To best understand the method for adding algebraic fractions with different denominators, we should first review how to add fractions from our arithmetic days. I am going to do this using three fractions rather than two as usually demonstrated because there is a short cut that can be used for two fractions but not for three. There is no sense in learning a short cut for a specific case until you understand and can use the method that will ALWAYS work.
First, a quick review of the terminology of fractions and the meanings of the parts of a fraction. We use fractions to indicate that "something" has been divided into equal parts and we are interested in some of those parts. Fractions do not exist by themselves. They represent a part of something else, so it is helpful to think of fractions as having the word "of" after them. For example: 1/2 of, 2/3 of, etc.
The bottom number of a fraction is called the denominator (no, I don't know why), and it tells us how many parts our "something" has been divided into. The top number of a fraction is called the numerator and tells us how many of those parts we are interested in. The numerator is always read as a counting number: one, two, five, etc., while the denominator is read as an ordinal (positional) number: third, fourth, ninth, etc. The denominator can be thought of as a "label" like "apples" and "oranges."
Let's use the fraction 2/5 as an example and let's assume the "something" being divided into parts is our allowance. This fraction would be read as two-fifths, and it is indicating that our allowance is being divided into five equal parts and we are interested in two of those parts. Perhaps we have to save 2/5 of our allowance for college.
Remember that for all addition, we must be adding identical items. We can add 3 apples to 2 apples and have 5 apples, we can add 3 oranges to 2 oranges and have 5 oranges, but we cannot add 3 apples to 2 oranges unless we can change the labels to something identical: 3 pieces of fruit plus 2 pieces of fruit gives us 5 pieces of fruit. With this thought in mind, it becomes obvious why you were taught that fractions can only be added if the denominators are the same: the denominator IS the label. Also, remember that when adding fractions with like denominators, we keep the same denominator (label) and add ONLY the numerators. A number example might look like: 1/7 + 3/7 + 2/7 = (1 + 3 + 2)/7 = 6/7. A simple algebraic example might look like: 1/x + 5/x + 2/x = (1 + 5 + 2)/x = 8/x. A slightly more complicated example: 2/y + a/y + 3/y = (2 + a + 3)/y = (5 + a)/y.
Our task is to add algebraic fractions with different denominators. Let's look at an arithmetic example first: 1/2 + 2/3 + 1/4. These cannot be added as written because they are not identical labels (denominators); so we need to change the labels to make them the same. How do we do that? Before you say "find the least common denominator or lcd," I'm going to tell you a secret--you do NOT have to find the lcd. Use the lcd only if you can immediately see what it is. Otherwise, you are wasting time to hunt for it. What do we use instead? It is surprisingly simple--just multiply all of the denominators together. That ALWAYS produces a number that can be divided evenly by each denominator.
For our example: 1/2 + 2/3 + 1/4, the LCD (Least Common Denominator) is 12, but the easiest denominator to find is (2)(3)(4) = 24. We need to change our problem from 1/2 + 2/3 + 1/4 to?/24 +?/24 +?/24. Be Careful! Students very often forget that the new fractions need to be "equivalent" to the original fractions. This means that even though they look different, they still represent the same value: 3/6 and 5/10 look very different but both have a value of 1/2. One of the most typical fraction mistakes happens right here when students change the denominator but forget to change the numerator as well.
(Caution! Caution! Caution! Up to this point, having to deal with slanted fractions has been for the most part annoying but understandable. However, from this point on, slanted fractions cause major confusion. I just have no other way to indicate fractions. So, to help fix this problem, I want you to go get a piece of paper and a pencil. Then every time you see a slanted fraction from here on, you need to re-write the same fraction correctly (vertically). What looks very confusing on the slant becomes much clearer written vertically. If you had any trouble understanding anything earlier, go back and re-write those fractions vertically. That will most likely clear up any confusion. When you have paper and pencil ready, you may continue.)
First, a quick review of the terminology of fractions and the meanings of the parts of a fraction. We use fractions to indicate that "something" has been divided into equal parts and we are interested in some of those parts. Fractions do not exist by themselves. They represent a part of something else, so it is helpful to think of fractions as having the word "of" after them. For example: 1/2 of, 2/3 of, etc.
The bottom number of a fraction is called the denominator (no, I don't know why), and it tells us how many parts our "something" has been divided into. The top number of a fraction is called the numerator and tells us how many of those parts we are interested in. The numerator is always read as a counting number: one, two, five, etc., while the denominator is read as an ordinal (positional) number: third, fourth, ninth, etc. The denominator can be thought of as a "label" like "apples" and "oranges."
Let's use the fraction 2/5 as an example and let's assume the "something" being divided into parts is our allowance. This fraction would be read as two-fifths, and it is indicating that our allowance is being divided into five equal parts and we are interested in two of those parts. Perhaps we have to save 2/5 of our allowance for college.
Remember that for all addition, we must be adding identical items. We can add 3 apples to 2 apples and have 5 apples, we can add 3 oranges to 2 oranges and have 5 oranges, but we cannot add 3 apples to 2 oranges unless we can change the labels to something identical: 3 pieces of fruit plus 2 pieces of fruit gives us 5 pieces of fruit. With this thought in mind, it becomes obvious why you were taught that fractions can only be added if the denominators are the same: the denominator IS the label. Also, remember that when adding fractions with like denominators, we keep the same denominator (label) and add ONLY the numerators. A number example might look like: 1/7 + 3/7 + 2/7 = (1 + 3 + 2)/7 = 6/7. A simple algebraic example might look like: 1/x + 5/x + 2/x = (1 + 5 + 2)/x = 8/x. A slightly more complicated example: 2/y + a/y + 3/y = (2 + a + 3)/y = (5 + a)/y.
Our task is to add algebraic fractions with different denominators. Let's look at an arithmetic example first: 1/2 + 2/3 + 1/4. These cannot be added as written because they are not identical labels (denominators); so we need to change the labels to make them the same. How do we do that? Before you say "find the least common denominator or lcd," I'm going to tell you a secret--you do NOT have to find the lcd. Use the lcd only if you can immediately see what it is. Otherwise, you are wasting time to hunt for it. What do we use instead? It is surprisingly simple--just multiply all of the denominators together. That ALWAYS produces a number that can be divided evenly by each denominator.
For our example: 1/2 + 2/3 + 1/4, the LCD (Least Common Denominator) is 12, but the easiest denominator to find is (2)(3)(4) = 24. We need to change our problem from 1/2 + 2/3 + 1/4 to?/24 +?/24 +?/24. Be Careful! Students very often forget that the new fractions need to be "equivalent" to the original fractions. This means that even though they look different, they still represent the same value: 3/6 and 5/10 look very different but both have a value of 1/2. One of the most typical fraction mistakes happens right here when students change the denominator but forget to change the numerator as well.
(Caution! Caution! Caution! Up to this point, having to deal with slanted fractions has been for the most part annoying but understandable. However, from this point on, slanted fractions cause major confusion. I just have no other way to indicate fractions. So, to help fix this problem, I want you to go get a piece of paper and a pencil. Then every time you see a slanted fraction from here on, you need to re-write the same fraction correctly (vertically). What looks very confusing on the slant becomes much clearer written vertically. If you had any trouble understanding anything earlier, go back and re-write those fractions vertically. That will most likely clear up any confusion. When you have paper and pencil ready, you may continue.)
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