Fractions

I'm not going to bore you with the basic grade school pizza examples. I've been there, done that, and cried a little. Probably died a bit inside while I was at it. I really disliked fractions once upon a time. It was a fear of the unknown. But once you understand fractions, they really aren't so bad.

There's much more to fractions than simply dividing pizza slices. If you understand that  represents 1 slice of a pizza that has 8 slices, you're good to go. Keep in mind that the other 7 slices of that pizza could have been exterminated by the Daleks; we don't care. We can only work with what we are given.

Quick review:

A fraction consists of two numbers: the numerator and the denominator. These numbers are called the terms of a fraction.

The numerator is the number on top. It tells you the number of equally sized parts of the whole.

The denominator is the number on the bottom. It tells you the whole, or total (how many parts are needed to make the whole pizza).

                                                         1 <---numerator
                                                         2 <---denominator

Proper Fractions

Proper fractions occur when the numerator is less than the denominator. It is a fraction whose value is less than 1.

Thus, is a proper fraction.

Proper fractions are what every fraction should be, but sometimes fractions get a little wild...

Improper Fractions

Fractions where the denominator is less then the numerator are called improper fractions. It is a fraction whose value is greater than 1. These pompous, misbehaved fractions have numerators that think they are better and more important than their denominators. So they are considered improper, and they are not welcome at family dinners.

is an example of an improper fraction.

You will see what we do to simplify improper fractions later on.

Equivalent Fractions

These fractions have the same value.


 
 All of the above fractions have the same value, namely, they can all be broken down to 1/4.

Whole Numbers

Do you realize that all whole numbers can be written as fractions? Take 5, for instance.


5 is a whole number but it can be written as and it is still equal to 5.

In that case, the denominator of the fraction is equal to 1. In other words, the fraction is equal to its numerator.

   because  

Sometimes you can write fractions as whole numbers.

         because  

This is because the fraction's numerator and denominator are the same.

Also, if you have a fraction where the numerator can be evenly divided by the denominator, you can also convert that number into a whole number:

  because 10/5 = 2

What if you want to divide a fraction that can't be converted into a whole number?

All you have to do is use the reciprocal (or inverse) of that fraction.

So, if we wanted to find the reciprocal of then we would multiply it by which makes the number 1


because 

More on that later...

Mixed Numbers

To mucky the waters even more, we also have mixed numbers. Mixed numbers occur when you have a whole number (the quotient) and a fraction. These result from trying to convert an improper fraction into a whole number. They are like a poor attempt by improper fractions to try and make themselves proper again. Turning an improper fraction into a mixed number is an attempt to simplify the fraction.

Also, when you divide a whole number by another whole number that does not reduce to a whole number, the result has a remainder and is a mixed number. For example, 
                                      
                                       quotient ---> 3 1 <---numerator
                                                               2 <---denominator

Mixed numbers are easier to create than they are to solve. Generally, if you are working with an equation that has mixed numbers in it, you are better off converting the mixed number to an improper fraction. Otherwise it is difficult to solve.

Example:



Keep in mind that even though the numerator has been changing, the denominator has stayed the same. This is because no matter how many times we increase or decrease the parts (numerator) the whole (denominator) stays the same.

Think of these denominators as if you have pizza boxes that hold 4 pizzas and pizza boxes that hold 5 pizzas. No matter how many slices you have, you can still only fit 4 pizzas in one box and 5 in another, so you might have to increase the number of boxes but the size of the boxes won't change.

How are we going to solve    if their denominators are different?

We're going to need a bigger box!

Which leads us to...

Finding the Common Denominator

Finding the common denominator for a fraction is similar to finding the least common multiple.You cross-multiply the fractions to create a common denominator and then add the fractions.

In this case:

; take the two denominators, 4 and 5, and multiply them together = 20

Now cross multiply

23 * 5 and 14 * 4 equal 115 and 56.

Our new equations looks like this: 


is an improper fraction. We could turn it into a mixed number by dividing 171/20 =

What if we had numbers with larger denominators? Would we simply multiply them together?

Let's try a new example:



If we were to multiply the denominators together, 45*55 = 2475. This is quite a large number.

What if we were to find the least common multiple?

45 = 5*3*3
55=5*11

3*3*5*11 = 495 <--the new common denominator. A much more manageable number.

But what do we do with the numerators now?

We have to figure out how many times each denominator goes into the new common denominator.

Therefore, 495/45 = 11 and 495/55 = 9

Now we increase the numerators by the same amount that the denominators were increased by. There is no cross multiplication once we have found the least common multiple.

123*11 =1353
112*9 =1008




How do we know this is correct?

We can always compare the original fraction to the new fraction by dividing them and seeing if the ratios are the same:

123/45 = 2.733             112/55 = 2.036
1353/45 = 2.733           1008/495 = 2.036

This is the beauty of fractions. No matter how much you increase or decrease the numbers, the ratios remain the same!

What if we want to simplify our answer,   ?

We need to...

Reduce the Terms of Fractions

Just as we want to find the least common multiple of fractions, we also want to reduce fractions to their lowest terms. When we reduce a fraction to its lowest terms, we are making the numbers of the numerator and denominator as small as possible. This way we don't have a bunch of clunky, large fractions.

There are two ways to reduce a fraction to its lowest terms.

In my opinion, the way you use depends on the difficulty of the fractions.

The Easy Way: divide the numerator and the denominator by the same number.

Let's say you have                          

You know that 20 and 4 are both divisible by 4, so

<-- we know that we have reduced the fraction to its lowest terms because the
denominator is 1. Can't get any lower than 1.

Try another:

<-- again, we know that the fraction is reduced to its lowest terms because now the 
numerator is equal to 1.

This easy method will not work very well on our unsimplified fraction, mainly because we will not know whether or not we have reduced this fraction to its lowest terms.

The Hard(er) Way: use prime factorization to decompose a fraction down to its prime numbers. We then cancel common factors to reduce the fraction.

Let's try to simplify our unsimplified fraction, using this method.
                                                                             
First of all, we know that 495 is at least divisible by 5 because it ends in 5. What the heck is 2361 divisible by?

Let's find its digital root. Finding out what 2361 is divisible by will save us time because we won't be plugging in numbers randomly trying to see what number will evenly divide it.

2+3+6+1 = 12 = 1+2 = 3

We now know that 2361 is divisible by 3 because it has a digital root of 3.

What about 495?

4+9+5 = 18 = 1+8 = 9

495 is also divisible by 3 because it has a digital root of 9!

Let's begin prime factorization.

Wait, what is 787 divisible by??

Not by 2, because it ends in an odd number. Not by 5, because it doesn't end in a 5 or 0.

Digital root time:
7+8+7 = 22 = 2+2 = 4 <---not divisible by 3
7-8+7 = 6 <--- not divisible by 11 because it doesn't end in 11,0, or -11

It looks like 787 is a prime number and we have factored 2361 as far as we can go!

2361 = 3*787
495 = 3*3*5*11

We were able to cross a 3 off the list of both numbers because it was a common factor for both the numerator and denominator.

Now we have:

  or    or, as a mixed fraction:                                                   

And now we know that we have simplified this fraction as far as it will go! This fraction has been reduced to its lowest terms baby!

Increase the Terms of Fractions

In general, you want to increase the terms of a fraction when you need to add or subtract them.

Let's take our previously simplified fraction and subtract it by something:

Find the least common multiple:

165 = 3*5*11
120 = 2*2*2*3*5

2*2*2*3*5*11 = 1320; this number is our least common multiple for 165 and 120.

1320/165 = 8                  787*8 = 6296
1320/120 = 11                200*11 = 1320

We have now increased the all terms of the fraction because it was necessary to be able to perform the operation.



Now we will reduce the resulting fraction to its lowest terms:

4096 = 2*2*2*2*2*2*2*2*2*2*2*2
1320 = 2*2*2*3*5*11 

The greatest common factor for 4096 and 1320 is 2*2*2 = 8

Therefore,
  or    
           

Note that we could have just as easily reduced the fraction by dividing 4096 by 1320. But now you understand it better, right?

We've learned that to add or subtract a fraction, we need to cross multiply and and find a common denominator.

The rules are a little different with multiplication and division.

Multiplying Fractions

I think multiplying fractions is a lot more fun than adding or subtracting them because you don't have to go through cross multiplying and finding common denominators. It's much easier.

All you do when multiplying fractions is multiply the numerators by the numerators and the denominators by the denominators. It's that simple!

 

Wait, it can get even simpler! What if we simplified the numbers before we multiplied them? We can cross cancel; reduce the fractions to their lowest terms before we even multiply them by cancelling out common factors in both the numerator and the denominator. This is also known as cross simplifying.

In the previous case, we can reduce our fraction to:




 Cross cancelling makes your life a lot easier, doesn't it?

Dividing Fractions

We can't actually divide fractions like dividing whole numbers. We multiply them instead, by their reciprocal.

The reciprocal, if you remember, is the multiplicative inverse of a fraction. To get the reciprocal of a fraction, we switch the numerator and the denominator so that the numerator is now at the bottom (and technically the denominator) and the denominator is at the top (and technically the numerator).

Take 2, for instance. It can be written as . Its reciprocal is .
                                                                                                  
It has really bothered me as to why we need to multiply a fraction by its reciprocal in order to divide it.  The explanation is going to get a bit algebraic.

When you think about it, we actually find the multiplicative inverse of a whole number when we divide it.
Take 2 divided by 4. It can be written as or as   


Let's say we have , which can be written as:
 
We still can't do much with the equation, even in that form, so we need to somehow get rid of the denominator:

When we multiply    the two numbers cancel each other out.

So we are left with       or just  
                               _____
                                   1


Let's try this with numbers:

 

= 

It's a lot easier to skip those steps and just write:






or, if we were to cross cancel the 10 and 12 by 2:




Operations with Mixed Numbers

Keep in mind that before you perform any operations with mixed numbers, you must convert them to an improper fraction first. Mixed numbers are annoying like that. They require that extra step, conversion, before you can do something with them.

To convert a mixed number, you multiply the denominator by its quotient and then add the numerator, like so:




This is because 4, the quotient, is equal to

   or  

Then we simply add the existing  to the formula to get  .

Then we take our new improper fraction and add, subtract, divide, multiply as though it had never been a mixed number. When we get our final result, we can leave it as an improper fraction or simplify it back to a mixed number or, if we are lucky, it will convert to a whole number!

Don't get scared if the problem looks daunting. Don't acknowledge the mixed numbers; just convert them to improper fractions and you will be fine.

Try this problem on for size:


                                                                      
OMG, that looks pretty bad! Convert! Convert! Exterminate the mixed numbers!

We already know the answer to the mixed number on the left because I just solved it above, so let's solve for the number on the right:



Now that we've converted both of the mixed numbers, let's see what our new equation looks like:



We've gone as far as we can go without finding a common denominator. As we've learned, we need to do some cross multiplication to make that happen...

5*8 = 40 <---new denominator
22*8 = 176
59 *5 = 295

Our new equation:



Yes! They have a common denominator so we can subtract!



We can convert the result back to a mixed number.


119/40 = 2 remainder 39

Our answer is 

Practice makes perfect! (don't think that I didn't have to print out a bunch of worksheets and solve them before I wrote this blog)

Have you noticed how fractions have been building on other concepts that we have learned, particularly prime factorization/factoring and finding the least common multiple?

I have found, during my second foray into mathematics, that all math is is a set of rules. These rules have reasons behind them; sometimes we don't understand these rules. Before we can continue on, we must try to understand why these rules are in place. Question what you don't understand, just as I questioned why we must multiply a fraction by its reciprocal when dividing. It is only when you fully understand what you are doing that you can move on to the next level. By seeking answers, you are building a strong foundation.

I believe that a strong foundation in the basics of math is essential to being able to attempt higher levels of mathematics.  Unfortunately, sometimes individuals who do not fully grasp the concepts at hand are passed over in their classes. This leads to a weakened understanding of more difficult math subjects, or no understanding at all and and unnecessary disdain for mathematics.

Looking at fractions a second time has been like looking at the world in a whole new light. They make sense this time around!

I am hoping that as the math gets more difficult, my understanding of it will improve since I have now built a strong foundation. The alternative is just giving up and resigning myself to never fully understanding mathematics. I won't quit and I hope that you won't either.