Add Fractions (unlike denominators)

When you add fractions, you need to make sure that the denominators are the same.  

Why? The denominators of a fraction tell you how many pieces the whole is cut into.  If the denominators are different, then the sizes of the pieces are different, so you can't just add them together.

For example, imagine two pizzas.  One pizza is cut into four pieces (each piece is $\dfrac{1}{4}$).  The other pizza is cut into 12 pieces (each piece is $\dfrac{1}{12}$).  The pizzas are the same size, but if you get a slice from the pizza cut into 4 pieces, you get a bigger slice than if you get a slice from the pizza cut into 12 pieces.

If someone handed you one piece from each pizza, and you wanted to find out how much pizza you had all together, you would do an addition problem:

$\dfrac{1}{4} + \dfrac{1}{12}$

But, if you add the  pieces together, you have two pieces but you don't have $\dfrac{2}{4}$ (you have less than that) and you don't have $\dfrac{2}{12}$ (you have more than that).  Although you might be tempted to add the denominators, you definitely don't have $\dfrac{2}{16}$!  $\dfrac{2}{16}$ means that a pizza is cut into 16 pieces, and you have 2 of them... that means you have two very small pieces, much less pizza than you actually have!

So, what you want to do is change the denominators of your fractions so that they are the same, then you can add them together! 

When you want to add (or subtract) fractions, it's easiest if you write them vertically.  Vertically, you have lots of room to figure out and write your new, equivalent fractions that have the same denominator.

$$\eqalign{\dfrac{1\;}{4\;}&\\\underline{+\dfrac{1}{12}}&}$$

Then, think about what common denominator $\dfrac{1}{4}$ and $\dfrac{1}{12}$ could share.  A common denominator is a number that both denominators can go into evenly (a multiple of both numbers).  

To use the pizza example, we want to think of that we could cut up both $\dfrac{1}{4}$s and $\dfrac{1}{12}$s so that they would be the same size (and once they are the same size, we can add them together!). So think about how you could group slices together so that you had the same portion of pizza.

As you can see, $\dfrac{1}{4}$ of a pizza is the same as $\dfrac{3}{12}$ of a pizza! We can convert fourths to 12ths and then talk about both pizzas in a comparable way.  We just need to convert every $\dfrac{1}{4}$ of a pizza into $\dfrac{3}{12}$ of a pizza.

You don't have to think about pizzas every time you want to add or subtract fractions. Mathematically, you think about the lowest common denominator.  In this case, our denominators are 4 and 12.  The lowest common denominator is the lowest number that can be divided evenly by both 4 and 12. Both 4 and 12 go into 12, so we can use 12 as our common denominator (note: a common denominator will never be smaller than either of the original numbers.  The lowest possible common denominator will be the same as one of the denominators).

$$\eqalign{\dfrac{1}{4}&=&&\dfrac{3}{12}\\+\dfrac{1}{12}&=&&\dfrac{1}{12}\\\\&\text{}&&\overline{\dfrac{4}{12}}=\dfrac{1}{3}}$$

Let's try a problem with no pizzas.

$$\eqalign{\dfrac{1}{6}+\dfrac{3}{8}\\ \text{Write it vertically  }&\left\{ \begin{array}{rcl} \dfrac{1}{6}&\\ +\dfrac{3}{8}& \\\hline \qquad \qquad\\ \qquad  \end{array}\right.  \\\text{Find the common denominator: 24  }&\left\{ \begin{array}{rcl}  \dfrac{1}{6}&=&&\dfrac{?}{24}\\-\dfrac{3}{8}&=&&\dfrac{?}{24}\\\hline \qquad \qquad \\ \qquad \end{array}\right.\\\begin{array}{rcl}\text{Multiply each fraction by a fractional version of 1  }\\\text{that will make each denominator equal 24  }\end{array}&\left\{ \begin{array}{rcl}\dfrac{1}{6}&\times&&\dfrac{4}{4}=&&&\dfrac{4}{24}\\+\dfrac{3}{8}&\times&&\dfrac{3}{3}=&&&\dfrac{9}{24}\\ \hline \qquad \qquad \end{array}\right.\\\text{Add the fractions with common denominators  }&=\begin{array}&\quad && \quad &&& \quad &&&&\quad\dfrac{13}{24}\end {array}}$$

Why does this work?

Remember, when you add fractions you need to have a common denominator.

To get a common denominator, you find a denominator that both denominators go into evenly, then you convert the denominators.  In the picture above, we used the pizza to find the number of 12ths that equaled 1/4. When you don't have a picture, you use math.  

You can multiply any number by 1.  And any fraction with the same number in the numerator and the denominator equals 1 (so, $\dfrac{3}{3} = 1$ and $\dfrac{68}{68}=1$, etc.).  So, if you figure out what you have to multiply your denomintor by to turn it into the common denominator that you've chosen (in this case, to turn 6 into 24 you need to muliply by 4 and to turn 8 into 24, you need to multiply by 3), and multiply each of your fractions by the fraction with those denominators that equals 1 (so, $\dfrac{4}{4}$ and $\dfrac{3}{3}$).

Then, you have two fractions, with the same denominator, that you can simply add together.