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. If you work through this carefully, you'll never have problems with fractions again.
Fractions are quite important - but nearly every adult has difficulties with them. Sure, you probably had it beaten into you in school - it probably took a whole year out of your life to learn them. But do you still know it all? You would if you used fractions every day - but if you don't, then everything you did in that year has gone from your head forever!
Someone once told me that anyone can memorize anything that can be written on a 3x5 file card - but they didn't say how small the lettering was allowed to be!
OK - that's all you'll have to learn - but you might need a bit more explanation if you want it to "stick"...and if you make it through to the end, I'll give you an even smaller "Cheat Sheet"!
In what follows, the actual rules are in bold-face type - all the rest is just explanation and some handy hints to save work.
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! Really, I work with fractions all the time, and I had to look it up.
I'll just call them "TOP" and "BOTTOM" - and I'll mostly 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.
1 ÷ 2 NUMERATOR ÷ DENOMINATOR
At it's heart - a fraction is just a division that we can't be bothered to do.
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.
So you probably know that four slices of an eight-slice pizza is just half of a pizza (unless you're one of those unprincipled "six-slicers"...I'm not sure I want to teach you fractions!).
--- = ---
You knew instinctively to "reduce" the 4/8ths fraction into the 1/2 fraction...but on a conscious level, what were you doing? You divided both the top and bottom of the 4/8 fraction by 4 and wound up with 1/2. So if I asked you to simplify 6/8ths you'd say 3/4ths because you divided top and bottom by two.
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.
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:
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 cold, hard mathematics don't give a damn whether they are simplified or not.
To simplify is human!
Sometimes - like if you're measuring a distance in ye-olde-worlde inches - you'll get a number like "ten and thee eighths inches":
That's called a "mixed number" because it's a mixture of a whole number and a fraction. Sometimes you need to turn a "top-heavy" fraction like "ten thirds" into a mixed number like "three and a third":
---- = 3 ---
Mixed numbers are easier on our poor brains - easier to get your head around.
Here's the rule:
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.
Sometimes the number on top is bigger than the number on bottom - so, for example 5/4ths of a dollar is really a buck and a quarter. A number with a 'whole number' part and a 'fraction' part is a "mixed number".
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:
Usually, multiplying numbers is harder than adding them...but with fractions, it's the other way around...sometimes math likes to joke around with us like that! So let's do multiplication of fractions first.
It's not always obvious when you'd want to multiply fractions - but if you have half of a cake and you have to share it three ways - then you need a third of a half of a cake - which is a third MULTIPLIED by a half...one sixth of a cake each. (HINT: Think BIG chocolate cake - with butter frosting. You can't do fractions with mere cupcakes!)
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 very hard problem:
123456 432454 53389041024 -------- x -------- = ------------- 1234560 4324540 5338904102400
...but you should be able to spot that by sheer luck (and the fact that I chose the numbers!) that both fractions could have been simplified to 1/10 so all we really need is to do:
1 1 ---- x ---- 10 10
...which is just:
1 x 1 --------- 10 x 10
1 ----- 100
But if you have a calculator handy, you can do the multiplications the hard way without simplifying first.
Dividing fractions is actually even easier than dividing whole numbers!
Turn the second fraction upside-down - then multiply them.
2 3 --- divided by --- 3 4
...is just this:
2 4 --- multiplied by --- 3 3
2 x 4 ------- 3 x 3
8 --- 9
It's not very common to need to divide fractions - but it comes up sometimes - and the rule is easy enough to remember.
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...
Weirdly, adding fractions is MUCH harder than multiplying or dividing them. I mean, who knows what a half a cake plus two fifths of a cake is? That's HARD! (OK I've gotta stop using food analogies - I'm starting to yearn for seven twentyninths of a cake right now!)
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 number 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.
Sometimes you can unsimplify one and simplify the other, maybe you can simplify the answer too?
4 10 8 5 13 --- + ---- = ---- + ---- = ---- 7 28 14 14 14
Now, in this case, we were lucky and we could easily see what to multiply 4/7 by two to make the bottom into 14 - and divide 10/28 by two to get 5/14 so that 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 unsimplify C/D by multiplying 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 B times D is the same number as D times B.
So, you can memorize this rule (or put it on the back of that 3x5 file card!):
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!!
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.
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.
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've 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.
It's really not easy to trip up mathematics...to confuse the system so badly that it can't give you a straight answer...to fundamentally wreck the fabric of the universe! But here it is!
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 are some weird and wonderful ways that our universe tries to avoid having fractions with zero underneath. The reason that nothing but light can travel at the speed of light, the reason black holes are black, the reason the big bang was able to be the source of everything...these are all places where the universe had to do something weird to "hide" the fact that it would otherwise have to divide by zero!
That's *IT*...a whole year of math classes squashed into ten handy rules! Actually, just nine if you've already forgotten what a "Denominator" is - and just 8 because you'll never really have to divide by zero. Multiplication and division are really just one rule - and so are addition and subtraction. Simplification and un-simplification are really the same rule, used backwards. So arguably, we're down to five rules.
So if you can remember just five rules - then you know everything there is to know about fractions. You're DONE.
Most adults find fractions VERY hard. But if you can understand and learn these rules, you won't have any problems. However, you'll still need lots of practice - so rule eleven is:
YOU NEED TO PRACTICE. Keep doing the homework!