Adding and Subtracting Fractions

Adding & Subtracting Fractions

To be able to directly compare two fractions (or add them together, or subtract one from the other) they need to have the same denominator. When fractions have the same denominators they are basically the same sorts of things, meaning they can be combined. It's like combining two piles of bananas, instead of a pile of pears and a pile of bicycles.

If the denominators are not the same we need to make them the same by multiplying by a unity fraction. The simplest way is using a unity fraction made from the denominator of the other fraction.

13 + 14 = ( 13 x 44 ) + ( 14 x 33 ) = 412 + 312

Once the denominators are the same the numerators can simply be added together while the denominator stays the same. This example has four "somethings" added to three "somethings" giving seven "somethings". In this case the "somethings" are twelfths, but they could easily be bananas.

412 + 312 = 712

Sometimes the denominators will have common factors, and we need to simplify the final fraction. For example:

12 + 14 = ( 12 x 44 ) + ( 14 x 22 ) = 48 + 28 = 68

Then simplify 68 = 34

Or we could use a better multiplying fraction to start with, so we can use the lowest common denominator.

12 + 14 = ( 12 x 22 ) + 14 = 24 + 14 = 34

Lowest Common Denominator

We know that multiplying by 1 won't change the value of a number, so if we multiply a fraction by a fraction which is equal to 1, it won't change in value. There are lots to chose from, and we can pick any we like.

 1 = 1⁄1 = 2⁄2 = 3⁄3 = ... = 6,782 = ... 6,782

Let's look at an example. Assuming you're hungry, would you rather have 29 of an ice cream cake or 524 of it? To be sure of which is bigger we'll have to convert them so they have the same denominator. The easiest way to choose one is to use the denominator of the other fraction.

29 x 2424 = 48216

524 x 99 = 45216

So 29 > 524 by 3216

But 216 is divisible by 3 (because it's 63 and 6 is divisible by 3, and because 2+1+6 is a multiple of 3) which means there were common factors in the denominators. In this case, both 9 and 24 can be divided by 3. Finding the lowest common multiple of the original two denominators will give us the lowest common denominator, which will give us the simplest fraction at the end of it.

29 x 88 = 1672

524 x 33 = 1572

29 - 524 = 1672 - 1572 = 172

So 29 > 524 by 172

172 of an ice cream cake isn't much, but it's still worth having.

"Lowest common denominator" is also used figuratively to describe how television sometimes creates their programmes to appeal to the lowest ideals of their viewing audience, for example by using the simplest, coarsest humour, instead of anything that might require careful consideration.

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