A Year of Learning Fractions in Ten Simple Rules

Working with fractions is quite tricky - and the nasty thing about them is that you really can't use a calculator or a computer to help out.

It's quite important to learn how to work with fractions - but nearly everyone has difficulties with them. You'll find that quite a lot of grownups won't be able to do things like adding two fractions.

The actual rules are in bold-face type - all the rest is just explanation and some handy hints to save work. If you work through this carefully, you'll never have problems with fractions again.

Here are the ten things you'll need to learn about fractions:

Writing and Naming Fractions

The number on the top of the fraction is called the NUMERATOR and the number on the bottom is called the DENOMINATOR.

OK - so now you know that, you can forget it again - that's something that only math teachers care about!

Oh - well I guess that means you'd better remember it after all. Sorry.

I'll be writing my fractions like this:

   1                 NUMERATOR
  ---              -------------
   2                DENOMINATOR

But that's the same thing as writing them like this:

   1 / 2          NUMERATOR / DENOMINATOR

It doesn't really matter which way you do it - but it's often better to think about 'the number on the top' and 'the number on the bottom' - and that's harder when you write them both on the same line. However, writing them both on the same line reminds us that a fraction is just a handy way of showing that one number is DIVIDED by the other.

Reducing a fraction to it's simplest form.

Find the biggest number that will divide EXACTLY into both the top and bottom of the fraction. Divide both top and bottom of your fraction by that number and it's as simple as it can ever be.

Sometimes you *think* you've found the biggest number to divide by - but it isn't utterly the biggest possible. Actually, that's OK, you just have to look again after you've simplified the fraction - and see if you can simplify it some more. Often it's easier to simplify in lots of little steps like that than it is to figure out what you should have divided by.

For example:

      60
    -----
     260

...you can easily see that you can divide them both by 10 because both numbers end in zero. This gives you:

       6
     ----
      26

...but when you check, you should see that you can divide both 6 and 26 by 2 and make the fraction even simpler:

       3
     ----
      13

It's almost always a good idea to simplify fractions when you can. If I told you that we had one hundred and twenty eight, two hundred and fifty sixths of a cake left over from supper yesterday, it wouldn't be as clear as if I had said that we had half a cake left - but they mean the same thing. We simplify fractions only to make life easier for us poor humans. Computers, calculators and mathematics don't care whether they are simplified or not.

Turning a fraction into a 'mixed' number.

Divide the top of the fraction by the bottom. You'll get an answer - usually with some remainder. The result of the division is the 'whole number' part of the mixed number - and the remainder from the division becomes the top part of the fraction. The number that was on the bottom of the fraction just stays there.

Example:

       210
      -----
       100

Dividing 210 by 100 gives you 2 remainder 10. So, our mixed number is:

                10
       2  and -----
               100

...but the fraction can be simplified by dividing top and bottom by 10 - so the answer is two and one tenth.

                1
       2  and ----
               10

Two things to notice about the original problem that can save you time:

• If the number on the top of the fraction is SMALLER than the one on the bottom - then the answer is just going to be a fraction with no whole number part - so there is no need to even try to turn it into a mixed number.
• If the remainder in that first division is zero - then the answer is a whole number with no fraction.

Multiplying two fractions.

Multiplying fractions is actually easier than adding them! So we'll do multiplication first. Here's how:

Multiply the two numbers on the tops of the fractions together and put them on the top of the answer, multiply the two numbers on the bottom of the fractions together and put them on the bottom of the answer.

When you're done, it's usually a good idea to simplify the answer in the usual way. You may be able to save yourself some work by simplifying the fractions BEFORE you multiply them.

For example, without simplifying, this would be a hard problem:

      123456       432454
     --------  x  --------
     1234560      4324540

...but you should easily be able to see that both fractions can be simplified to:

         1         1
       ----   x  ----
        10        10

...which is just:

         1 x 1
       ---------
        10 x 10

...which is:

           1
         -----
          100

Dividing two fractions.

Dividing fractions is actually even easier than dividing whole numbers!

Turn the second fraction upside-down - then multiply them.

So this...

        2                   3
       ---    divided by   ---
        3                   4

...is just this:

        2                   4
       ---  multiplied by  ---
        3                   3

...which is:

        2 x 4
       -------
        3 x 3

...which is:

          8
         ---
          9

A handy rule to know

You can "un-simplify" a fraction by multiplying it top and bottom by the same number.

You already know that you can simplify a fraction by dividing the top and bottom by the same thing - right?

         300        3
        -----  =  ----      (We just divided top-and-bottom by 100)
        1400       14

...doing that doesn't change how big the fraction is. Well, if simplifying it doesn't change how big it is - then un-simplifying it is also OK:

         3       300
       ----  =  -----    (We just multiplied top and bottom by 100)
        14      1400

That doesn't sound very useful - but wait for the next rule...

Adding two fractions.

Weirdly, adding fractions is MUCH harder than multiplying or dividing them.

The numbers on the BOTTOM of the two fractions have to be the same before we can add them. Once the numbers on the bottom ARE the same, we can just add the numbers on the top and leave the numbers on the bottom alone.

Here is an easy one where the numbers on the bottoms happen to be the same before we start work:

         5      4      5 + 4       9
       ---- + ----  =  -----  =  ----
        14     14       14        14

But what if the numbers underneath aren't the same? Well, we have to make them be the same using our handy simplifying and unsimplifying rules.

So, for example:

        4      5
       --- + ----
        7     14

We can't simplify 5/14 - but we can un-simplify 4/7 by multiplying top and bottom by 2 to make 8/14. Now we have two fractions which both have 14 underneath and we can just add them:

        4      5       8      5       8+5     13
       --- + ----  = ---- + ----  =  ----  = ----
        7     14      14     14       14      14

...as usual, you should try to simplify the answer - but in this case, it's already as simple as it can be.

Now, in this case, we were lucky and we could easily see what to multiply 7 by to make it into 14 so we could do the addition. You can't ALWAYS see how to do that - but fortunately, there is a rule you can use that always works - although it sometimes takes quite a bit more work...

Let's do it with letters instead of numbers:

       A     C
      --- + ---
       B     D

If we unsimplify A/B by multiplying both top and bottom by D - and then multiply C/D top-and-bottom by B then we get:

       A x D     C x B

      ------- + -------
       B x D     D x B

Now, the two fractions can always be added because BxD is the same as DxB.

So, it is always true that:

       A       C      A x D  +  C x B
      ---  +  --- =  -----------------
       B       D           B x D

WOW! We end up having to do three multiplications just so we can add two fractions! That's why it's worth looking for a short-cut way to make the bottom halves of the fractions be the same...but if you can't find a way, use the equation.

Let's see if it works...we all know that a half plus a quarter is three-quarters - right?

       1     1
      --- + ---
       2     4

...so changing the letters in the Equation into numbers gives us:

       1     1       1 x 4  +  1 x 2      6
      --- + ---  =  -----------------  = ---
       2     4            2 x 4           8

...and 6/8 can be simplified by dividing top-and-bottom by two - which gives us 3/4 - three quarters! Yeaaaahhhh!!

Subtracting two fractions.

This is exactly the same kind of thing as adding - and it's just as hard.

You have to get the bottom halves of the two fractions to be the same - then you can subtract the numbers on the top.

So:

        4      5       8      5       8-5      3
       --- - ----  = ---- - ----  =  ----  = ----
        7     14      14     14       14      14

Just like with adding, making the numbers on the bottom be the same isn't easy - so we need an Equation with letters in it.

       A       C      A x D  -  C x B
      ---  -  --- =  -----------------
       B       D           B x D

You can see it's exactly the same as the 'adding fractions' equation - except that there are '-' signs in this one instead of '+' signs.

Adding, Subtracting, Multiplying and Dividing 'mixed' numbers.

The trick here is to turn the mixed number back into an ordinary fraction again. To do that:

Multiply the whole number part by the bottom of the fraction - and then add that to the top part of the fraction.

For example, if you had:

            5
     3 and ---
            8 

You'd multiply the 3 by the 8 - and add it to the 5 to give:

      3 x 8  +  5      24 + 5       29
      -----------  =  --------  =  ----
           8              8          8

Once you have turned all your mixed numbers back into fractions, you can do the adding, subtracting, multiplying and dividing using all the rules you already know - and when you have the answer, you can simplify it and turn it back into a mixed number if you need to.

A fraction can't have Zero underneath.

It's not OK to have a fraction with zero underneath.

If you ever come across a fraction with zero underneath - like:

        1234
      -------
         0

...then you should probably scream and run away from it! It's NEVER possible for this to happen unless you have made a horrible mistake somewhere - because this peculiar fraction doesn't have 'an answer'. Try dividing 1234 by zero on your calculator. It'll probably show a little flashing 'E' somewhere - that means "ERROR".

Some people will tell you that this fraction is infinity - but even that isn't right. The answer is that this fraction just isn't allowed in mathematics.

There Is No Rule Eleven!

That's *IT*...a whole year of math classes squashed into ten handy rules! If you can remember them all, you can amaze your teacher. Most people are going to find fractions VERY hard - most adults have forgotten all about adding fractions. But if you can understand and learn these rules, you won't have any problems. However, you'll still need lots of practice - and you have to pass tests and make grades in school - so rule eleven is:

Keep doing the homework!