Why do we add fractions the way we do- by getting the common denominator? A legitimate question! The following is a visual explanation of why we need to do so…
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When we add two fractions with different denominators, we have been taught to “find the common denominator”, then add the numerators only. But why not add the denominators too?
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Let’s try to add 2/7 + 3/5. We know from our standard algorithm that we would change both denominators to 35, then change the numerators by making sure to keep the ratio between the numerator and the denominator in tact. We’d end up with 10/35 + 21/35 = 31/35.
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But how can we see why this works? Let’s first look at both as pictures:
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The picture on the left represents “2 out of 7”, and the picture on the right, “3 out of 5”.
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To add means to combine, and fractions- with the exception of improper fractions- represent amounts less than one. So, we want to combine these two shaded regions into one, or, if we can make more than one, we want to see how many “ones” we can make. But the shaded bars aren’t the same size. And how big is “one”?
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In the algorithm we’d “find the common denominator”, but what does this mean and look like? It means we have to change the look of these two fractions so that their numerators represent portions of whole broken up into the same amount of pieces. To do this, we break the picture on the left up into fifths and the picture on the right up in to sevenths.
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Now our two fraction pictures are broken up into the same amount of pieces, and each piece is the same size.
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To justify this, let’s look at the dimensions of each area…
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The fraction picture on the left has an area of (7×5). The fraction picture on the right has an area of (5×7). Because of the commutative property, we know that 7×5 = 5×7, so both fraction pictures have an equal total area (denominator), and that area is 35 spaces. For this same reason, the shaded spaces in both pictures are also all the same size.
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But now we have 21 (out of 35) shaded pieces in the fraction picture on the left and 10 (out of 35) shaded pieces in the fraction picture on the right. Can we do this? Is 21/35 the same as 3/5? Is 10/35 the same as 2/7?
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Because we can break a “whole” up into as many pieces as we want, these fractions are equal. For example, if two people both had a liter of soda each, one could give small cups of soda to 21 friends, and the other friend could give larger cups of soda to 10 friends. If both empty their bottles, both gave out the same amount of soda despite giving it out to different numbers of friends.
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It’s the same with these two fractions. Once we have set shaded regions (3/5 and 2/7), we can break these regions up into an infinite number of pieces and still have 3/5 and 2/7.
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Now we can begin adding one to the other.
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A total of 31 spaces are filled in when we take the shaded spaces from the fraction picture on the left and add them to the picture on the right. So, 31 out of 35 spaces are now filled in, or “31/35”.
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If we had used the standard algorithm, we would have added the numerators of the fractions (after we found the common denominators) to get this 31.
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2/7 + 3/5 = 10/35 + 21/35 = 31/35, or 4/35 less than a whole.
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For why we need to first find the common denominator, see two or three posts down…
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