#PreAlgebra - How to add fractions (and why do I need a common denominator...)

Question

What's ?

Answer

Analysis

First, some fraction stuff! The top number in a fraction is the numerator and the bottom number is the denominator:



Think of a pizza. The denominator tells us the number of pieces there is in 1 whole pizza. The numerator tells us the number of pieces of that pizza that we're concerned with. Notice that if the numerator and the denominator are the same number, we'll end up with the value 1 (or, in other words, the entire pizza):



What our question is asking us to do is to add one slice from the pizza with 4 pieces and 1 slice from the pizza with 6 slices:



Now - if we were simply adding things up, like doing 1+1, all we'd be saying is that I have one slice of pizza here and one there and together they add up to 2:

1+1=2

But with the fractions, we have a bit more information. We can't simply add the numerators together to see that we have 2 pieces of pizza - we need to put more information into our answer. We need to compare the two pizzas on equal terms. To do that, we want the pizzas to both have the same number of pieces in each one - and that is what the Common Denominator is - making the pizzas have the same number of pieces.

Ideally, we'd like to make as few cuts as possible (that pizza is getting cold and I know we're all hungry for some), so we'd like to find the Lowest Common Denominator (LCD). So how do we do that?

I do it by breaking down the two denominators into its prime factors (those numbers that are prime that multiplied together equal the denominator):

4 = 2 X 2
6 = 2 X 3

So now what do we do? The Lowest Common Denominator is the same as the Lowest Common Multiple - we need to include all the primes from each of the biggest groupings. For instance, there are two 2's in the 4, so we need both of those. And there's a 3 in the 6, so we need that too. So we have:

LCD = 2 X 2 X 3 = 12

And now we can scale up our fractions, using clever forms of the number 1, to find the sum.

First remember that anything times 1 equals itself (so we can multiply by 1 and not change the value of the fraction):



And any fraction that has it's numerator and denominator the same is equal to 1. We want to pick values of "1" that when multiplied with our denominators, gives the LCD:





Let's stop here for a second. Notice that with the pizza that had 4 slices, if I cut the pizza into 12 slices, my having 3 of those slices is the same amount of pizza. The fractions are equal. Same with the pizza that had 6 slices - if I cut those slices in half so that there are 12 slices of pizza, and I have 2 of them, it's the same amount of pizza.

Ok, let's finish up:



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Questions and comments always welcome!