## Introduction: Adding Fractions

Grand Prize in the

Burning Questions Round 6.5

Congratulations! You clicked on the link to go to this instructable! (that was obvious.....) Anyway, this hopefully will teach you the basics of adding fractions. Well, here's Fraction 101.

The top number is the numerator and the bottom number is the denominator. To add fractions, the denominators (bottom numbers) have to be the same. One easy way to remember that the denominator is the bottom number is to remember that it's the one "Down under" (since they both begin with D you're all set).

Here's an example: ^{3}/_{7} & ^{5}/_{7} can be added, but ^{4}/_{7} & ^{4}/_{9} cannot.

Well, they can, but a few changes have to be made. We'll go over these later.

## Step 1: Adding Simple Fractions

**Adding simple fractions**

Here's an example of adding a fraction.

^{1}/

_{4}+

^{2}/

_{4}= _

First, check to see if the denominators, the bottom numbers, are the same. If they are, you're lucky because the fractions are easy to add. To add fractions with common denominators, all you have to do is add the two numerators leaving the denominator the same.

In this example you'd add 1 + 2 = 3 which will be the numerator of our answer.

So,

^{1}/

_{4}+

^{2}/

_{4}=

^{3}/

_{4}. Note that we don't add the denominator. Our added fraction will end up being

^{3}/

_{4}.

The reason for this is that a fraction is basically a division problem. When you add them you add the two numerators together to make one division problem, or a fraction. You can do this easily when the denominators are the same.

For example, think of a pie (not pi):

- If you divide the pie into 4 pieces, each piece is one of four pieces, or

^{1}/

_{4}of the pie.

^{2}/

_{4}would be two of the four pieces of the pie, etc. (see first two pictures below)

- If you give one person

^{1}/

_{4}and another

^{2}/

_{4}you've given away three of four pieces of the pie, or

^{3}/

_{4}and you have

^{1}/

_{4}of the pie left. (see picture below)

- If you add the

^{1}/

_{4}, the pie you have left, and the

^{3}/

_{4}, the amount of pie you gave away, you get one, because adding

^{1}/

_{4}and

^{3}/

_{4}gives you

^{4}/

_{4}or a whole pie!

- If the numerator and denominator of a fraction are the same, then it equals 1. Like

^{2}/

_{2}, which is 2 halves, or one whole.

## Step 2: Improper Fractions

**Improper Fractions and Mixed Numbers**

Here's another example:

^{4}/

_{7}+

^{5}/

_{7}= _

When you check to see if the denominators are the same, you find that they are. So you just add the numerators.

4+5=9 , so our numerator is 9. Our added fraction will end up being

^{9}/

_{7}.

Hmm, doesn't that look weird? This is called an improper fraction. An improper fraction is when the numerator is larger than the denominator.

Let's change the improper fraction into a mixed number. A mixed number is a number with both an integer (number) part and a fraction part. Integers are whole numbers, that means no fractions or decimals. In algebra, they are called integers so you don't get them mixed with decimals and/or fractions.

Anyway, back to our problem:

We have the improper fraction

^{9}/

_{7}that we want to change into a mixed number. You do this by seeing how many denominators fit into the numerator. In this case, 7 fits into 9 once. The remainder, 2 is the new numerator like this: 1

^{2}/

_{7}

As we saw in the previous step, a fraction with the same numerator and denominator is equal to 1. By finding out how many groups of 7 pieces, in this example, (or how many "wholes") are available, we can separate them out and see how many fractional parts we have left.

This works with any fraction where the numerator is larger than the denominator.

- Take

^{123}/

_{10}. While this looks daunting at first, it's really pretty easy.

- As we did earlier, figure out how many times 10 goes into 123 (divide 123 by 10).

- You get 12 with 3 left over.

- Since our denominator (and therefore the number we're dividing by) is 10, our answer is 12 whole groups of 10 and 3 tenths, or 12

^{3}/

_{10}

Sometimes, you'll find that after you divide, there is no remainder. For example

^{20}/

_{10}. When you divide 20 by 10 you get 2 wholes with no remainder. In this case your answer is 2 with no fraction.

**In summary**, to change an improper fraction into a mixed number:

1) Find out how many whole times the denominator goes into the numerator by dividing-- this is the whole number part of your mixed number.

2) Take the remainder of your division and use that as the new numerator of the fraction with the same denominator.

## Step 3: Adding Fractions With Different Denominators

**Adding Fractions with Different Denominators**

To add fractions with different denominators you need to change the fractions so they have a common denominator before adding them.

To find a common denominator, you can either find a common multiple, or you can multiply each fraction (top and bottom) by the denominator of the other fraction.

Here's an example of the latter:

^{1}/

_{2}+

^{1}/

_{3}= _

First, we'll do

^{1}/

_{2}.

Since the denominator of the other fraction is 3, we multiply both the numerator by 3, and the denominator by 3, which gives us:

^{1}/

_{2}=

^{3}/

_{6}. This works because if you multiply both the numerator and the denominator by the same number, the fraction will have the same value.

Now we have

^{3}/

_{6}+

^{1}/

_{3}=_.

Let's do the second one,

^{1}/

_{3}. We multiply both the top and bottom by 2 (the original denominator of the first fraction) giving us:

^{2}/

_{6}.

So our equation is now

^{3}/

_{6}+

^{2}/

_{6}=.

Now add them up and we get (Drum roll please)

^{5}/

_{6}!!!

## Step 4: Adding More Than Two Fractions

**Adding more than two fractions**

Adding more than two fractions is simple. You can add them the same way as previously described.

If they all have the same denominator, just add all of the numerators and put the result over the denominator.

For example:

^{1}/

_{9}+

^{2}/

_{9}+

^{1}/

_{9}+

^{2}/

_{9}=

^{6}/

_{9}

Just like before, if you add the numbers and the numerator is larger than the denominator, you can change the fraction to a mixed number following the same steps.

**Finding a Common Denominator**

If the denominators are not the same, you still need to convert them before adding.

For example:

^{1}/

_{2}+

^{1}/

_{4}+

^{1}/

_{3}+

^{1}/

_{12}=

You can convert them all at once or you can convert them a pair at a time.

Since it's typically easiest to convert them all at once, we'll do that with our example. In this case, it's easiest to find a common multiple, or a number that is a multiple of all of the denominators.

In our example, the multiples of:

2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, ...

4 are 4, 8, 12, 16, 20, 24, ...

3 are 3, 6, 9, 12, 15, 18, 21, 24, ...

12 are 12, 24, ...

So, looking for a multiple that's common to all four of our denominators, we see that 12 and 24 both appear on all four lists. We can use either, but typically the smallest, or lowest, is used, which is why it's often called the lowest common multiple.

Once we've found a common multiple, we'll use 12 in our case, you need to convert each of our fractions to an equivalent one with that denominator.

So for

^{1}/

_{2}, to get 12 as our denominator, we need to multiply both the numerator and denominator by 6 (multiplying the top and bottom both by the same number gives us an equivalent fraction since its the same as multiplying the fraction by 1 (e.g.

^{2}/

_{2}= 1)).

This gives

^{1}/

_{2}=

^{6}/

_{12}.

Doing the same with

^{1}/

_{4}(multiplying top and bottom by 3 to give us a denominator of 12) gives:

^{1}/

_{4}=

^{3}/

_{12}.

And subsequently with

^{1}/

_{3}(multiplying top and bottom by 4 to give us a denominator of 12) gives:

^{1}/

_{3}=

^{4}/

_{12}.

With

^{1}/

_{12}, we're all set since it's denominator is already 12.

This makes our problem:

^{1}/

_{2}+

^{1}/

_{4}+

^{1}/

_{3}+

^{1}/

_{12}=

^{6}/

_{12}+

^{3}/

_{12}+

^{4}/

_{12}+

^{1}/

_{12}=

^{14}/

_{12}= 1

^{2}/

_{12}

Note that you can use the lowest common multiple technique to add 2 fractions, as well. But it's often easier to just multiply as we did in Step 3. Multiplying the two denominators automatically gives you a common multiple of the two numbers. It's easy and will always work. The only minor drawback is that your answer may require a little more simplification which won't be a problem as soon as you complete Step 5.

## Step 5: Simplifying Fractions

**Simplifying Fractions**

Often after you add, the fraction can be simplified. A fraction whose numerator and denominator both have a factor in common can be simplified. (Factors are numbers that the original number is evenly divisible by, or numbers that can be multiplied to give the original number.)

We can demonstrate this using the results of the two examples in Step 4:

^{6}/

_{9}and 1

^{2}/

_{12}.

If we examine

^{6}/

_{9}, and find the factors of the numerator and the factors of the denominator, we find:

factors of 6: 1, 2, 3, and 6 (because 1 x 6 = 6 and 2 x 3 = 6)

factors of 9: 1, 3, and 9

Since 3 is a common factor of both numbers, we can divide both the numerator and denominator by 3 and get a simplified fraction. As with multiplying both the top and bottom of the fraction by the same number, dividing them each by the same number results in an equivalent, or equal, fraction.

So 6 divided by 3 = 2,

and 9 divided by 3 = 3,

so our resulting simplified fraction =

^{2}/

_{3}.

Likewise with 1

^{2}/

_{12}. Looking at the fractional part of the answer and finding the factors we find:

factors of 2: 1 and 2

factors of 12: 1, 2, 3, 4, 6, and 12

Dividing both the numerator and denominator of our fraction by 2 (the greatest common factor), gives:

2 divided by 2 = 1

12 divided by 2 = 6

so our resulting simplified answer is 1

^{1}/

_{6}.

## Step 6: Adding Mixed Numbers

**Adding Mixed Numbers**

After learning what you have so far, you may wonder how to add mixed numbers. At this point, adding mixed numbers will be a snap.

To add mixed numbers, you add the whole numbers, then add the fractions and simplify them, and finally add the two parts together!

For example:

1

^{5}/

_{6}+ 2

^{3}/

_{6}=

add the whole numbers: 1 + 2 = 3

add the fractions:

^{5}/

_{6}+

^{3}/

_{6}=

^{8}/

_{6}

simplify the fraction:

^{8}/

_{6}= 1

^{2}/

_{6}(change to a mixed number)

continue simplifying: 1

^{2}/

_{6}= 1

^{1}/

_{3}(divide numerator and denominator each by 2)

add them together: 3 + 1

^{1}/

_{3}= 4

^{1}/

_{3}

You're done!

## Step 7: Congratulations!!!!!

Congratulations, you have survived 6 entire steps of fraction learning!

You now know how to add fractions, find common denominators, what the heck improper fractions and mixed numbers mean.

You even know how to add mixed numbers. I hope that you like my Instructable and please vote for it and rate it!!!_{I love comments}

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## 12 Comments

your awesome

I love ice cream!!!

meh301 gets a point!!!!!!

You make damn sense

Wow, I can't believe my luck. I merely stumbled on your fantastic instructions here on the day my daughter has to study for a fractions test! So thakns so much. I really have been struggling to explain these logically to her! Now I can with ease! Well done on your deserving win too!

Glad I could help!

or 1 1/6 i thought you were simplifying all these :P

I love ice cream!

Nice Instructable and congratulations on your win! By the way, I

ice cream.loveThis is a good instructible.

I only have one small criticism:

It is not good practice to write mixed fractions without the addition operator. A mixed fraction is a sum, and it should be written as such, e.g.

3 +

^{1}/_{2}The notation:

3

^{1}/_{2}is confusing because there is another convention that says putting two expressions side by side like that means multiplication; i.e.

3

^{1}/_{2}= 3 *^{1}/_{2}=^{3}/_{2}This may seem nit-picky to many of you, but this is math after all. It is good to be clear about what your expressions mean.

See also:

http://mathworld.wolfram.com/MixedFraction.html