Learn Fractions NOW:
Fractions for the Person Who Has Never Understood Math!
By Justin Ascott
NOW Books
An Imprint of Minute Help Press
www.minutehelpguides.com
©2011. All Rights Reserved.
Table of Contents
Introduction
Different Types of Fractions
How To Multiply and Divide Fractions
Multiplying Fractions
Dividing Fractions
Invisible Denominators
Reciprocals
Back to Dividing Fractions
Working with Complex Fractions
Simplification
Conclusion
How To Add and Subtract Fractions
Adding Fractions with Like Denominators
Subtracting Fractions with Like Denominators
Finding a Common Denominator
Adding Fractions with Unlike Denominators
Subtracting Fractions with Unlike Denominators
Simplification
Conclusion
Working with Mixed Numbers
Adding Mixed Numbers
Subtracting Mixed Numbers
Adding Mixed Numbers When the Fractions Add up to More Than 1
Subtracting Mixed Numbers When the Second Fractions Is “Too Big”
Multiplying Mixed Numbers
Dividing Mixed Numbers
Conclusion
Working with Fractions and Decimals
Converting Decimals to Fractions
Converting Fractions to Decimals
All fractions don't divide neatly into a decimal number.
Adding and Subtracting Decimal Numbers
Dividing Decimal Numbers
Conclusion
Conclusion
Introduction
Maybe you and fractions are not exactly best friends. It could be that you aren't even on speaking
terms. In fact, if you see fractions coming, you hide.
You aren't alone. Many people have had trouble with fractions.
But you have to understand fractions so that you can pass math. So what are you going to do?
Relax. Help is on the way.
Here's the beginning. Stop saying “I can't do fractions.” Say this instead: “I'm learning how to do
fractions.” It's true. You haven't finished learning yet.
But you will. You can learn fractions. You haven't done it yet, but you can and you will.
Different Types of Fractions
Words Used in This Lesson:
•
Fraction
•
Proper Fraction
•
Improper Fraction
•
Vulgar Fraction
•
Mixed Number
•
Invisible Denominator
•
Complex Fraction
You already know more about fractions than you think.
Did you ever share a cookie with a friend? Each of you got half, right? That's a fraction.
A fraction is about pieces of things like a cookie.
You divided your 1 cookie by 2 (for you and your friend).
1
2
So 1 divided by 2 equals one half.
1÷ 2=
That's what a fraction is. It's one number divided by another number that is not zero. We show it by
putting the number you want to divide – your one cookie – on top of the number you want to divide it
by – two friends (you and your friend).
The number on the top is the numerator,
and the number on the bottom is the denominator.
Why can't you use zero? Well, have you ever divided anything by nothing? How would you get an
answer? Nobody knows how it would happen, so a denominator can't be zero.
There are different kinds of fractions. We'll put the explanation of those different kinds right here, and
you can come back to them if you forget which one is which.
A proper fraction is probably the first kind of fraction that comes to your mind. In a proper fraction,
the numerator is smaller than the denominator.
1 5 3
Here are some examples of proper fractions: 2 6 7
If a proper fraction has a smaller numerator than denominator, what do you think an improper
fraction is? It's probably just what you guessed – a fraction where the numerator is larger than the
denominator.
5 9 11
Here are some examples of improper fractions: 2 5 10
A vulgar fraction is either a proper fraction or an improper fraction. In this case, “vulgar” doesn't
mean rude, crude, and socially unacceptable. It means “commonly used.” The examples you just saw
of proper fractions and improper fractions are examples of vulgar fractions, too. Nowadays we use the
terms “proper fraction” and “improper fraction,” so you won't hear the term “vulgar fraction” very
often.
If you had two cookies and divided them between yourself and three friends, each one would get half,
2
right? But what you're really doing is taking 2 and dividing it by 4. That's 4 . It's really the same
1
1
2
2 1
thing as 2 . So 4 = 2 . 4 and 2 are equivalent fractions, because they mean the same thing.
4 1 4 2 2 6
Here are some examples of equivalent fractions: 8 , 2 6 , 3 5 , 15
A mixed number is a little bit different, but you have still seen them before. It has a whole number,
5
5
like 3, and then a fraction, like 6 . So it means the whole number plus the fraction: 3 6 .
2
Here are some examples of mixed numbers: 2 3
4
3
7
12
9
10
When you multiply a number by its reciprocal, you get 1. You haven't learned how to do this yet, but
you can come back to this later just to get things solid in your mind.
2 5
1 2
3 4
,
,
Here are some examples of numbers and their reciprocals: 5 , 2
2 1
4 3
Did you know that every number can be a fraction? No, wait, don't thump your head on your desk. It's
not difficult to get this.
A fraction is any number divided by any other number, right? (Except for zero – don't forget, you can't
divide by zero.) And if you divide a number by 1, it stays itself.
Let's think about that. If you have two oranges, and you aren't going to share them, because you want
them both, you still have two oranges, don't you? You're really dividing those 2 oranges by 1 person.
And 2 divided by 1 = 2. That's why you call 1 an invisible denominator. If you see a whole number,
and you need to work with it as if it were a fraction, just divide that whole number by 1.
Here are some examples of whole numbers and how they look with invisible denominators:
6,
6
1
3,
3
1
15,
15
1
A complex fraction looks very strange at first. It's a fraction that has a fraction in the numerator or
denominator or both.
But wait! How can a fraction have another fraction inside it? Well, a fraction is just one number
divided by another, right? So why not divide a fraction by another number, or divide a number by a
fraction? (Don't worry about how to work them out yet. You'll learn how to deal with them later.)
1
1
5
3 2
2
7
Here are some examples of complex fractions: 1
8
3
2
Now that you have seen what different kinds of fractions are, you can go on to learning how to work
with them. You can always come back and look at this section if you need help!
How To Multiply and Divide Fractions
Words Used in This Lesson:
•
Invisible Denominator
•
Reciprocal
When you learned arithmetic, you learned how to add first, and then you learned how to subtract. After
that, you learned how to multiply and then to divide.
You add, subtract, multiply, and divide fractions, too. And there's good news!
Multiplying and dividing fractions are the easiest things to do with fractions!
First you'll go through a little background, and then you'll learn how to do fraction multiplication and
division problems. It's going to be a piece of cake! See, a piece of cake is a fraction, right? And it's
easy, right? And . . .
Okay, never mind. Back to fractions.
Multiplying Fractions
You remember how to multiply whole numbers – you just plain multiply, like this: 4× 3= 12
In other words, let's say you have a set of stickers. Each set has 4 stickers. You want 3 sets. How
many stickers would you have if you had 3 sets? 4 stickers x 3 sets = 12 stickers.
It's the same thing with fractions.
1
3
Imagine that you have 4 of a liter of soda, but you really want 2 of that. When you hear of with a
fraction, think multiply. (You'll need to remember this when you get to percentages. Just tuck it in the
back of your mind for now.)
1
3
3 1
So you need to multiply 4 by 2 – and it looks like this: 4 × 2
Okay, so you multiply them. The only thing that is different about multiplying fractions is that you
multiply the numerator by the numerator and multiply the denominator by the denominator.
Remember which is which?
The number on the top is the numerator,
and the number on the bottom is the denominator.
3 1 3× 1
Therefore, 4 × 2 = 4× 2
So far, that's good. But you probably knew that you couldn't leave it like that. You need to say what
3 x 1 is and what 4 x 2 is.
3× 1 3
=
4× 2 8
Seriously. That's all you do to multiply fractions.
Now do another problem just to get the whole process stuck firmly in your head.
3 2 3× 2
× =
7 5 7× 5
3× 2 6
=
7× 5 35
As you have seen, multiplying fractions is really easy.
Dividing Fractions
Dividing fractions is almost as easy. If you divide by a fraction, you're really multiplying by that
fraction's reciprocal.
What?
Okay, let's go over this bit by bit. You'll get it.
12
Think about how it works with whole numbers again. If you divide 12 by 3, it's like saying 3 . You
put the 12 in the numerator and the 3 in the denominator. Of course, 12÷ 3= 4 , so you're ending up
with a whole number.
The dividend is the numerator, or the number you start with. The divisor is the denominator, or the
number you divide by. The quotient is the answer.
12
1
Dividing 12 by 3 is also like multiplying 1 by 3 –
12 12 1
=
×
3 1 3
12 1 12
× =
1 3 3
Invisible Denominators
Remember the invisible denominator? When you divide any number by 1, it stays the same.
2
5
3
For example, 1 = 2 , 1 = 5 , and 1 = 3 – that last one looks familiar, doesn't it?
12 12 1
When you divided 12 by 3, you turned the problem into 3 = 1 × 3
In other words, you left the dividend alone. Well, okay, you gave it an invisible denominator, but
12
that didn't change the value of the dividend, because 1 = 12 .
3
Then you took the divisor, gave it an invisible denominator 1 = 3
3 1
and flipped it. 1 → 3
12 1 12
Then you multiplied the two numbers: 1 × 3 = 3
Reciprocals
When you flip a fraction, you get the fraction's reciprocal. You learned in “Kinds of Fractions” that
when you multiply a number by its reciprocal, you get 1.
Don't just take our word for it. Check it out!
5
Take the fraction 7 . Find its reciprocal. How do you do that? You flip it.
5 7
So the reciprocal of 7 is 5 .
5
7
If 7 and 5 are reciprocals, then multiply them and what do you get?
5 7 35
× =
7 5 35
35
But the answer is 35 . Is that 1?
Think about it. If you had a pizza with 35 pieces, and you had all 35 pieces, how many pizzas do you
have? Just 1 pizza.
If you divide a number by itself, you get 1.
35
=1
35
1,956
=1
1,956
6
=1
6
There's just one exception. You can't divide anything by 0. It doesn't work. Sorry, but that's life.
Back to Dividing Fractions
Try it with a problem where both the dividend and divisor are fractions.
2 7
÷
5 8
2
1. The first number (the dividend) just stays the same: 5
7
8
2. Take the second number (the divisor) and flip it: 8 → 7
3. Now you have a multiplication problem, so turn the division sign into a multiplication
2
8
sign: 5 × 7
4. Finally, just do the math:
2 8 2× 8
× =
5 7 5× 7
2× 8 16
=
5× 7 35
In other words, keep the dividend (the first number) the same and flip the divisor (the second number).
Then just multiply the two numbers. And remember – multiplying fractions is easy!
Working with Complex Fractions
Remember complex fractions? That's a fraction that has a fraction or mixed number in the numerator,
denominator, or both. They look like this:
2
5
3
7
2
9
7
239
3
7
Really, a complex fraction is just one number divided by another. So that first complex fraction is just
2
÷
5
2
÷
5
3
7 – and you know how to deal with that. Leave the dividend alone, flip the divisor, multiply.
3 2 7
= ×
7 5 3
2 7 2× 7
× =
5 3 5× 3
2× 7 14
=
5× 3 15
Try the second example –
2 2
9 9
=
7 7
1 There's that invisible denominator again.
2
9 2 1
= ×
7 9 7
1
Flip that divisor!
2 1 2× 1
× =
9 7 9× 7
2× 1 2
=
9× 7 63
You probably know how to do the third example, but let's go through it anyway.
239
239
1
=
3
3
7
7 It's the invisible denominator. We do keep running into them.
239
1
239 7
=
×
3
1
3
7
Flip that divisor!
239 7 239× 7
× =
1
3
1× 3
239× 7 1673
=
1× 3
3 The answer is an improper fraction, but we'll get to those later.
Simplification
When you divide by a fraction, you get a fraction for the quotient (the answer). Sometimes the
fraction is more complicated than it needs to be, so you have to simplify it.
Why would you want to do that? Well, like so many other things in math, it helps if you draw a
picture.
A pizza was cut into four pieces.
2
You get two of them. So you have 4 of a pizza.
2
Wait. Sure, you have 4 of a pizza. But are you going to call it that?
“Hey, man, whatcha got?”
“I got two fourths of a pizza, man.”
Probably not. You have half a pizza. I know it, you know it, everybody knows it.
1
2
You simplified 4 and you got 2 . Here's how.
It's all about factors. Remember factors? As in “the factors of 6 are 2 and 3”? Here's another way that
they come in handy: simplification. It works like this:
2 2× 1
=
4 2× 2
2× 1 2 1
= ×
2× 2 2 2
1× 6= 6
2
But remember – 2 is just another name for 1. And 1 x any number = that number.
1× 11= 11
1× 21,402= 21,402
So why bother to keep that 1 around? You wouldn't call 6 “1 x 6,” would you? No. You just call it 6.
2 1 1
× =
2 2 2 There. You've simplified it.
So how does this work when you multiply and divide fractions? Try it step by step.
First, multiply the numerator by the numerator and the denominator by the denominator.
5 2 5× 2
× =
6 7 6× 7
Then factor all the numbers. Make sure that every number is prime – that it has no other factors but
itself and 1. Keeping all the factors in place makes it easier to simplify later.
5× 2
5× 2
=
6× 7 2× 3× 7
Now put all the factors in numerical order.
5× 2
2× 5
=
2× 3× 7 2× 3× 7
This way, you can really see which numbers are in both the numerator and the denominator – and you
can factor them out.
2× 5
2
5
= ×
2× 3× 7 2 3× 7
2
You know that 2 is just a name for 1. You don't need a factor of 1 – it doesn't do anything useful. Get
rid of it!
2
5
5
×
=
2 3× 7 3× 7
Almost finished! Just multiply the factors that are left, and you're finished.
5
5
=
3× 7 21
Just to make sure you get it, try a division problem. Remember, multiplication and division are very
similar. The only thing you do differently is flipping the divisor (the second number). Start out by
doing that now.
7 5 7 6
÷ =
×
12 6 12 5
Multiply the numerator by the numerator and the denominator by the denominator.
7 6 7× 6
× =
12 5 12× 5
Now factor all the numbers that you can.
7× 6
7× 2× 3
=
12× 5 2× 2× 3× 5
Put the factors in numerical order.
7× 2× 3
2× 3× 7
=
2× 2× 3× 5 2× 2× 3× 5
Now that the factors are in order, you can really see which numbers are in both the numerator and the
denominator – and you can factor them out.
2× 3× 7
2 3
7
= × ×
2× 2× 3× 5 2 3 2× 5
2
3
2 and 3 are both names for 1. Why keep them around? That factor of 1 doesn't do anything useful.
Throw them out!
2 3
7
7
× ×
=
2 3 2× 5 2× 5
Now all you have to do is multiply the factors that are left!
7
7
=
2× 5 10
Conclusion
Some of you may be saying that you will never have to use this, because you will always have a
calculator. And it's true that usually, you will have a calculator. But calculators occasionally break. In
some cases, you aren't even sure that they break, because only a part of the calculations are wrong.
That's why you need to know how to check a calculator's answer.
For example, imagine that you are in charge of a motorcycle race. It's far away from cities where
nobody else ever goes. Everybody transports their motorcycles to the race with no gas in their tanks,
because that's better for these expensive machines. All the racers and spectators are there. You're all set
to start the race, but you find that there are only 8.5 quarts of gasoline for 6 motorcycles. That cuts
down the length of the race, of course – but more importantly, it means that you don't have enough gas
for everybody to fill their motorcycle's tanks. You have to divide 8.5 quarts between 6 motorcycles.
You'd better know how to check your calculator's answer so that all the bikers and spectators will be
satisfied, hadn't you?
How To Add and Subtract Fractions
Words Used in This Lesson:
•
Like Denominators
•
Unlike Denominators
•
Common Denominator
Adding Fractions with Like Denominators
Because fractions are numbers, they can be added to each other and subtracted from each other. But
how will you do that?
Start with something easy. That's a good way to do things. Pictures help, too.
1
1
You have 2 of an apple and your friend Emma has 2 of an apple.
1
This 2 is yours
1
and this 2 is Emma's.
1
2
You're really hungry, and Emma isn't. Emma gives you her 2 of an apple. Now you have 2 of an
apple. Put them together, and the apple looks like this:
1 1 2
You just showed that 2 + 2 = 2
2
2
You also know that 2 is the whole thing, 1 apple. So 2 = 1 .
5
7
103
Any number over the same number = 1. So 5 = 1 and 7 = 1 and 103 = 1
That's because you're dividing that number by itself.
If you are adding two fractions with the same denominator (also called like denominators),you just
add the numerators together and keep the denominator the same.
1 1 1+ 1
+ =
2 2
2
The two fractions have like denominators, so you can just add the numerators
and keep the denominator the same
1+ 1 2
=
2
2
Add the numerators
2
=1
2
2 halves equal 1 whole
As you have seen, when two fractions have the same denominator, we call that like denominators.
Think of “like” as the opposite of “unlike.”
1
3
For example, think of 5 and 5 .
These two fractions have like denominators, so you can add them the same way
1
1
that you added 2 and 2 .
1 1 1+ 1
+ =
2 2
2
The two fractions have like denominators,
1 3 1+ 3
+ =
5 5
5
1+ 1 2
=
2
2
1+ 3 4
=
5
5
so you can just add the numerators
and keep the denominator the same.
2
=1
2
4
5
But in the example on the right, the fractions
don't add up to a whole.
The answer is still a fraction.
Subtracting Fractions with Like Denominators
You do the same thing when you subtract fractions with like denominators – but, of course, you
subtract instead of add. Here's an example:
7 2 7− 2
− =
9 9
9
The two fractions have like denominators,
7− 2 5
=
9
9
so you can just subtract the second numerator from the first numerator
and keep the denominator the same.
Finding a Common Denominator
As you might imagine, the denominators of fractions that you are adding and subtracting are not always
2
1
the same or like. For example, you might need to add fractions like this: 3 + 12 The denominators
are not the same. What will you do?
In math, if you are not sure how to do a problem, do a simpler problem of the same kind. Drawing a
picture often helps, too. So here's a simpler problem.
1
1
You have 2 pizza and your mom gives you 4 pizza. How much pizza do you have now?
That's a simple problem. Now here are the pictures to go with it.
← Here's what you have
1
now – 2
1
Here's what your mom is giving you – 4
→
Put them together and you get this:
3
You know what this is. It's 4 pizza. But how do you figure this out mathematically? If you could
figure out how to do that, you could add any fractions.
You need to find a common denominator. “Common” doesn't mean “ordinary” here. It has the same
meaning as in “a friend in common.” If you have a friend in common with somebody else, you have
the same friend. So a common denominator is one that is the same for two (or more) fractions, as
many as you need to add or subtract.
1 1
Here's the pizza problem in mathematical form: 2 + 4 = ❑
You can't add these two fractions, because they don't have like denominators – their denominators
aren't the same. But when you look at the picture, you know what the answer is. How will you do it
mathematically?
You have to get like denominators. In other words, the denominators must be the same. Here's what
you can do:
5. Find the prime factors of both denominators.
1
2 already has only a prime factor in its denominator, 2.
1
1
1
=
4 has two prime factors in its denominator: 4 2× 2 Those factors are 2 and 2.
1
1
If you could get 2 to have the same denominator as 4 , you could add them.
1
6. Remember that if you multiply any number by 1, it stays the same? If you multiply 2 by a
1
name for 1, that fraction will still equal 2 , and the two fractions will have a common
denominator.
2
7. Any number divided by itself is 1. If you use 2 as that name for 1, you can get that common
denominator, because then both denominators will equal 2 x 2.
1 2 2
× =
2 2 4
Now both fractions have a common denominator: 4. So you can add them.
2 1 2+ 1
+ =
4 4
4
2+ 1 3
=
4
4
Adding Fractions with Unlike Denominators
You sneaked into this topic when you found a common denominator for that last problem!
Let's review what you just did, but with a different problem.
1. You looked at the fractions to add and saw that
5 2
+ =❑
12 9
the two denominators were not the same – they
were unlike denominators.
2. You don't like unlike denominators. Denominators
need to be the same – they need to be like denominators.
5
5
=
12 2× 2× 3
So you factored each denominator to see what factors
2
2
=
9 3× 3
needed to be in this like denominator – this common denominator.
3. You look at the factors.
12 = 2 x 2 x 3
9=3x3
You need to have all these factors in the common denominator.
12 has these factors: 2, 2, 3
9 has these factors: 3, 3
Your common denominator needs to have these factors: 2, 2, 3, 3
That way, it will have all the factors that 12 and 9 have.
So the common denominator for this problem is 2 x 2 x 3 x 3 = 36
4. To use this common denominator, multiply each denominator by a name for 1 that has the
factors it needs.
5
5
2
2
=
=
12 2× 2× 3
9 3× 3
5
5
3
2
2
2 2
=
×
=
× ×
2× 2× 3 2× 2× 3 3 You multiply by a name for 1.
3× 3 3× 3 2 2
In this case, you had to do that twice!
You choose the name or names for 1 that will
give both fractions a common denominator.
5.
6.
5
3 15
× =
2× 2× 3 3 36
15 8 15+ 8
+
=
36 36
36
15+ 8 23
=
36
36
Multiply the fractions.
2
2 2 8
× × =
3× 3 2 2 36
Now that both fractions have a common denominator,
you can add them!
Subtracting Fractions with Unlike Denominators
You do the same thing when you subtract fractions with unlike denominators – but, of course, you
subtract instead of add. Here's an example:
1. Look at the fractions' denominators. Are they the same?
No, they aren't. The two fractions have unlike denominators.
2. You don't like unlike denominators. Denominators
need to be the same – they need to be like denominators.
You must factor each denominator to see what factors
19 3
−
=❑
25 10
19 19
=
25 5× 5
3
3
=
10 2× 5
needed to be in this like denominator – this common denominator.
3. You look at the factors.
25 = 5 x 5
10 = 2 x 5
You need to have all these factors in the common denominator.
25 has these factors: 5, 5
10 has these factors: 2, 5
Your common denominator needs to have these factors: 2, 5, 5
That way, it will have all the factors that 25 and 10 have.
So the common denominator for this problem is 2 x 5 x 5 = 50
4. To make both factors have this common denominator, you will multiply each factor
by a name for 1.
19 19
3
3
=
=
25 5× 5
10 2× 5
19
19 2
3
3
5
=
×
=
×
5× 5 5× 5 2
2×
5
2×
5
5
You multiply by a name for 1.
In this case, you had to do that twice!
You choose the name or names for 1 that will
give both fractions a common denominator.
7.
8.
19 2 38
× =
5× 5 2 50
38 15 38− 15
−
=
50 50
50
38− 15 23
=
50
50
You multiply the fractions.
3
5 15
× =
2× 5 5 50
Now that both fractions have a common denominator,
you can do the subtraction!
Simplification
Remember simplification from multiplying and dividing fractions? Here it comes again. You already
know how to simplify, so this section won't be very long.
When you add or subtract fractions, you get a fraction for the sum or difference (answer). Sometimes
the fraction is more complicated than it needs to be, so you have to simplify it. Just as you learned
1
2
before, you don't call two quarter-pieces of anything 4 . Hardly ever, anyway. You call them 2 ,
2 1 2
because 4 = 2 × 2 . You have to find factors that are in both the numerator and denominator of the
2
fraction so that you can turn them into a name for one like 2 and get rid of them.
Here's an example of how simplification works when you add or subtract fractions.
1 1
1. You know how to start on this. The fractions' denominators are 2 + 6 = ❑
unlike. You must find a common denominator.
2. You factor each denominator to see what factors
need to be in the common denominator.
1
1
=
2 2× 1
1
1
=
6 2× 3
3. You look at the factors.
2 = 2 x 1 – 2 is prime, so it has no other factors.
6 = 2 x 3 – 6 one has two factors, 2 and 3.
Your common denominator needs to have these factors: 2, 3
That way, it will have all the factors that 2 and 6 have.
So the common denominator for this problem is 2 x 3 = 6.
4. To use this common denominator, multiply each denominator by a name for 1 that has the
factors it needs.
1 1 3
1
= ×
2 2 3
6 is just fine the way it is.
1 3 3
× =
2 3 6
5.
3 1 4
+ =
6 6 6
6.
4 2× 2
=
6 2× 3
1
For 2 , multiply by the name or names for 1
that will give both fractions a common denominator.
Now you have a common denominator, you can do the addition!
There is just one more step. You need to see if you can simplify the answer.
To do this, factor the numerator and denominator and see if they have any
factors in common.
Make sure that every number is prime – that it has no other factors but
itself and 1. Keeping all the factors in place makes it easier to simplify later.
7.
4
=
6
2
×
2
2
×
2
2
=
3
2
3
2
3
2
You know that 2 is just a name for 1. You don't need a factor of 1 –
it doesn't do anything useful. Get rid of it!
Conclusion
You can now add, subtract, multiply, and divide fractions. You can add and subtract fractions with like
and unlike denominators, and you can simplify the answer if necessary. You've learned a lot!
If you feel shaky about any of it, go back and do the steps again. Just as you become a better bike-rider
when you practice, practicing math will make it easier for you, too.
Working with Mixed Numbers
Words Used in This Lesson:
•
mixed numbers
You already know almost everything that mixed numbers do. Mixed numbers do all the same things
that fractions do, but you add whole numbers to the mix.
Adding Mixed Numbers
As you may remember, mixed numbers have both a fraction and a whole number in them.
They are numbers like this:
1
1
2
5
6
13
13
17
25
To add mixed numbers, add the fractions, then add the whole numbers, then put them together. Hold
on, don't fuss – of course there's going to be more explanation than that!
First, think about this in pictures. That's a good idea when you begin to work with a mathematical
concept. And make them simple – another good idea.
By the way, you should remember this when you are taking a test. If you can't remember exactly how
to do a problem, it might help to draw a picture of the problem. It can also help to do a simpler
problem to show how it's done, and then do the test problem the way the way the simpler problem is
done.
So here come the pictures:
1
1
You can probably see what this is – 1 2 + 1 3
1
1
First, add the fractions. 2 + 3
It's just a picture. We can move the fractions around any way we want to. Why not smash them
together? You could always pretend it's a video game or something. Okay, it's pretty lame for a video
game. Smash them together anyway.
1
1
Whoa, it even changed color! – All right, it didn't, that's just for fun. But this really is 2 + 3 so that
you can see it more clearly. You can't really tell what fraction it is, but you can see that it is close to 1.
That will help later on, when you work with this as a mathematical problem.
Now add the whole numbers. That part is pretty easy, right?
1+ 1= 2 So the answer is 2 . . . and a fraction. A big fraction. You'll see how to work that out on this
example. It's a lot like the picture.
1
1
1 +2 =❑
2
3
1 1
+ =❑
2 3
Start with the fractions.
Do the fractions have like denominators?
No, so you need to find a common denominator.
1
1
=
2 2× 1
1
=
2
1
×
2
1
×
2
3
=
3
3
3
3
6
3 2 3+ 2
+ =
6 6
6
3+ 2 5
=
6
6
You need a denominator with the factors 2 and 3.
Multiply each fraction by a name for 1 that includes
the factor that you are missing.
1
1
=
3 3× 1
1
=
3
1
×
3
Great, you found the common denominator!
Now add the fractions.
5
All right! You know that the fraction part of the answer is 6
1
×
3
2
=
2
2
2
2
6
– so go back to the whole numbers and add them.
1+ 2= 3
5
5
3+ = 3
6
6
Add the fraction, and you're finished with the problem!
Subtracting Mixed Numbers
Subtracting works just the same way, except that you subtract instead of add.
Adding Mixed Numbers When the Fractions Add up to More Than 1
Suppose that you add mixed numbers and you get a fraction that is too big for one number? Let's see
how that works.
This example will use mixed numbers where the fractions have common denominators. You know
how to add fractions with unlike denominators already. It will be the same deal for those, but let's start
with something a little easier.
2
4
Here's the example: 3 5 + 2 5 = ❑
2 4 2+ 4
+ =
5 5
5
2+ 4 6
=
5
5
Start with the fractions.
Hmm. That's a big fraction. It's bigger than 1 – it's an improper fraction.
Now what?
6 5 1
− =
5 5 5
You know it's larger than 1, so why not subtract a name for 1? Be sure to use
a name for 1 that has a common denominator with the big fraction.
1 5 1
+ = +1
5 5 5
1
1
+ 1= 1
5
5
You have to add that 1 back in, right? So there's your answer!
Subtracting Mixed Numbers When the Second Fractions Is “Too Big”
You can probably guess what you need to do when you have to do a problem like this:
1
4
3 −1 =❑
5
5
You're going to need to “borrow” a 1, just as you borrow in whole number subtraction.
1 4
− =❑
5 5
Do the fractions first. They're on the left.
3− 1= ❑
Put the whole numbers on the right.
5 1 4
( + )− = ❑
5 5 5
5+ 1 4 6 4
− = −
5
5 5 5
6 4 1
− =
5 5 5
1
1
+ 1= 1
5
5
Add a name for 1 to the first number. Be sure to
(3− 1)− 1= 2− 1
use the right common denominator. And don't
(3− 1)− 1= 2− 1
forget to subtract it from the first whole number.
2− 1= 1
Now add the fraction and the whole number – and you're finished!
Multiplying Mixed Numbers
To multiply mixed numbers, you need to convert them into improper fractions. Then you multiply the
improper fractions – as you know, multiplying fractions is the simplest thing about fractions. To finish
up, you make the improper fraction back into a mixed number.
2
1
Let's start with a simple example, as usual. 1 3 × 3 4 = ❑
2
1
1 =❑
3 =❑
First, convert each mixed number into an
3 3
4 4
improper fraction. Put the first number on the
left and the second number on the right.
1= ❑
3
1=
3
3
Each whole number needs to become part of the
fraction. You will convert the whole number
to an improper fraction with the same
common denominator as the proper fraction.
So, on the left, turn 1 into an improper fraction.
How many thirds are in 1? You know that!
On the right, turn 3 into an improper fraction.
This is a little tougher, but you can still do it.
3= ❑
4
4
If 1= 4 then how much is 3?
4 3× 4
=
4
4
3× 4 12
=
4
4
3×
3 2 3+ 2
12 1 12+ 1
+ =
+ =
Add
the
proper
fractions
back
to
the
whole
3 3
3
4 4
4
3+ 2 5
12+ 1 13
=
=
numbers that were turned into improper fractions.
3
3
4
4
5 13 5× 13
× =
Now set the problem up for multiplication.
3 4 3× 4
5× 13 65
=
3× 4 12
Now you have the answer – but you need to put it back into mixed number form.
65
= 5.4166667
Here's the easy way to do that. Use your calculator to do the division.
12
5× 12
12
5× 12 60
=
12
12
65 60 5
−
=
12 12 12
5=
5
5
5+
=5
12
12
In this problem, as with many others, you will get a fraction that never ends.
Don't worry about that. Just keep the whole number, which is 5.
12
Now figure out how many that is over 12. Remember that 1= 12
Subtract that number from the improper fraction,
and you have the fraction that goes with this mixed number.
Add 5 to the fraction, and there you go!
Dividing Mixed Numbers
As an example for dividing mixed numbers, let's use the mixed numbers we already converted to
improper fractions in the last section.
2
1 5 13
1 ÷3 = ÷
3
4 3 4
5 13 5 4
÷
= ×
3 4 3 13
Leave the dividend alone and flip the divisor.
5 4 5× 4
×
=
3 13 3× 13
5× 4 20
=
3× 13 39
Do the math – and you're finished!
Conclusion
Look at that! You've learned how to add, subtract, multiply, and divide mixed numbers. There are a
lot of steps to some of this stuff, but you got it! And if you didn't, you can go over these steps again.
Use pictures, use simple problems, use whatever you have to – you can learn it!
Working with Fractions and Decimals
Words Used in This Lesson:
•
Decimal
•
Decimal Point
•
Decimal Place
You already know what a fraction is – and you don't know it, but you already know what a decimal is,
too. Any time you talk about or work with dollars, you're really talking about or working with
decimals.
Decimals are numbers with a period in them. We call that period a decimal point.
Here are some examples of decimals:
1.7 $6.98 3.14159
You can call numbers like this either decimals or decimal numbers.
Okay, but what do they really mean?
A number like 3.8 really has two parts. It has the 3, and you know what that means. Then there's the
.8, which you can read either as “point 8” or “and eight tenths.”
8
4
.
4
Hun
dred
ths
Tho
usa
ndth
s
Ten
-tho
usa
ndth
s
Ten
ths
6
s
One
1
s
Ten
s
dred
Hun
nds
usa
Tho
nds
usa
-tho
Ten
3
Decimal point
Here's a picture to help you understand more about decimals.
7
9
2
The number shown here is read like this:
“Thirty-one thousand, six hundred eighty-four and four thousand seven hundred ninety-two tenthousandths” or
“Thirty-one thousand, six hundred eighty-four point four seven nine two”
The words and and point are the way the decimal point is read.
Converting Decimals to Fractions
Look at the picture again – look at the labels for the numbers to the right of the decimal point again.
Tenths
Hundredths
Thousandths
Ten-thousandths – they're all fractions. They're written differently, but they're the same numbers.
Take the numbers to the right of the decimal point apart and you'll see it even more clearly.
4,792
.4792=
10,000
Look at a few more examples and you'll understand how decimals can translate into fractions.
36
367
3,671
3
.3=
.36=
.367=
.3671=
100
1,000
10,000
10
Remember that 0 can hold a place, too.
3
10
.3=
.03=
3
100
.003=
3
1,000
.0003=
3
10,000
If you count the numbers after the decimal point, you'll see how many decimal places the number has.
For example, 3.76 has 2 decimal places.
4,697.2 has 1 decimal place.
134.43027 has 5 decimal places.
The number of decimal places shows you what fraction you would use either to read the number or to
convert it into a fraction. For every decimal place, you need one zero in the fraction's denominator –
put a 1 in front of all the zeroes and you'll get the right fraction.
Look at that line at the top of the page again.
.3=
3
10 = three tenths
.36=
36
100 = thirty-six hundredths
.367=
367
1,000 = three hundred sixty-seven thousandths
.3671=
3,671
10,000 = three thousand six hundred seventy-one ten-thousandths
So converting decimals to fractions just depends on how many decimal places there are. Use as many
0s as there are decimal places, then put a 1 in front of the 0s.
Converting Fractions to Decimals
Going the other way around, from fractions to decimals, is a different matter, but in some ways it's
even easier. All you need to do is divide – then, if necessary, round off to the number of decimal
places you want.
3
For example, to convert 4 to a decimal number, just divide 3 by 4. If you use a calculator, this part is
5
3
really easy. 4 = .75 And to convert 8 to a decimal number, pull that calculator out again and you'll see
5
that 8 = .625 – but
All fractions don't divide neatly into a decimal number.
2
Try converting 3 to a decimal. You get .666666666 . . . it just goes on and on. It isn't ever going to
stop. What do you do?
You have to decide how many decimal places you need, and then you round to that number of places.
Let's say you want 3 decimal places. In that case, .666666666... turns into .667.
5
Here's another example. To convert 7 to a decimal number with three places, first do the division.
You get .714285714285, and those last 6 decimal places repeat forever. All you want is 3 decimal
places, so you round .714285714285 to .714.
Adding and Subtracting Decimal Numbers
You already know how to add whole numbers, and you've learned to add fractions. Adding decimal
numbers is more like adding whole numbers than adding fractions, but there are a couple of things to
remember.
The first thing to do is keep the decimal points lined up. So if you are adding 3.14 and 14.82, just do
this:
3.14
+14.82
17.96
See how the second decimal point is right underneath the first one? And the decimal point in the
answer is lined up, too.
Keep them lined up even if the numbers do not have the same number of decimal places. Here's how
that works: 2.983
25.872
+32.9542
-3.36
35.9372
22.512
If your addition or subtraction gives you an answer that ends in 0, most math books tell you to drop it.
Math teachers differ on this. However, 3.10 = 3.1 and 2.567000 = 2.567, so dropping or keeping the 0s
makes no real difference in the number.
Here are two examples of how you could start off with (for example) 3 decimal points but end up
with 2.
19.55
83.92
+7.05
-78.02
26.60 or 26.6
5.90 or 5.9
Multiplying Decimal Numbers
Another task that is similar to the same operation with whole numbers is multiplying decimal numbers.
Your calculator can do all this, of course, but it's useful to know how to do it yourself.
When you multiply decimal numbers, do exactly what you do with whole numbers, but – add up the
number of decimal places in both original numbers, then put that number of places after the decimal
point. Here's an example:
248.82
x12.3
Do the math as usual, keeping the numbers lined up correctly by putting
74646
placeholder 0s in the second and third multiplication lines.
497640
2488200
3060486
When you have the answer, count the number of decimal places in the original
numbers. 248.82 has 2 decimal places, and 12.3 has 1 decimal place. 1 + 2 = 3.
3060.486
321
You had a total of 3 decimal places in the original numbers. Starting from the
right end, count over 3 decimal places, then put your decimal point there.
Try another one.
8372.29
x382.268
6697832
50233740
167445800
1674458000
66978320000
251168700000
320045855372
54321
3200458.55372
Yes, it's pretty big! You need to know how to do the big problems, too.
How many decimal points are in the original 2 numbers?
2 decimal points are in the first number
+3 decimal points are in the first number
5
Count 5 decimal places from the right, and put the decimal point there.
Dividing Decimal Numbers
When you divide, decimal numbers behave a little differently. The divisor needs to have no decimal
places at all. To do that, your decimal place will bounce over as many places to the right as it needs to
in both the divisor and the dividend. Then you do the math.
That sounds a little complicated. Let's see an example. The dividend is the number under the division
sign; the divisor is the number outside it. The quotient is the answer.
_____
1.2 ) 16.8
____
12 ) 168
We need the divisor (1.2) to be a whole number, to have no decimal
places at all. So let's bounce that decimal place in 1.2 to the right 1 place.
To match that, we need to bounce the decimal point in 16.8 to the right
1 place, too.
All set! Now do the math.
7_
12 ) 168
You probably know what's next – let's try another example! 7.4 * 20.25 = 157.2825
____
2.4 ) 27
_____
2.4 ) 27.000
150.9/
The divisor needs to be a whole number, so we need to bounce its
decimal place over to the right 1 place.
But wait! We can't! There are no decimal places in the dividend
(27) at all! What will we do?
Remember that 27 = 27.0 = 27.00 = 27.00000000. We can keep
putting 0s after a decimal point forever, and it's still the same number.
So let's put some decimal places on it.
______
24 ) 270.00
11.25
24 ) 270.00
Now we have enough decimal places! Let's bounce that decimal
place in the divisor (2.4) over to the right one place. Then it will
be a whole number.
Now – do the math!
Conclusion
Now you know how to do the following tasks:
1. Turn decimals into fractions
2. Turn fractions into decimal numbers
3. Round off fractions that you turned into decimals, when necessary
4. Multiply using decimal numbers
5. Divide using decimal numbers
You're really making progress!
Conclusion
Congratulations! You now know a lot more about fractions than you did before.
If you take fractions (or anything) in little bites, you probably find yourself learning more than you had
expected was even possible.
So remember, when you want to learn something new, especially if this is something that has given you
trouble in the past – do two things.
1. Do it in little pieces. Don't try to do too much at one time.
2. Keep telling yourself that you can learn it and that you are learning it. That will help you a
great deal.
Good work!
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