denominator. The pieces are all the same size because the denominators are all the same size. We start with 2/5, or 2 out of 5 pieces shaded in, and we add 1/5, or 1 piece, which gives us a total of 3/5, or 3 out of 5 pieces shaded in.

We can subtract the two fractions in the same way. When you subtract fractions, we subtract the numerators. Here, 2 minus 1 is 1. And again, the denominators stay the same. Looking at the bar representation, we start with 2 out of 5 pieces, or 2/5 shaded in. And we subtract 1 piece, or 1/5, which leaves us with 1 piece, or 1/5 of the bar shaded in.

Here's an example of how to add or subtract fractions with uncommon denominators. We want to add 1/2 and 3/4. Looking at the picture representation of these two fractions, we see we have a problem. We can't simply combine the pieces together as we did in our last example, because the pieces are different sizes. However, if the denominators were the same, the pieces would be the same size, and we could add or subtract our numerators together.

We can make our two fractions to have common denominators by writing equivalent fractions. To find a common denominator, we start by multiplying our two denominators together. In this case, our common denominator will be 8, because 2 times 4 is 8. This method of finding a common denominator will always work, but it may not give you the least or smallest common denominator, and some simplification might be necessary at the end.

To get a denominator of 8 in our first fraction, we multiply the 2 in the denominator by 4. However, if we multiply by 4 in the denominator, we have to multiply by 4 in the numerator as well so that we do not change the value of the fraction. Instead, we now have an equivalent fraction 4/8. Again, we can multiply by 4 in the denominator and the numerator without changing the value of the fraction because 4/4 is the same as 1, and multiplying by 1 does not change the value of our fraction.

To get a denominator of 8 in our second fraction, we multiply the 4 in the denominator by 2. If we multiply by 2 in the denominator, we have to multiply by 2 in the numerator as well. Now we have another equivalent fraction, 6/8. Again, we can multiply by 2 in the denominator and numerator without changing the value of the fraction because 2/2 is the same as 1, and multiplying by 1 does not change the value of our fraction.

Now that our denominators are both 8, we can add our numerators together and leave our denominator the same, giving us an answer of 10/8. 10/8 can be simplified, or reduced, by dividing the numerator and the denominator by 2, giving us a final answer of 5/4.

Let's look at that last example again to see how we can find the least common denominator. Finding the least common denominator between factions can eliminate some common factors, making simplification easier. To find the least common denominator, we find the smallest number that is a multiple of the denominators of each fraction.

Let's look at the multiples of our first denominator, 2. The multiples of 2 are 2, 4, 6, 8, 10, 12, and so on. The multiples of our second denominator, 4, are 4, 8, 12, 16, 20, 24, and so on. Looking at both lists, the smallest common multiple is 4. Therefore, 4 is our least common denominator. So now we can rewrite our fractions using 4 as our common denominator.

To get the fraction 1/2 to have a denominator of 4, we multiply by 2 in the denominator and numerator. This gives us an equivalent fraction of 2/4. We notice that the second fraction, 3/4, already has a denominator of 4, so we can leave it unchanged. Now that our denominators are the same, we can add the numerators together and leave the denominators as 4, giving us a final answer of 5/4. And this is our final answer. Again, notice that this time we did not need to simplify our answer because we had used the least common denominator.

Let's look at our last example to see how we use order of operations, or PEMDAS, with fractions. The problem is 3/4 plus 2 times 1/3 minus 1/2. We start with multiplication and multiply 2 times 1/3. The whole number 2 can be written as a fraction with a denominator of 1, or 2/1. When we multiply 2/1 and 1/3, the numerator is 2 and the denominator is 3.

So now our problem is 3/4 plus 2/3 minus 1/2. Now we can add and subtract our fractions from left to right. We know we need a common denominator, and looking at our denominators 4, 3, and 2, the least common denominator will be 12, because 12 is the least common multiple of 4, 3, and 2.

For the first fraction, we need to multiply by 3 in the denominator and the numerator. In the second fraction, we need to multiply by 4 in the denominator and numerator. And for the last fraction, we need to multiply by 6 in the denominator and numerator.

Now our problem is 9/12 plus 8/12 minus 6/12. Since the denominators are all the same, we can simply add and subtract our numerators. So we have 9 plus 8 minus 6, which is 11, and we leave our denominator as 12. This gives us a final answer of 11/12. And 11/12 is in simplest form, so we do not have to reduce.

Let's go over our important points from today. Make sure you get these in your notes so you can refer to them later. To add or subtract fractions, fractions must have a common or same denominator. To find a common denominator, you can always multiply the denominators together, but you may need to simplify your answer. When adding or subtracting fractions with common denominators, simply add or subtract the numerators and keep the denominator the same.

An order of operations must be used when simplifying all expressions, including expressions with fractions. So I hope that these key points and examples helped you understand a little bit more about adding and subtracting fractions. Keep using you notes and keep on practicing, and soon you'll be a pro. Thanks for watching.