Last week, in Fundamental Understanding of Mathematics XLI, we took a look at the definition of a fraction, and showed how to add two fractions with like denominators, or bottom numbers. The bottom number tells us into how many pieces the unit length is divided, and gives us a sense of the size of each piece. We saw that adding fractions on a number line was a lot like adding whole numbers, except the pieces are smaller.
But, when we try to add fractions when the bottom numbers are not the same, we have a problem. The pieces are different sizes. How do we reconcile, say, a 1/3rd sized piece with a 1/5th sized piece? If we take them and place them end to end, to add them up, the end of the pair will not fall on any third division mark, nor will it fall on any fifth division mark.
As you can see, whether we add 1/3 to 1/5, or add 1/5 to 1/3, we get a length that is somewhere between 1/3 and 2/3, or between 2/5 and 3/5. How do we divide up our unit length so that when we add these two fractions their length will end exactly on some division mark?
In order to find the answer to this puzzle, we need the idea of equivalent fractions. Equivalent fractions are fractions that look different when written, but when placed on the number line, are exactly the same length.
For example, 2/6 and 1/3 are equivalent fractions, since, even though they look different when written, they are the same length on the number line. Likewise, 4/6 and 2/3 are equivalent, as are 6/6 and 3/3.
Mathematically, we can define two fractions as equivalent if we get one of the fractions by multiplying the top number and the bottom number of the other fraction by the same non-zero whole number. To get 2/6ths then, we start with 1/3rd, multiply the top number by 2, giving us twice as many pieces, and we also multiply the bottom number by 2, making each piece half as long, or 2/6ths.
Notice that multiplying the top and bottom numbers by the same whole number is the same as multiplying it by a fraction with the same number on top as on bottom, in this case, 2/2. Also notice that 2/2 is the same as 1.
Likewise, 3/3 is equal to 1, 6/6 is also equal to 1. Any number divided by itself is equal to 1. This makes sense: division is repeated subtraction -- so how many times can you subtract a number from itself? Once. And it comes out even, with no remainder. Guaranteed, every time.
The way this observation fits into the big picture is this: when we multiply a number by 1, the number doesn't change. 1/3 x 1 = 1/3. But, as we can see, 1/3 x 1 can be written 1/3 x 2/2 which is equal to 2/6. Since 1/3rd and 2/6ths are equal to the same thing (1/3 x 1), they are equal to each other. 1/3 = 2/6. They are the same number. They are the same length on the number line.
Getting back to our original puzzle, adding 1/5 and 1/3. We need to find equivalent fractions for these numbers such that the length of the pieces (the denominator, or bottom number) is the same. A number that is both a multiple of five and a multiple of three. The simplest way to do this is to multiply three and five, to get fifteen.
Now we have equivalent fractions for both 1/3rd and 1/5th that have the same denominator: 5/15ths and 3/15ths.
These equivalent fractions have the same bottom number, and we already know how to add fractions with the same bottom number, also known as "common denominators." We just add the top numbers and place the sum over the "common denominator."
Here is what it looks like on a number line. We changed 1/5 into 3/15, by multiplying by 3/3. We changed 1/3 into 5/15 by multiplying by 5/5. We chose 3/3 and 5/5 because they are both equal to 1, and because, when we multiplied, we would get, in both cases, 15 as the bottom number. Where did we find the 3 and the 5? The bottom number of the other fraction.
Here is what it looks like written down using fraction notation. In practice, we'd put together a formula:
and do it all in fewer steps.
Have fun in the comments.