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 You Can Do Math:
Working With Fractions
Copyright & Other Notices
Published in 2014-2015 by Answers 2000 Limited
Copyright © 2014-2015, Sunil Tanna
Sunil Tanna has asserted his right to be identified as the author of this Work
in accordance with the Copyright, Designs, and Patents Act 1988.
Information in this book is the opinion of the author, and is correct to the
best of the author's knowledge, but is provided "as is" and without warranty
to the maximum extent permissible under law. Content within this book is
not intended as legal, tax, financial, medical, or any other form of
professional advice.
While we have checked the content of this book carefully, in any
educational book there is always the possibility of typographical errors, or
other errors or omissions. We apologize if any such errors are found, and
would appreciate if readers inform of any errors they might find, so we can
update future editions/updates of this book.
Answers 2000 Limited is a private limited company registered in England
under company number 3574155. Address and other information about
information about Answers 2000 Limited can be found at
www.ans2000.com
Updates, news & related resources from the author can be found at
http://www.suniltanna.com/fractions
Information about other math books by the same author can be found at
http://www.suniltanna.com/math

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Introduction
For some time now, I have tutored both children and adults in math and
science. This book is based on my own personal experience as a tutor, and
is one of a series of books on different topics:

If you want to find out about the other math books that I have
written, please visit: http://www.suniltanna.com/math
For science books that I have written, visit:
http://www.suniltanna.com/science

Anyway, let's get started with the topic of this book... understanding and
working with fractions – also sometimes known as "simple fractions ",
"common fractions ", or "vulgar fractions ".

Required Skills Before Learning Fractions:

Some very basic math skills are required before starting to learn fractions.
These are:

(1) Being able to count confidently in whole numbers, preferably up to at

least 1,000.

(2) Being able to add, subtract, multiply, and divide whole numbers.

(3) Familiarity with times tables – the better and more fluent that a student
is at their times tables, the easier that they will find fraction calculations.
To learn times tables, I suggest my book, Teach Your Kids Math:
Multiplication Times Tables , but frankly, it doesn't matter how a student
learns times tables – just so long as they do learn them.

(4) The last two chapters of this book explain how to convert fractions into
decimal numbers and vice-versa. Thus, although an understanding of
decimal numbers is not essential for most of the book, it is needed for these
two chapters. If you (or your child) isn't yet familiar with decimal numbers,
my suggestion would be to skip these topics for now, and return to them at
some point in the future after learning decimals .

You should therefore make sure that you (or your child) is familiar and
comfortable with these basics, before attempting to learn (or teach your
child) fractions!

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The Structure of This Book:
This book is intended as a guide to help adults and teens to learn and master
fractions, and for parents and teachers wishing to help younger children
learn fractions.

In the case of younger children - the introduction to this book,

which you are reading now, is obviously for the parent, guardian
or teacher.

The rest of the book is written for the learner – although in the
case of younger children, my suggestion would be that the
teacher/parent works carefully through each chapter together
with the child. These chapters address the learner/the child
directly as "you". I would recommend that the parent or teacher
slowly read aloud from each chapter to the child – with a pen
and paper handy, so any difficult points can be gone over
immediately.

When using this book, it is important to practice each technique

that you learn – as you learn it – so I have included questions
within each chapter at appropriate points. The answers to these
questions can be found at the end of that chapter.

There are many online resources, as well as books and other products that
may help with learning fractions. I have placed more information about, and
links to, many of these resources at http://www.suniltanna.com/fractions
Learning Objectives for This Book:
By the end of this book, you (or your child) should be able to:

Know what a fraction is

Understand and work with both mixed numbers and top heavy
("improper") fractions – including being able to convert between
the two
Understand and work with equivalent fractions
Compare and simplify fractions
Perform additions involving fractions
Perform subtractions involving fractions
Perform multiplications involving fractions
Perform divisions involving fractions (including simplifying

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 "complex" fractions)
Convert a fraction to a decimal number*
Convert a decimal number to a fraction*

* If you (or your child) is not yet familiar with decimal numbers, I would
suggest you omit these topics for now – you can always return to them in
future after learning decimals .

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Chapter 1: What is a Fraction?

A. Introducing Fractions
A fraction, sometimes known as a "common fraction ", "simple fraction "
or a "vulgar fraction " is the way in math that we represent dividing
something up into equal-sized pieces.

Imagine, for example that four friends (Aaron, Byron, Charles, and
Dominic) decide to share a pizza. They would cut the pizza into four equal
slices as shown below, and give one slice to each of the friends as shown in
the diagram below:

If we wanted to represent the amount of pizza that each of the friends got in
mathematics then we would use fractions.

Here is the amount of pizza that Aaron got:

In mathematics,
the amount of pizza that Aaron got is written as (sometimes also written
as 1/4). This means the pizza was divided into 4 equal-sized pieces, and
Aaron got 1 of those 4 pieces.

Here is the amount of pizza that Byron got:

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In mathematics, the amount of pizza that Byron got is also written as .
Again, this is because the pizza was divided into 4 equal-sized pieces, and
Byron got 1 of those 4 pieces.

Here is the amount of pizza that Charles got:

In mathematics, the amount of pizza that Charles got is also written as .

Again, this is because the pizza was divided into 4 equal-sized pieces, and
Charles got 1 of those 4 pieces.

Here is the amount of pizza that Dominic got:

In mathematics, the amount of pizza that Dominic got is also written as .

Again, this is because the pizza was divided into 4 equal-sized pieces, and
Dominic got 1 of those 4 pieces.

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How much pizza did Aaron, Byron and Charles get between them? Here is
the amount of pizza that these three friends got:

In mathematics, the amount of pizza that these three friends got is written as
. This is because the pizza was divided into 4 equal-sized pieces, and
the three friends got 3 of the 4 pieces.

In each case:

The bottom number in the fraction, known as the "denominator ",

represents the number of equal-sized pieces that the pizza is
divided into.
The top number in the fraction, known as the "numerator ",
represents the number of those pieces that the particular person
(or group of people) gets.

Imagine now that the 4 friends are joined by 2 more people, Edward and
Frank, just in time for a second round of pizza. This new pizza is served
and is cut into 6 slices.

What fraction of a pizza will each person now get?

Here's the amount of pizza that Aaron will now get:

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 In mathematics,
the amount of pizza that Aaron now gets is written as (sometimes written
as 1/6). This means the pizza was divided into 6 equal-sized pieces, and
Aaron got 1 of those 6 pieces.

What about Byron? How much pizza will he now get?

Once again the

amount of pizza that Byron now gets is written as – meaning that the pizza
was divided into 6 equal-sized pieces, and Byron got 1 of those 6 pieces.

I could repeat the process for each of the other people – Charles, Dominic,
Edward, and Frank – but hopefully you will be able to see that each of these
people also will now get of a pizza. In each case:

The bottom number in the fraction, the denominator, 6, indicates

that the pizza has been divided into 6 equal pieces.
The top number in the fraction, the numerator, 1, indicates that the
person gets 1 of those pieces.

Let's imagine that Frank says that he isn't hungry, and gives his piece to
Edward. What fraction of the pizza will Edward now get?

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Edward now gets (also sometimes written as 2/6).

The bottom number in the fraction, the denominator, 6, indicates

the pizza has been divided into six equal pieces.
The top number in the fraction, the numerator, 2, indicates that
Edward gets 2 of those pieces.

Let's review what we have learned so far, by trying some questions.

Questions:

1. Shade the part of the shape that corresponds to the fraction.

Example Question: Shade of this shape:

Answer: (any answer with 3 sections shaded, and 2 unshaded is

correct)

Now you try:

1. (a) Shade of this shape:

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1. (b) Shade of this shape:

1. (c) Shade of this shape:

1. (d) Shade of this shape:

1. (e) Shade of this shape:

1. (f) Shade of this shape:

1. (g) Shade of this shape:

1. (h) Shade of this shape:

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1. (i) Shade of this shape:

1. (j) Shade of this shape:

1. (k) Shade of this shape:

1. (l) Shade of this shape:

1. (m) Shade of this shape:

1. (n) Shade of this shape:

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1. (o) Shade of this shape:

2. Identify the shaded fraction.

Example Question: What fraction of the shape is shaded?

Answer:

Now you try:

2. (a) What fraction of the shape is shaded?

2. (b) What fraction of the shape is shaded?

2 . (c) What fraction of the shape is shaded?

2. (d) What fraction of the shape is shaded?

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2. (e) What fraction of the shape is shaded?

2. (f) What fraction of the shape is shaded?

2. (g) What fraction of the shape is shaded?

2. (h) What fraction of the shape is shaded?

2. (i) What fraction of the shape is shaded?

2. (j) What fraction of the shape is shaded?

2. (k) What fraction of the shape is shaded?

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2. (l) What fraction of the shape is shaded?

2. (m) What fraction of the shape is shaded?

2. (n) What fraction of the shape is shaded?

2. (o) What fraction of the shape is shaded?

B. Names of Fractions

Now you know how to write fractions such as , , , etc., you may be
wondering how to say the names of these fractions.

The most common way to say the names of fractions is to first say the value
of the numerator (the top number), and then say the value of the denominator
(the bottom number). However, the denominator is usually given as an
ordinal number (third, fifth, sixth, etc.).

For example:

is pronounced as "one sixth".

is pronounced as "two sixths".

is pronounced as "one eighth".

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 is pronounced as "three eighths".

There are also a couple of special cases however to be aware of:

If the denominator is 2, instead of saying "second", we say "half"

(plural: "halves").
If the denominator is 4, many people will say "quarter" rather
than "fourth". "Quarter" is generally preferred in British and
World English, however in American English both "fourth" and
"quarter" are acceptable.

Here is a list of the words used for some of the most common values of the
denominator:

2 – "half" (plural: "halves")

3 – "third"
4 – "quarter" (British and World English) or "fourth" (American
English)
5 – "fifth"
6 – "sixth"
7 – "seventh"
8 – "eighth"
9 – "ninth"
10 – "tenth"
12 – "twelfth"
15 – "fifteenth"
16 – "sixteenth"
20 – "twentieth"
32 – "thirty second"
100 – "hundredth"
1000 – "thousandth"
10000 – "ten thousandth"
1000000 – "millionth"

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Now let's try some questions.

Questions:

3. Write the fraction corresponding to the English phrase.

Example Question: Write three tenths as a fraction.

Answer:

Now you try:

3. (a) Write three eighths as a fraction.

3. (b) Write one eighth as a fraction.
3. (c) Write nine tenths as a fraction.
3. (d) Write three quarters as a fraction.
3. (e) Write one half as a fraction.
3. (f) Write one quarter as a fraction.
3. (g) Write two fifths as a fraction.
3. (h) Write four sevenths as a fraction.
3. (i) Write five ninths as a fraction.
3. (j) Write two thirds as a fraction.

4. Write the English phrase corresponding to the fraction.

Example Question: What is the English phrase for ?

Answer: "five sevenths"

Now you try:

4. (a) What is the English phrase for ?

4. (b) What is the English phrase for ?

4. (c) What is the English phrase for ?

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4. (d) What is the English phrase for ?

4. (e) What is the English phrase for ?

4. (f) What is the English phrase for ?

4. (g) What is the English phrase for ?

4. (h) What is the English phrase for ?

4. (i) What is the English phrase for ?

4. (j) What is the English phrase for ?

5. Write the English phrase corresponding to the fraction.

Example Question: What is the English phrase for the shaded part of the
shape?

Answer: "five sixths"

Now you try:

5. (a) What is the English phrase for the shaded part of the shape?

5. (b) What is the English phrase for the shaded part of the shape?

5. (c) What is the English phrase for the shaded part of the shape?

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5. (d) What is the English phrase for the shaded part of the shape?

5. (e) What is the English phrase for the shaded part of the shape?

5. (f) What is the English phrase for the shaded part of the shape?

5. (g) What is the English phrase for the shaded part of the shape?

5. (h) What is the English phrase for the shaded part of the shape?

5. (i) What is the English phrase for the shaded part of the shape?

5. (j) What is the English phrase for the shaded part of the shape?

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C. Fractions as Division
When we think of fractions we often think of them as being the same as
dividing an object into equal-sized pieces, and then taking one or more of
those pieces.

The fraction is the same as dividing an object into 4 equal

pieces, and then taking 1 of those pieces. In other words, is
the same as dividing 1 by 4.

Of course, we can choose to take any of the four pieces, so each of the
following is also true:

The fraction is the same as dividing an object into 4 equal

pieces, and then taking 3 of the pieces. In other words, is the
same dividing 1 by 4 (to get ), and then multiplying by 3 (to get
).

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 Of course, we can choose any three pieces, so each of the following
is also true:

But there is also another way to look at things when the numerator (the top
number) is more than 1. Instead of thinking of as being division of 1
object into 4, followed by multiplication by 3, we can instead think of it as
division of 3 objects by 4:

If we divide each of 3 objects into 4 pieces, then we get from

each of the 3 objects.

Combining (adding) these pieces back together, gives us .

This is illustrated in the following diagram:

We can write these relationships in mathematical symbols:

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Or, in words:

One divided by four is one quarter (or one fourth).

One quarter added to another quarter added to another quarter is
three quarters (one fourth added to another fourth added to
another fourth is three fourths)
Three divided by four is three quarters (or three fourths)

One way to visualize the fact that is the same as 3 ÷ 4 is to consider the
following diagram. In the picture, you can see that there are 3 objects
divided equally between four colors (red, blue, yellow and green).

As you can see, the red colored pieces form of one object:

Likewise, the blue colored pieces can be put together to form of one
object:

Likewise the yellow colored pieces can be put together to form of one
object:

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And, of course, the green colored pieces form of one object:

Questions:

6. Convert fractions into divisions.

Example Question: convert to a division

Answer: 3 ÷ 8

Now you try:

6. (a) Convert into a division.

6. (b) Convert into a division.

6. (c) Convert into a division.

6. (d) Convert into a division.

6. (e) Convert into a division.

6. (f) Convert into a division.

6. (g) Convert into a division.

6. (h) Convert into a division.

6. (i) Convert into a division.

6. (j) Convert into a division.

7. Convert divisions into fractions

Example Question: Convert 5 ÷ 9 into a fraction.

Answer:

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Now you try:

7. (a) Convert 6 ÷ 11 into a fraction.

7. (b) Convert 5 ÷ 12 into a fraction.
7. (c) Convert 3 ÷ 5 into a fraction.
7. (d) Convert 2 ÷ 7 into a fraction.
7. (e) Convert 3 ÷ 10 into a fraction.
7. (f) Convert 3 ÷ 4 into a fraction.
7. (g) Convert 7 ÷ 8 into a fraction.
7. (h) Convert 3 ÷ 7 into a fraction.
7. (i) Convert 4 ÷ 5 into a fraction.
7. (j) Convert 4 ÷ 7 into a fraction.

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Answers to Questions in Chapter 1

1. (a) (any answer with 1 section shaded and 3 unshaded is

correct)

1. (b) (any answer with 1 section shaded and 2 unshaded is

correct)

1. (c) (any answer with 3 sections shaded and 1 unshaded is

correct)

1. (d) (any answer with 2 sections shaded and 3 unshaded is

correct)

1. (e) (any answer with 3 sections shaded and 2 unshaded is

correct)

1. (f) (any answer with 2 sections shaded and 2 unshaded is

correct)

1. (g) (any answer with 3 sections shaded and 1 unshaded is

correct)

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1. (h) (any answer with 1 section shaded and 5 unshaded is
correct)

1. (i) (any answer with 5 sections shaded and 3 unshaded is

correct)

1. (j) (any answer with 4 sections shaded and 2 unshaded is

correct)

1. (k) (any answer with 1 section shaded and 1 unshaded is

correct)

1. (l) (any answer with 1 section shaded and 1 unshaded is

correct)

1. (m) (any answer with 2 sections shaded and 1 unshaded is

correct)

1. (n) (any answer with 4 sections shaded and 1 unshaded is

correct)

1. (o) (any answer with 4 sections shaded and 1 unshaded is

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correct)

2. (a)

2. (b)

2. (c)

2. (d)

2. (e)

2. (f)

2. (g)

2. (h)

2. (i)

2. (j)

2. (k)

2. (l)

2. (m)

2. (n)

2. (o)

3. (a)

3. (b)

3. (c)

3. (d)

3. (e)

3. (f)

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3. (g)

3. (h)

3. (i)

3. (j)
4. (a) two ninths
4. (b) three sevenths
4. (c) one fifth (or a fifth)
4. (d) two quarters (or two fourths)
4. (e) four tenths
4. (f) one quarter (or a quarter)
4. (g) one third (or a third)
4. (h) one half (or a half)
4. (i) four sevenths
4. (j) eight ninths
5. (a) one half (or a half)
5. (b) one half (or a half)
5. (c) three quarters (or three fourths)
5. (d) three quarters (or three fourths)
5. (e) three quarters (or three fourths)
5. (f) five eighths
5. (g) three fifths
5. (h) three fifths
5. (i) two thirds
5. (j) two thirds
6. (a) 7 ÷ 10
6. (b) 1 ÷ 4
6. (c) 5 ÷ 8
6. (d) 1 ÷ 2

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6. (e) 2 ÷ 3
6. (f) 3 ÷ 5
6. (g) 3 ÷ 7
6. (h) 3 ÷ 8
6. (i) 6 ÷ 7
6. (j) 6 ÷ 11

7. (a)

7. (b)

7. (c)

7. (d)

7. (e)

7. (f)

7. (g)

7. (h)

7. (i)

7. (j)

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Chapter 2: Mixed Numbers and Top Heavy Fractions

A. Introducing Mixed Numbers

Please take a look at the following diagram:

If we want to represent this as a number, we can think of it as wholes

plus . We say this as "two and three quarters", and we write this as .

This type of combination of a whole number and a fraction is known as a

"mixed number ". This is because it contains a mixture of a whole number
and a fraction.

Questions:

1. Identify the fraction as a mixed number.

Example Question: Identify the mixed number represented by the shaded

parts of the shapes:

Answer:

Now you try:

1. (a) Identify the mixed number represented by the shaded parts of the
shapes:

1. (b) Identify the mixed number represented by the shaded parts of the

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shapes:

1. (c) Identify the mixed number represented by the shaded parts of the
shapes:

1. (d) Identify the mixed number represented by the shaded parts of the
shapes:

1. (e) Identify the mixed number represented by the shaded parts of the
shapes:

1. (f) Identify the mixed number represented by the shaded parts of the
shapes:

1. (g) Identify the mixed number represented by the shaded parts of the
shapes:

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1. (h) Identify the mixed number represented by the shaded parts of the
shapes:

1. (i) Identify the mixed number represented by the shaded parts of the
shapes:

1. (j) Identify the mixed number represented by the shaded parts of the
shapes:

B. Introducing Top Heavy Fractions

There is also another way to think of mixed numbers.

Take another look at this diagram:

By simply adding a few subdivision lines, we could get this diagram:

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Nothing has really changed – just how we subdivided the numbers up. In
other words – both diagrams represent exactly same number - they are
simply different ways of thinking about this same number.

Likewise, there are two ways to represent this number in math symbols:

One way, as already described, is a mixed number - two wholes

and three quarters - written in mathematical symbols as .
The other way is to count up the number of quarters (there are
eleven), and then simply say they are 11 quarters - which is
written in mathematical symbols as .

Fractions like are called "top heavy fractions ", which refers to the fact
that the top number in the fraction (the numerator) is bigger than the bottom
number (the denominator).

Please note: Some people refer to top heavy fractions as "improper

fractions ". This does not mean there is anything wrong with top heavy
fractions, but is simply used to contrast them from "proper fractions "
where the numerator (top number) is less than the denominator (bottom
number).

Questions:

2. Identify the fraction as a top heavy fraction.

Example Question: Identify the top heavy fraction represented by the

shaded parts of the shapes:

Answer:

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Now you try:

2. (a) Identify the top heavy fraction represented by the shaded parts of the
shapes:

2. (b) Identify the top heavy fraction.

2. (c) Identify the top heavy fraction represented by the shaded parts of the
shapes:

2. (d) Identify the top heavy fraction represented by the shaded parts of the
shapes:

2. (e) Identify the top heavy fraction represented by the shaded parts of the
shapes:

2. (f) Identify the top heavy fraction represented by the shaded parts of the
shapes:

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2. (g) Identify the top heavy fraction represented by the shaded parts of the
shapes:

2. (h) Identify the top heavy fraction represented by the shaded parts of the
shapes:

2. (i) Identify the top heavy fraction represented by the shaded parts of the
shapes:

2. (j) Identify the top heavy fraction represented by the shaded parts of the
shapes:

C. Converting Mixed Numbers to Top Heavy Fractions

There is a simple method of converting a mixed number to a top heavy
fraction:

Step 1: Multiply the whole number by the denominator (the

bottom number) of the fraction part.

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 Step 2: Add the numerator (top number) of the fraction part of the
mixed number.
Step 3: The total becomes the new numerator (top number) in the
top heavy fraction.
Step 4: The denominator (bottom number) in the top heavy
fraction as the denominator of the fraction part of the mixed
number.

Let's try an example:

Suppose we want to convert into a top heavy fraction. Let's follow

the steps:

Step 1: We multiply 2 by 4, giving 8.

Step 2: We add 1 to 8, giving 9.
Step 3: This means the numerator (top number) of the top heavy
fraction will be 9.
Step 4: The denominator (bottom number) of the top heavy
fraction will be the same as the denominator in the fraction part
if the mixed number, namely 4.

Result: The result is thus

Questions:

3. Convert the mixed numbers to a top heavy fraction.

Example Question: Convert to a top heavy fraction.

Answer:

Now you try:

3. (a) Convert into a top heavy fraction.

3. (b) Convert into a top heavy fraction.

3. (c) Convert into a top heavy fraction.

37
3. (d) Convert into a top heavy fraction.

3. (e) Convert into a top heavy fraction.

3. (f) Convert into a top heavy fraction.

3. (g) Convert into a top heavy fraction.

3. (h) Convert into a top heavy fraction.

3. (i) Convert into a top heavy fraction.

3. (j) Convert into a top heavy fraction.

D. Converting Whole Numbers to Top Heavy Fractions

There are occasions when you may wish to convert a whole number into a
fraction.

This is actually an incredibly easy process:

Step 1: The numerator (top number) in the fraction is the same as

the whole number that you started with.
Step 2: The denominator (bottom number) in the fraction is
always 1.

Thus for example, if we want to convert 8 into a fraction, it simply becomes

.

Likewise, if we want to convert 5 into a fraction, it becomes .

And, if we convert 1 into a fraction it becomes .

Questions:

4. Convert whole numbers to fractions.

Example Question: Convert 12 into a fraction.

Answer:

38
Now you try:

4. (a) Convert 4 into a fraction.

4. (b) Convert 6 into a fraction.
4. (c) Convert 2 into a fraction.
4. (d) Convert 3 into a fraction.
4. (e) Convert 7 into a fraction.
4. (f) Convert 1 into a fraction.
4. (g) Convert 5 into a fraction.
4. (h) Convert 8 into a fraction.
4. (i) Convert 9 into a fraction.
4. (j) Convert 10 into a fraction.

E. Converting Top Heavy Fractions to Mixed Numbers

If you have a top heavy fraction and want to convert it to a mixed number
(or where applicable a whole number) this is a simple process too:

Step 1: Divide the numerator (top number) of the fraction by the

denominator (bottom number), and calculate the number of times
it goes in, and also any remainder.
Step 2: The result of the division gives the whole number part of
the answer.
Step 3: If the remainder is 0, there is no fraction part in the
answer.
Step 4: If the remainder is not 0, the remainder becomes the
numerator (top number) of the fraction part of the answer. The
denominator (bottom number) of the fraction part of the answer is
the same as in the top heavy fraction.

Let's try an example:

Imagine that we want to convert to a mixed or whole number.

Step 1: We divide 15 by 3. This gives a result of 5 with a

remainder of 0.

39
 Step 2: The whole number part of the answer is therefore 5.
Step 3: Since the remainder was 0, there is no fraction part in the
answer (and Step 4 can be skipped).

Result: So is equivalent to .

Let's try another example:

Imagine that we want to convert to a mixed or whole number.

Step 1: We divide 13 by 4. This gives a result of 3 with a

remainder of 1.
Step 2: The whole number part of the answer is therefore 3,
Step 3 does not apply – since there is a remainder, the final
result will include both a whole number and a fraction part.
Step 4: Since the remainder was 1, the numerator (top number) in
the fraction part of the answer is 1. The denominator (bottom
number) in the fraction part of the answer is the same as in the
original top heavy fraction, namely 4.

Result: So is equivalent to

Questions:

5. Convert these top heavy fractions into their mixed or whole number
equivalents.

Example Question: convert into an equivalent whole or mixed number.

Answer:

Now you try:

5. (a) Convert into an equivalent whole or mixed number.

5. (b) Convert into an equivalent whole or mixed number.

5. (c) Convert into an equivalent whole or mixed number.

40
5. (d) Convert into an equivalent whole or mixed number.

5. (e) Convert into an equivalent whole or mixed number.

5. (f) Convert into an equivalent whole or mixed number.

5. (g) Convert into an equivalent whole or mixed number.

5. (h) Convert into an equivalent whole or mixed number.

5. (i) Convert into an equivalent whole or mixed number.

5. (j) Convert into an equivalent whole or mixed number.

5. (k) Convert into an equivalent whole or mixed number.

5. (l) Convert into an equivalent whole or mixed number.

5. (m) Convert into an equivalent whole or mixed number.

5. (n) Convert into an equivalent whole or mixed number.

5. (o) Convert into an equivalent whole or mixed number.

5. (p) Convert into an equivalent whole or mixed number.

5. (q) Convert into an equivalent whole or mixed number.

5. (r) Convert into an equivalent whole or mixed number.

5. (s) Convert into an equivalent whole or mixed number.

5. (t) Convert into an equivalent whole or mixed number.

41
Answers to Questions in Chapter 2

1. (a)

1. (b)

1. (c)

1. (d)

1. (e)

1. (f)

1. (g)

1. (h)

1. (i)

1. (j)

2. (a)

2. (b)

2. (c)

2. (d)

2. (e)

2. (f)

2. (g)

2. (h)

2. (i)

2. (j)

3. (a)

42
3. (b)

3. (c)

3. (d)

3. (e)

3. (f)

3. (g)

3. (h)

3. (i)

3. (j)

4. (a)

4. (b)

4. (c)

4. (d)

4. (e)

4. (f)

4. (g)

4. (h)

4. (i)

4. (j)

5. (a)

5. (b)

5. (c)

43
5. (d)

5. (e)

5. (f)

5. (g)

5. (h)

5. (i)

5. (j)

5. (k)

5. (l)

5. (m)

5. (n)

5. (o)

5. (p)

5. (q)

5. (r)

5. (s)

5. (t)

44
Chapter 3: Simplifying and Comparing Fractions

A. Equivalent Fractions
Look at the following:

Can you see that the proportion (or fraction) of each figure that is shaded is
exactly the same in each case?

What this means is that all the different fractions are equivalent , (we
therefore say they are "equivalent fractions "), or in mathematical symbols:

The general rule is that we can transform any fraction into an equivalent
fraction by:

Either, multiplying the numerator (top number) and denominator

(bottom number) by the same whole number. We can use any
whole number that we like, but it must be the same for both
numerator (top) and denominator (bottom).
Or, by dividing the numerator (top number) and denominator
(bottom number) by the same whole number. Again, we can use

45
 any whole number that we like, but it must be the same for both
numerator (top) and denominator (bottom).

Thus, for example, if we started with , we could multiply both top and
bottom by 3, which would result in . And we could thus say and are
equivalent fractions, or simply:

Likewise, if started with , we could divide both top and bottom by 2,

which would give us . Or, we could divide both top and bottom by 4,
which would give us . And thus we can say that , , and are all
equivalent fractions, or simply:

We can also chain together a series of multiplications and divisions to reach

an equivalent fraction. For example, if we wanted to convert into sixths
(a fraction with 6 as the denominator or bottom number), we could first
divide the top and bottom of by 4, giving us , then multiply top and
bottom of by 3, giving us . And thus we can say that , , and are
all equivalent fractions, or simply:

Questions:

1. Fill in the missing number in the equivalent fraction.

Example Question:

Answer: (3 was the missing number)

Now you try:

1. (a) Fill in the missing number in

46
1. (b) Fill in the missing number in

1. (c) Fill in the missing number in

1. (d) Fill in the missing number in

1. (e) Fill in the missing number in

1. (f) Fill in the missing number in

1. (g) Fill in the missing number in

1. (h) Fill in the missing number in

1. (i) Fill in the missing number in

1. (j) Fill in the missing number in

1. (k) Fill in the missing number in

1. (l) Fill in the missing number in (Hint: use a chain of divisions

and multiplications)

1. (m) Fill in the missing number in

1. (n) Fill in the missing number in

1. (o) Fill in the missing number in

1. (p) Fill in the missing number in

1. (q) Fill in the missing number in

1. (r) Fill in the missing number in

1. (s) Fill in the missing number in (Hint: use a chain of divisions

and multiplications)

1. (t) Fill in the missing number in

1. (u) Fill in the missing number in

1. (v) Fill in the missing number in

47
1. (w) Fill in the missing number in

1. (x) Fill in the missing number in

1. (y) Fill in the missing number in

B. Simplifying Fractions
If you are asked to simplify a fraction or express a fraction its simplest
terms or express a fraction in its lowest terms , what you are being asked
to do is find the equivalent fraction with the lowest possible denominator
(bottom number). Additionally, after doing this, if you end up with a top
heavy fraction, you would then normally want convert it to a mixed number
(or a whole number if applicable).

Here is the process to simplify a fraction:

Step 1: Look at the denominator (bottom number) and numerator

(top number), and see if you can find a whole number that they
are both divisible by. If you find such a number, divide both the
denominator and numerator by this number.
Step 2: Look at the result and see if you can repeat the process
perhaps using the same or a different whole number. Keep
repeating the process until you are sure that no more divisions
are possible.
Step 3: If you end up with top heavy fraction, convert it into a
mixed number (or if applicable, a whole number).

Let's try an example:

Imagine that we want to simplify

Step 1: We notice that top and bottom are both divisible by 2

since both are even numbers. We therefore divide 88 by 2 to get
44, and 528 by 2 to get 264. This gives an equivalent fraction of
.
Step 2: We notice that top and bottom are again both divisible by
2 since both are even numbers. We therefore divide 44 by 2 to
get 22, and 264 by 2 to get 132. This gives an equivalent fraction

48
 of .
Step 2 (again): We notice that top and bottom are again both
divisible by 2 since both are even numbers. We therefore divide
22 by 2 to get 11, and 132 by 2 to get 66. This gives an
equivalent fraction of .
Step 2 (again): We now notice that top and bottom are both
divisible by 11 (since both are on the 11 times table). We
therefore divide 11 by 11 to get 1, and 66 by 11 to get 6. This
gives an equivalent fraction of .
Step 2 (again): We now observe that it is not possible to repeat
the dividing process further from .
Step 3: We do not have a top heavy fraction, so we do not need
to convert it into a mixed number, the answer is still .

Result: This means the final answer is . expressed in its

lowest terms, or simplest form, is .
Note: We could also write out the series of simplifications that
we performed in mathematical symbols like this:

Let's try another example:

Imagine that we want to simplify .

Step 1: We notice that top and bottom are both divisible by 3

since both are on the 3 times table. We therefore divide 27 by 3
to get 9, and 6 by 3 to get 2. This gives an equivalent fraction of
.
Step 2: We now observe that it is not possible to repeat the
dividing process further from .

Step 3: We convert to a mixed number, namely .

49
 Result: The final answer is . expressed in its simplest
form as a mixed number is .
Note: We could also write out the series of steps that we
performed in mathematical symbols like this:

What about if you start with a mixed number? In this case, you only need to
simplify the fraction part - you can leave the whole number part unchanged.

Let's try an example of that:

Imagine you are asked to simplify

Step 1: Looking only at the fraction part, we notice that top and
bottom are both divisible by 4 since both are on the 4 times
table. We therefore divide 8 by 4 to get 2, and 12 by 4 to get 3.
This gives an equivalent fraction of .
Step 2: Still looking only at the fraction part, we now observe
that it is not possible to repeat the dividing process further from
.
Step 3: Does not apply since we are dealing with the fraction
part of a mixed number.

Result: We simply put the new fraction part ( ) together with

the whole number part of the original number (5), giving us an
answer of .
Note: We could also write out the step that we performed in
mathematical symbols like this:

Questions:

2. Express as these numbers in their simplest form: any top heavy

fractions should be converted into whole numbers or mixed numbers

50
with the fraction part in its lowest terms.

Example Question: Express as a fraction in its simplest form.

Answer:

Now you try:

2. (a) Express as a fraction in its simplest form.

2. (b) Express as a fraction in its simplest form.

2. (c) Express as a fraction in its simplest form.

2. (d) Express as a fraction in its simplest form.

2. (e) Express as a fraction in its simplest form.

2. (f) Express as a fraction in its simplest form.

2. (g) Express as a fraction in its simplest form.

2. (h) Express as a fraction in its simplest form.

2. (i) Express as a fraction in its simplest form.

2. (j) Express as a fraction in its simplest form.

2. (k) Express as a fraction in its simplest form.

2. (l) Express as a fraction in its simplest form.

2. (m) Express as a fraction in its simplest form.

2. (n) Express as a fraction in its simplest form.

2. (o) Express as a fraction in its simplest form.

C. Comparing Fractions
If you want to compare two fractions to determine which is larger and
which is smaller, you can only easily do this if both have the same

51
denominator (bottom number). If they have the same denominator, then you
can simply compare the numerators (top numbers) – the smaller fraction
will have the smaller numerator, and the larger fraction will have the larger
numerator.

For example, if we want to compare and , we can see both have the
same denominator (8), so we can then go ahead and compare the numerators
(5 and 7). We can thus say is smaller, and is larger.

But how would compare fractions where the denominators (bottom

numbers) are different?

The answer is to use a simple trick: we use the idea of equivalent fractions
to find equivalents to one or both fractions, so we can make the comparison
between fractions in which the denominators (bottom numbers) are the same
(they have a common denominator ).

Let's try an example:

Imagine that we want to compare and .

First of all, we observe that the denominators (bottom numbers)

in the two fractions are different. Fortunately, we spot that if we
multiply 4 by 2 it will produce 8, so we multiply the top and
bottom of by 2, it will produce an equivalent fraction ( ) but
with the same common denominator as the other fraction ( ).

We can now compare and .

Result: Clearly is the smaller of the two fractions. And, since

is an equivalent fraction of , this means that is smaller of
the two fractions, and is the larger of the fractions.

Let's try another, slightly more difficult, example:

Imagine that we want to compare and .

52
 First of all, we observe that the denominators (bottom numbers)
in the two fractions are different. We also notice that we can't
simply multiply one fraction's denominator to produce the other
fraction's. This means that we are going to need to find
equivalents of both fractions with the same "common
denominator ". The common denominator needs to be a number
that is both in the 6 times table (so we can convert to it), and in
the 9 times table (so we can convert to it).
For the common denominator, we can choose any number that is
in both the 6 and 9 times tables – for example, 18, 36, 54, etc.
Usually it's easiest to choose the lowest number possible (18 in
this case) – but it doesn't matter if you choose one of the other
alternatives. Let's say we choose 36...

We need to convert to a fraction with a denominator of 36. We

do this by multiplying top and bottom by 6, thus producing .

Likewise, we also need convert to a fraction with a

denominator of 36. We do this by multiplying to top and bottom
by 4, thus producing .

Result: We compare and . Clearly is the smaller of the

two fractions. And, since is an equivalent fraction of , this
means that is the smaller of the two fractions, and is the
larger of the two fractions.

As you can see, the process for comparing fractions is relatively simple.
However, the toughest part for many people is finding a common
denominator . If you struggle with this step, I would strongly recommend
that you practice your times tables as this will help you find the smallest
(and hence easiest) common denominators. In any case, you can always
simply multiply the two denominators together - this will always produce a
working common denominator , although it might be larger (and hence
requiring more working out) than the ideal choice.

Let's repeat the comparison between and , but using this method –
multiplying the denominators together to find a common denominator .

53
 First, we observe that the denominators (bottom numbers) in the
two fractions are different. We also notice that we can't simply
multiply one fraction's denominator to produce the other
fraction's. This means that we are going to need to find
equivalents of both fractions with the same "common
denominator ". In this example, we'll choose 54 (the result of
multiplying the two denominators, 6 and 9, together) as our
common denominator.

We need to convert to a fraction with a denominator of 54. We

do this by multiplying top and bottom by 9 (since there are 9 X 6
= 54), thus producing .

Likewise, we also need to convert to a fraction with a

denominator of 54. We do this by multiplying to top and bottom
by 6 (since 6 X 9 = 54), thus producing .

Result: We compare and . Clearly is the smaller of the

two fractions. And, since is an equivalent fraction of , this
means that is the smaller of the two fractions, and is the
larger of the two fractions.

Questions:

3. Which is the larger fraction?

Example Question: Which is the larger of or ?

Answer:

Now you try:

3. (a) Which is the larger of or ?

3. (b) Which is the larger of or ?

3. (c) Which is the larger of or ?

3. (d) Which is the larger of or ?

54
3. (e) Which is the larger of or ?

3. (f) Which is the larger of or ?

3. (g) Which is the larger of or ?

3. (h) Which is the larger of or ?

3. (i) Which is the larger of or ?

3. (j) Which is the larger of or ?

3. (k) Which is the larger of or ?

3. (l) Which is the larger of or ?

3. (m) Which is the larger of or ?

3. (n) Which is the larger of or ?

3. (o) Which is the larger of or ? (Hint: Could this be a trick question?)

55
Answers to Questions in Chapter 3

1. (a) (4 was the missing number)

1. (b) (9 was the missing number)

1. (c) (14 was the missing number)

1. (d) (70 the missing number)

1. (e) (3 was the missing number)

1. (f) (3 was the missing number)

1. (g) (8 was the missing number)

1. (h) (40 was the missing number)

1. (i) (80 was the missing number)

1. (j) (1 was the missing number)

1. (k) (3 was the missing number)

1. (l) (6 was the missing number)

1. (m) (15 was the missing number)

1. (n) (10 was the missing number)

1. (o) (9 was the missing number)

1. (p) (2 was the missing number)

1. (q) (4 was the missing number)

1. (r) (6 was the missing number)

1. (s) (9 was the missing number)

1. (t) (9 was the missing number)

1. (u) (15 was the missing number)

56
1. (v) (16 was the missing number)

1. (w) (12 was the missing number)

1. (x) (10 was the missing number)

1. (y) (5 was the missing number)

2. (a)

2. (b)

2. (c)

2. (d)

2. (e)

2. (f)

2. (g)

2. (h)

2. (i)

2. (j)

2. (k)

2. (l)

2. (m)

2. (n)

2. (o)

3. (a)

3. (b)

3. (c)

57
3. (d)

3. (e)

3. (f)

3. (g)

3. (h)

3. (i)

3. (j)

3. (k)

3. (l)

3. (m)

3. (n)
3. (o) They are equivalent fractions – so they are both the same!

58
Chapter 4: Adding fractions

A. Adding Two Fractions Together

As with comparing fractions, adding fractions together is only directly
possible if the fractions all have the same denominator (bottom number).
Fortunately however, adding fractions with different denominators is still
possible, although it does requires the use of a trick.

Once again, the trick is to use the technique of equivalent fractions. Using
equivalent fractions, we can find equivalents to the fractions which we wish
to add, but all with the same denominator (bottom number). This same
denominator is known as the "common denominator ".

Here are the steps for adding two (or more) fractions together:

Step 1: Convert any mixed numbers or whole numbers to top

heavy fractions.
Step 2: If the fractions that you are adding already have the same
denominator (bottom number), then this is the denominator of the
result.
However, if they do not have the same denominator, you need use the
equivalent fractions method to change one or more of the fractions so
that they do have the same denominator (a "common denominator ").
You can choose the value of common denominator by simply picking a
number which is in the same times table as all the denominators of the
original fractions. The common denominator then becomes the
denominator of the result.
As before when comparing fractions, if you are struggle to find a
common denominator , you can always simply multiply the two
denominators together to find a working one (although it may be
larger, and require more working out, than would otherwise be the
case).

Step 3: Add the numerators (top numbers) of the fractions. This

is the numerator of the result.
Step 4 (optional): Simplify the result, and then, if applicable,
express it as a whole number or mixed number.

59
Let's try an example:

Imagine that we wish to add and .

Step 1: Step 1 is not applicable for this example, as there are no

mixed or whole numbers.
Step 2: Both fractions already have the same denominator
(bottom number) of 5. Therefore, we do not need to use
equivalent fractions. The denominator of the result is therefore 5.
Step 3: We add the numerators (3 + 1). The numerator of the
result is thus 4, and the answer is .
Step 4: The answer is neither top heavy, nor can be simplified,
so step 4 is not applicable for this example.

Result: The final answer is .

Let's try another example:

Imagine that we wish to add and .

Step 1: Step 1 is not applicable for this example, as there are no

mixed or whole numbers.
Step 2: Both fractions do not already have the same denominator
(bottom number). Therefore, we do need to use equivalent
fractions. We choose 10 as the common denominator since this is
in both the 10 and 5 times tables. We note that the common
denominator, 10, will also be the denominator of the result.

We do not need to use the equivalent fractions method on since this

already has the desired denominator (bottom number) of 10.

We do however need to use the equivalent fractions method on

since this does not yet have the desired denominator (bottom number)
of 10. We thus multiply the top and bottom of by 2, and becomes
.

The overall calculation has now become .

60
 Step 3: We add the numerators (3 + 2). The numerator of the
result is thus 5, and the answer is .

Step 4: We note that we can simplify by dividing top and

bottom by 5. Giving .

Result: The final answer is .

Imagine that we wish to add and :

Step 1: Step 1 is not applicable for this example, as there are no

mixed or whole numbers.
Step 2: Both fractions do not already have the same denominator
(bottom number). Therefore, we do need to use equivalent
fractions. We choose 18 as the common denominator since this is
in both the 6 and 9 times table. We note that the common
denominator, 18, will also be the denominator of the result.

We need to use the equivalent fractions method on since this does

not yet have the desired denominator (bottom number) of 18. We thus
multiply the top and bottom of by 3, and becomes .

We need to use the equivalent fractions method on since this does

not yet have the desired denominator (bottom number) of 18. We thus
multiply the top and bottom of by 2, and becomes .

The overall calculation has now become .

Step 3: We add the numerators (15 + 8). The numerator of the

result is thus 23, and the answer is .

Step 4: converts into which is our final answer.

Result: is the final answer.

Questions:

1. Add the fractions. Where applicable, express the answer as a whole

61
number or as mixed number with the fraction part in its lowest terms.

Example Question:

Answer:

Now you try:

1. (a)

1. (b)

1. (c)

1. (d)

1. (e)

1. (f)

1. (g)

1. (h)

1. (i)

1. (j)

1. (k)

1. (l)

1. (m)

1. (n)

1. (o)

B. Adding More Than Two Fractions Together

The method that we have already described for adding two fractions can
also be used for adding together any number of fractions – the only thing that

62
we must remember is that we need to use a common denominator for all the
fractions being added.

Here's an example: Imagine that we wish to add to to

Step 1: Step 1 is not applicable for this example, as there are no

mixed or whole numbers.
Step 2: We notice that the fractions do not already have the same
denominator (bottom number). Therefore, we do need to use
equivalent fractions.
We need to choose a common denominator that is in all of the 2, 4 and
6 times tables. The obvious number to choose as a common
denominator is 12...

We need to use the equivalent fractions method on since this does

not yet have the desired denominator (bottom number) of 12. We thus
multiply the top and bottom of by 6, and becomes .

Likewise, we need to use the equivalent fractions method on since

this does not yet have the desired denominator (bottom number) of
12. We thus multiply the top and bottom of by 3, and becomes .

Likewise, we need to use the equivalent fractions method on since

this does not yet have the desired denominator (bottom number) of
12. We thus multiply the top and bottom of by 2, and becomes
.

The overall calculation has now become .

Step 3: We add the numerators (6 + 9 + 10). The numerator of

the result is thus 25, and the answer is .

Step 4: converts into which is our final answer.

Result: is the final answer.

Sometimes the numbers and the multiplications can get quite tricky when
you are adding three or more fractions together. If you struggle with such

63
calculations, you might find it easier to add two of the fractions together,
simplify this intermediate result if possible, then add the third fraction on to
the intermediate result to get the final answer.

Let's compare doing the same calculation two different ways: Adding to
to :
First, adding all three fractions in one go – as you will see, the calculations
can be quite challenging:

Step 1: Step 1 is not applicable for this example, as there are no

mixed or whole numbers.
Step 2: We notice that the fractions do not already have the same
denominator (bottom number). Therefore, we do need to use
equivalent fractions.
We need to choose a common denominator that is in all of the 6, 9 and
4 times tables. The lowest value that we could choose for the
common denominator is 36, but you might not spot this. You can in
fact choose any number that is in all of the 6, 9 and 4 times tables –
and if you were to struggling to find of one, you could generate one by
doing 6 X 9 X 4 – giving 108.
So let's proceed on the basis of choosing 108 as our common
denominator...

We first need to use the equivalent fractions method on since this

does not yet have the desired denominator (bottom number) of 108.
We thus multiply the top and bottom of by 18 (since 6 X 18 = 108),
and becomes .

We also need to use the equivalent fractions method on since this

does not yet have the desired denominator (bottom number) of 108.
We thus multiply the top and bottom of by 12 (since 9 X 12 = 108),
and becomes .

We also need to use the equivalent fractions method on since this

does not yet have the desired denominator (bottom number) of 108.
We thus multiply the top and bottom of by 27 (since 4 X 27 = 108),

64
 and becomes .

The overall calculation has now become .

Step 3: We add the numerators (90 + 24 + 27). The numerator of

the result is thus 135, and the answer is .
Step 4: We notice that both 141 and 108 are on the 3 times table
since in both cases the digits add up to a multiple of 3 (for more
tricks like this, see Teach Your Kids Math: Multiplication Times
Tables ). We therefore can divide both top and bottom by 3,
simplifying into . We then convert this into the mixed
number

Result: is the final answer.

Alternatively, we can add the fractions bit by bit:

We are initially going to add to to get an intermediate result.

Step 1: Step 1 is not applicable for this example, as there are no
mixed or whole numbers.
Step 2: We notice that the fractions do not already have the same
denominator (bottom number). Therefore, we do need to use
equivalent fractions.
We need to choose a common denominator that is in the 6 and 9 times
tables. The lowest value that we could choose for the common
denominator is 18, but you might not spot this. You can in fact choose
any number that is both of the 6 and 9 times tables – and if you were
struggling to find one, you could generate one by doing 6 X 9 – giving
54.
So let's proceed on the basis of choosing 54 as our common
denominator... this will also be denominator of our intermediate
result.

We need to use the equivalent fractions method on since this does

not yet have the desired denominator (bottom number) of 54. We thus
multiply the top and bottom of by 9, and becomes .

65
We also need to use the equivalent fractions method on since this
does not yet have the desired denominator (bottom number) of 54.
We thus multiply the top and bottom of by 6, and becomes .

The first addition has now become .

Step 3: We add the numerators (45 + 12). The numerator of the

result is thus 57, and the intermediate answer is .
Step 4: We notice that both 57 and 54 are on the 3 times table
since in both cases the digits add up to a multiple of 3 (again, for
more tricks like this, see Teach Your Kids Math: Multiplication
Times Tables ). We therefore can divide both top and bottom by
3, simplifying into . Since is a top heavy fraction, we
could convert it into a mixed number, but since we wish to do
further calculations on it, we need not do so – thus is our
intermediate result.

Intermediate result: .

We now need to repeat the adding process, adding to to

produce our final answer.
Step 1: Step 1 is not applicable for this example, as there are no
mixed or whole numbers.
Step 2: We notice that the fractions do not already have the same
denominator (bottom number). Therefore, we do need to use
equivalent fractions.
We need a common denominator that is both a multiple of 18 and 4.
We thus choose a common denominator of 36, which will also be the
denominator of the result of this addition.

We need to use the equivalent fractions method on since this does

not yet have the desired denominator (bottom number) of 36. We thus
multiply the top and bottom of by 2 (since 18 X 2 = 36), and
becomes .

Likewise, we also need to use the equivalent fractions method on

66
 since this does not yet have the desired denominator (bottom number)
of 36. We thus multiply the top and bottom of by 9, and becomes
.

The second addition has now become .

Step 3: We add the numerators (38 + 9). The numerator of the

result is thus 47, and the result is .

Step 4: We convert into the mixed number .

Result: is the final answer.

Questions:

2. Add the fractions. Where applicable, express the answer as a whole

number or as mixed number with the fraction part in its lowest terms.

Example Question:

Answer:

Now you try:

2. (a)

2. (b)

2. (c)

2. (d)

2. (e)

2. (f)

2. (g)

2. (h)

67
2. (i)

2. (j)

68
Answers to Questions in Chapter 4

1. (a)

1. (b)

1. (c)

1. (d)

1. (e)

1. (f)

1. (g)

1. (h)

1. (i)

1. (j)

1. (k)

1. (l)

1. (m)

1. (n)

1. (o)

2. (a)

2. (b)

2. (c)

2. (d)

2. (e)

2. (f)

69
2. (g)

2. (h)

2. (i)

2. (j)

70
Chapter 5: Subtracting Fractions

As with adding, subtracting one fraction from another is only directly

possible if both of the fractions have the same denominator (bottom
number). Fortunately however, subtracting fractions with different
denominators is still possible, although, like adding, it does require the use
of equivalent fractions. Using equivalent fractions we can find equivalents
to the fractions we wish to subtract, but all with the same denominator
(bottom number). Once again, this is known as the "common denominator
".

Here are the steps for subtracting one fraction from another:

Step 1: Convert any mixed numbers or whole numbers to top

heavy fractions.
Step 2: If the fractions that you are subtracting already have the
same denominator (bottom number), then this is the denominator
of the result.
However, if they do not have the same denominator, you need use the
equivalent fractions method to change one or both of the fractions so
that they do have the same denominator (a "common denominator ").
You can choose the value of common denominator by simply picking a
number which is in the same times tables as the denominators of the
original fractions. The common denominator then becomes the
denominator of the result.

Step 3: Perform the subtraction on the numerators (top numbers)

of the fractions. This is the numerator of the result.
Step 4 (optional): Simplify the result, and then, if applicable,
express it as a whole number or mixed number.

Let's try an example:

Imagine that we wish to calculate minus .

Step 1: Step 1 is not applicable for this example, as there are no

mixed or whole numbers.
Step 2: Both fractions already have the same denominator

71
 (bottom number) of 5. Therefore, we do not need to use
equivalent fractions. The denominator of the result is therefore 5.
Step 3: We subtract the numerators (3 - 1). The numerator of the
result is thus 2, and the answer is .
Step 4: Step 4 is not applicable for this example.

Result: The final answer is .

Let's try another example:

Imagine that we wish to calculate minus .

Step 1: Step 1 is not applicable for this example, as there are no

mixed or whole numbers.
Step 2: Both fractions do not already have the same denominator
(bottom number). Therefore, we do need to use equivalent
fractions. We choose 10 as the common denominator since this is
in both the 10 and 5 times tables. We note that the common
denominator, 10, will be the denominator of the result.

We do not need to use the equivalent fractions method on since this

already has the desired denominator (bottom value) of 10.

We do need to use the equivalent fractions method on since this

does not yet have the desired denominator (bottom value) of 10. We
thus multiply the top and bottom of by 2, and becomes .

The overall calculation has now become .

Step 3: We subtract the numerators (3 - 2). The numerator of the

result is thus 1, and the answer is .
Step 4: Step 4 is not applicable for this example, as no
simplification is possible.

Result: The final answer is .

Imagine that we wish to calculate minus

72
 Step 1: Step 1 is not applicable for this example, as there are no
mixed or whole numbers.
Step 2: Both fractions do not already have the same denominator
(bottom number). Therefore, we do need to use equivalent
fractions. We choose 18 as the common denominator since this is
in both the 6 and 9 times tables. We note that the common
denominator, 18, will be the denominator of the result.

We need to use the equivalent fractions method on since this does

not yet have the desired denominator (bottom value) of 18. We thus
multiply the top and bottom of by 3, and becomes .

We need to use the equivalent fractions method on since this does

not yet have the desired denominator (bottom value) of 18. We thus
multiply the top and bottom of by 2, and becomes .

The overall calculation has now become .

Step 3: We subtract the numerators (15 - 8). The numerator of

the result is thus 7, and the answer is .
Step 4: Step 4 is not applicable for this example, as no
simplification is possible.

Result: The final answer is .

Questions:

1. Subtract the fractions. Where applicable, express the answer as a

whole number or as mixed number with the fraction part in its lowest
terms.

Example Question:

Answer:

Now you try:

1. (a)

73
1. (b)

1. (c)

1. (d)

1. (e)

1. (f)

1. (g)

1. (h)

1. (i)

1. (j)

74
Answers to Questions in Chapter 5

1. (a)

1. (b)

1. (c)

1. (d)

1. (e)

1. (f)

1. (g)

1. (h)

1. (i)

1. (j)

75
Chapter 6: Multiplying Fractions

A. Basic Multiplication of Fractions

If you want to multiply a fraction by another number (whether by another
fraction or by a whole number), this is the process:

Step 1: Convert any whole numbers or mixed numbers into top

heavy fractions.
Step 2: Multiply the numerators (top numbers) together to
produce the numerator of the result.
Step 3: Multiply the denominators (bottom numbers) together to
produce the denominator of the result.
Step 4 (optional): Simplify the result, and then, if applicable,
express it as a whole number or mixed number.

Let's try an example:

Multiply by

Step 1: Does not apply since both numbers are already fractions.
Step 2: We multiply the numerators (top numbers). Multiplying 2
times 3 produces 6. This is the result's numerator.
Step 3: We multiply the denominators (bottom numbers).
Multiplying 3 times 4 produces 12. This is the result's
denominator.

Step 4: The answer is , however we can simplify this (by

dividing top and bottom by 6) to the equivalent fraction of .

Result: The final answer is .

Let's try another example:

Imagine that we want to multiply by

Step 1: We convert to a top heavy fraction . So the

76
 calculation has now become .
Step 2: We multiply the numerators (top numbers). Multiplying 7
times 2 produces 14. This is the result's numerator.
Step 3: We multiply the denominators (bottom numbers).
Multiplying 4 times 5 produces 20. This is the result's
denominator.

Step 4: The answer is , however we can simplify this (by

dividing top and bottom by 2) to the equivalent fraction of .

Result: The final answer is .

Let's try one more example:

Imagine that we want to multiply by .

Step 1: We convert to a top heavy fraction . We also

convert 8 into a top heavy fraction . So the calculation has
now become .
Step 2: We multiply the numerators (top numbers). Multiplying 7
times 8 produces 56. This is the result's numerator.
Step 3: We multiply the denominators (bottom numbers).
Multiplying 4 times 1 produces 4. This is the result's
denominator.

Step 4: The answer is , however we can simplify this (by

dividing top and bottom by 4) to the equivalent fraction of ,
which in turn can be converted to the whole number 14.
Result: The final answer 14.

Questions:

1. Multiply the numbers. Where applicable, express the answer as a

whole number or as mixed number with the fraction part in its lowest
terms.

77
Example Question: Multiply by

Answer:

Now you try:

1. (a) Multiply by

1. (b) Multiply by (Hint: Convert everything, including whole

numbers, into fractions – revise section D of Chapter 2 if you can't
remember how to do this)

1. (c) Multiply by

1. (d) Multiply by

1. (e) Multiply by

1. (f) Multiply by

1. (g) Multiply by

1. (h) Multiply by

1. (i) Multiply by

1. (j) Multiply by

1. (k) Multiply by

1. (l) Multiply by

1. (m) Multiply by

1. (n) Multiply by

1. (o) Multiply by

B. Multiplying Several Fractions

If you want to multiply three or more numbers together, including one or
more fractions, this is also a simply an extension of the multiplication

78
process that we have already looked at:

Step 1: Convert any whole numbers or mixed numbers into top

heavy fractions.
Step 2: Multiply the numerators (top numbers) of all the fractions
together to produce the numerator of the result.
Step 3: Multiply the denominators (bottom numbers) of all the
fractions together to produce the denominator of the result.
Step 4 (optional): Simplify the result, and then, if applicable,
express it as a whole number or mixed number.

Let's try an example:

Imagine that we want to multiply , and .

Step 1: Does not apply since all the numbers are already
fractions.
Step 2: We multiply all the numerators (top numbers) together.
Multiplying 3 times 2 times 5 produces 30. That means the
numerator of the result is 30.
Step 3: We multiply all the denominators (bottom numbers)
together. Multiplying 4 times 7 times 6 produces 168. That
means the denominator of the result is 168.

Step 4: The answer is , however we can simplify this (by

dividing top and bottom by 6) to the equivalent fraction of .

Result: The final answer is .

Let's try another example:

Imagine that we want to multiply , and

Step 1: We need to make sure all numbers are fractions, by

converting any mixed numbers or whole numbers into fractions.
In this example, converts into , and converts into .

79
 In other words, the calculation has now become .

Step 2: We multiply all the numerators (top numbers) together.

Multiplying 2 times 9 times 5 produces 90. That means the
numerator of the result is 90.
Step 3: We multiply all the denominators (bottom numbers)
together. Multiplying 3 times 2 times 1 produces 6. That means
the denominator of the result is 6.

Step 4: The answer is , however we can simplify this (by

dividing top and bottom by 6) to the equivalent fraction of ,
which in turn is equivalent to the whole number 15.

C. Simplifying Multiplications
As you have seen, when we multiply two or more fractions together, we
often find ourselves being required to work with fairly large numbers.

Fortunately, there is a way to simplify many fraction multiplications: Just as

we can simplify any individual fraction by dividing the numerator (top
number) and denominator (bottom number) by the same number, we can
simplify fraction multiplications by dividing any one of the numerators and
any one of the denominators by the same number. The
numerator/denominator pair do not need to be in the same fraction (but we
must always be dividing one numerator, and one denominator). Moreover,
we can repeat the process more than once.

Let's try an example:

Imagine that we want to do this calculation:

The first thing that we might spot is that the numerator (top number) of
and the denominator (bottom number) of are both divisible by 9. So, we
can simplify the calculation, if we divide these two numbers by 9. If we go
ahead and do this simplification, the calculation now becomes:

80
Note: When doing these types of calculations with pen and paper, many
people will often cross out the numbers exactly as shown above. You may
find it helpful to do so too. However, if you prefer, you can also write the
new calculation completely separately like this:

We might perhaps next spot that the numerator (top number) of and the
denominator (bottom number) of are both divisible by 7. So, we can
simplify the calculation, if divide these two numbers by 7. If we go ahead
and do this simplification, the calculation now becomes:

Once again, when writing out the simplified calculation, you can either use
crossing out method as shown above, or simply write the new calculation in
full which would be:

We might next perhaps spot that the numerator (top number) of and the
denominator (bottom number) of are both divisible by 9. So, we can
simplify the calculation, if divide these two numbers by 9. If we go ahead
and do this simplification, the calculation now becomes:

Or, written as a new calculation:

Lastly, we might spot that both numerator (top number) and the denominator
(bottom number) of are both divisible by 2. If we go ahead and do this
simplification, the calculation now becomes:

81
Or, written as a new calculation:

And if we evaluate this: the numerator (top number) of the result is 3 (by
multiplying 3 by 1 by 1), and the denominator (bottom number) of the result
is 250 (by multiplying 50 by 5 by 1). So, the final answer is .

It is important to note that we do not change the result by simplifying in this

way. If we had omitted some of the simplifications during multiplication, or
done them in a different order, or simplified only after completing the
multiplications, we would have still produced the same answer, ,
although perhaps expressed as an equivalent (and not yet fully simplified)
fraction.

Questions:

2. Multiply the numbers. Where applicable, express the answer as a

whole number or as mixed number with the fraction part in its lowest
terms.

Example Question: Calculate

Answer:

Now you try:

2. (a)

2. (b)

2. (c)

2. (d)

2. (e)

82
2. (f)

2. (g)

2. (h)

2. (i)

2. (j)

83
Answers to Questions in Chapter 6

1. (a)

1. (b)

1. (c)

1. (d)

1. (e)

1. (f)

1. (g)

1. (h)

1. (i)

1. (j)

1. (k)

1. (l)

1. (m)

1. (n)

1. (o)

2. (a)

2. (b)

2. (c)

2. (d)

2. (e)

2. (f)

84
2. (g)

2. (h)

2. (i)

2. (j)

85
Chapter 7: Dividing fractions

A. Division of Fractions
If you want to perform a division calculation where either the number to be
divided, or the number to divide it by (the divisor), or both, are fractions,
then this is the process:

Step 1: If either (or both numbers) are whole numbers or mixed

numbers, convert them to top heavy fractions.
Step 2: Convert the division into a multiplication by swapping
("flipping") the numerator (top number) and denominator (bottom
number) of the divisor (the number that you are dividing by).
Step 3: Perform the multiplication.
Step 4 (optional): Simplify the result, and then, if applicable,
express it as a whole number or mixed number.

Let's try an example:

Imagine that we want to calculate

Step 1: We convert to a top heavy fraction . So we are

now calculating .
Step 2: We change the division to a multiplication, and flip the
divisor , so it becomes . The overall calculation has now
become .

Step 3: We multiply by giving us .

Step 4: We convert into the mixed number .

Result: The final answer is .

Questions:

1. Divide the numbers. Where applicable, express the answer as a

whole number or as mixed number with the fraction part in its lowest
terms.

86
Example Question:

Answer:

Now you try:

1. (a)

1. (b)

1. (c)

1. (d)

1. (e)

1. (f)

1. (g)

1. (h)

1. (i)

1. (j)

1. (k) (Hint: Convert everything, including whole numbers, into

fractions – revise section D of Chapter 2 if you can't remember how to do
this)

1. (l)

1. (m)

1. (n)

1. (o)

B. Complex Fractions
You may sometimes come across what is known as "complex fractions ".
These are fractions where the numerator (top number) and/or denominator

87
(bottom number) is itself a fraction.

Here are some examples of the complex fractions:

You may not realize it yet, but you actually already know how to convert
complex fractions into normal simple fractions. You simply need remember
that all fractions are simply another way of writing division – so all you
need to do is divide the numerator (top number) of the complex fraction by
the denominator (bottom number).

Let's try for each of the above complex fractions:

Then

Then

Then

Questions:

2. Simplify the complex fractions. Where applicable, express the answer

as a whole number or as mixed number with the fraction part in its
lowest terms.

Example Question: Simplify

Answer:

88
Now you try:

2. (a) Simplify

2. (b) Simplify

2. (c) Simplify

2. (d) Simplify

2. (e) Simplify

2. (f) Simplify

2. (g) Simplify

2. (h) Simplify

2. (i) Simplify

2. (j) Simplify

89
Answers to Questions in Chapter 7

1. (a)

1. (b)

1. (c)

1. (d)

1. (e)

1. (f)

1. (g)

1. (h)

1. (i)

1. (j)

1. (k)

1. (l)

1. (m)

1. (n)

1. (o)

2. (a)

2. (b)

2. (c)

2. (d)

2. (e)

2. (f)

90
2. (g)

2. (h)

2. (i)

2. (j)

91
Chapter 8: Converting Fractions to Decimals

Sometimes it is necessary to convert fractions into decimals. In this chapter,

we explain how you can do this. If you are not yet familiar with decimal
numbers, we suggest you leave this chapter for now and return to it once you
have learned about them .

A. Converting When the Denominator is a Power of Ten

Powers of ten are numbers like 10, 100 (which is 10 times 10), 1000
(which is 10 times 10 times 10), 10000 (which is 10 times 10 times 10
times 10), etc.

Converting fractions where the denominator (bottom number) is a power of

ten to decimals is quite straightforward. Here are the steps:

Step 1: If dealing with a top heavy fraction, convert it to a mixed

number. The whole number part of the mixed number will go to
the left of the decimal point. When not dealing with a top heavy
fraction, simply place 0 to the left of the decimal point.
Step 2: Count the number of zeroes in the denominator (bottom
number). This is the same as the number of digits as there will be
to the right of the decimal point.
Step 3: The numerator (top number) is written to the right of the
decimal point. But, if it does not have enough digits, then it is
padded with extra zeroes placed between the decimal point and
the numerator.

Let's try an example:

Imagine that we wish to convert to a decimal.

Step 1: We are not dealing with a top heavy fraction, so the digit
to the left of the decimal point will be 0.
Step 2: There are two zeroes in the denominator (100), so we
need two digits to the right of the decimal point.
Step 3: There are already two digits in the numerator (71), so
that means we can simply place it to the right of the decimal
point without inserting any extra zeroes. The answer is therefore

92
 0.71.

Let's try another example:

Imagine that we wish to convert to a decimal.

Step 1: We are not dealing with a top heavy fraction, so the digit
to the left of the decimal point will be 0.
Step 2: There are two zeroes in the denominator (100), so we
need two digits to the right of the decimal point.
Step 3: There is only one digit in the numerator (4), so that means
we need to insert one extra zero between the decimal point and
the numerator. The answer is therefore 0.04.

Let's try another example:

Imagine that we wish to convert to a decimal.

Step 1: We are not dealing with a top heavy fraction, so the digit
to the left of the decimal point will be 0.
Step 2: There are three zeroes in the denominator (1000), so we
need three digits to the right of the decimal point.
Step 3: There is only one digit in the numerator (6), so that means
we need to insert two extra zeroes between the decimal point and
the numerator. The answer is therefore 0.006.

Let's try another example:

Imagine that we wish to convert to a decimal.

Step 1: We are not dealing with a top heavy fraction, so the digit
to the left of the decimal point will be 0.
Step 2: There are three zeroes in the denominator (1000), so we
need three digits to the right of the decimal point.
Step 3: There are only two digits in the numerator (23), so that
means we need to insert one extra zero between the decimal
point and the numerator. The answer is therefore 0.023.

93
Let's try one final example:

Imagine that we wish to convert to a decimal.

Step 1: We are dealing with a top heavy fraction, so begin by

converting it to a mixed number, . The number to the left
of the decimal point will be the same as the whole number part
of the mixed number, namely .
Step 2: There are three zeroes in the denominator (1000), so we
need three digits to the right of the decimal point.
Step 3: There are only two digits in the numerator (45), so that
means we need to insert one extra zero between the decimal
point and the numerator. The answer is therefore 23.045.

Questions:

1. Convert the following fractions to decimals.

Example Question: Convert to a decimal.

Answer: 0.65

Now you try:

1. (a) Convert to a decimal.

1. (b) Convert to a decimal.

1. (c) Convert to a decimal.

1. (d) Convert to a decimal.

1. (e) Convert to a decimal.

1. (f) Convert to a decimal.

1. (g) Convert to a decimal.

1. (h) Convert to a decimal.

94
1. (i) Convert to a decimal.

1. (j) Convert to a decimal.

1. (k) Convert to a decimal.

1. (l) Convert to a decimal.

1. (m) Convert to a decimal.

1. (n) Convert to a decimal.

1. (o) Convert to a decimal.

B. Other Easy Fractions to Convert to Decimals

We've already the method to convert fractions where the denominator
(bottom number) is a power of 10 into a decimal number. In a few
moments, we will look at how you can convert fractions with any
denominator into a decimal, but before we do that let's consider whether
there are any additional cases where we could use the method we've
already learnt.

It turns out there is: For some fractions, we can simply use the technique of
equivalent fractions to convert the denominator (bottom number) to a power
of 10 – that is to a number like 10, 100, 1000, 10000, etc.

Let's try an example:

Imagine that we wanted to convert into a decimal number.

The easiest way to do this would be to convert into the equivalent fraction
of (by multiplying top and bottom by 2). The denominator (bottom
number) is now a power of 10, so we can then easily convert into the
decimal 0.4.

Let's try another example:

Imagine that we wanted to convert into a decimal number.

95
As we are dealing with a mixed number, we know that the whole number
part (3) will go the left of the decimal point, so we only need to deal with
the fraction part, which corresponds to the numbers to right of the decimal
point.

Once again, the easiest way to deal with is to convert it into the
equivalent fraction of (by multiplying top and bottom by 2). And we
know that corresponds to a 4 to the right of the decimal point. Thus, the
overall answer is 3.4.

Let's try another example:

Imagine that we wanted to convert into a decimal number.

The easiest way to do this would be to convert into the equivalent fraction
of (by multiplying top and bottom by 25). The denominator (bottom
number) is now a power of 10, so we can easily convert into the decimal
0.75.

Let's try another example:

Imagine that we wanted to convert into a decimal number.

The easiest way to do this would be to convert into the equivalent fraction
of (by multiplying top and bottom by 5). The denominator (bottom
number) is now a power of 10, so we can easily convert into the decimal
0.35.

Let's try one more example:

Imagine that we wanted to convert into a decimal number.

The easiest way to do this would be to convert into the equivalent fraction
of (by multiplying top and bottom by 4). The denominator (bottom

96
number) is now a power of 10, so we can easily convert into the decimal
0.36.

Don't worry if you can't immediately spot which number you need to
multiply the top and bottom by – you will get better with practice. Also,
you should remember that this technique is a shortcut method of converting
fractions into decimals, so even if you struggle to apply this particular
technique, you can still solve the problem using the general method of
converting fractions into decimals that I will show you in the next part of
this chapter.

That said, here is a table you can use to see what to multiply the top and
bottom by to produce an equivalent fraction with a denominator which is a
power of 10:

Questions:

2. Convert the following fractions to decimals.

Example Question: Convert to a decimal.

Answer: 1.8

Now you try:

2. (a) Convert to a decimal.

97
2. (b) Convert to a decimal.

2. (c) Convert to a decimal.

2. (d) Convert to a decimal.

2. (e) Convert to a decimal.

2. (f) Convert to a decimal.

2. (g) Convert to a decimal.

2. (h) Convert to a decimal.

2. (i) Convert to a decimal.

2. (j) Convert to a decimal.

2. (k) Convert to a decimal.

2. (l) Convert to a decimal.

2. (m) Convert to a decimal.

2. (n) Convert to a decimal. (Hint: You may find it helpful to convert the
fraction to a mixed number before converting it to a decimal)

2. (o) Convert to a decimal.

C. Converting Any Fraction to a Decimal

So far we have looked at converting fractions where the denominator
(bottom number) is a power of 10 (or can which can easily be converted
into an equivalent fraction where the denominator is a power of 10) into
decimal numbers. But what if you wish to convert a fraction where the
denominator is not a power of 10, and can not easily be made into a power
of 10?

There is a method to do this: We rely on the fact that writing two fractions
are equivalent to division (it might seem a long time ago now, but we
looked at this way back in section C of Chapter 1 – feel free to revise this
section if necessary before continuing).

98
For example, do you remember that is equivalent to 3 ÷ 8?

A consequence of this fact is that if we want to convert into a decimal we

can simply calculate 3 ÷ 8, for example using the "bus stop method ".

Here are the steps:

Step 1: Draw the bus stop, with the number 8 in front of the bus stop, and the
number 3 underneath it.

Step 2: Since we are calculating a decimal number, we draw a decimal

point to the right of the number 3, and add some zeroes to the right of this.
We also add a decimal point above the bus stop, making sure it is lined up
with the decimal point below.

Step 3: We divide the first digit under the bus stop (3) by 8. We write the
number of times that 8 goes into 3 above the bus stop (0), and we write the
remainder before the next digit under the bus stop (shown as the red number
3 ):

Step 4: We divide the number under the bus stop (3 0) by 8. We write the
number of times that 8 goes into 3 0 above the bus stop (3), and we write the
remainder before the next digit under the bus stop (shown as the red number
6 ):

Step 5: We divide the number under the bus stop (6 0) by 8. We write the

99
number of times that 8 goes into 6 0 above the bus stop (7), and we write the
remainder before the next digit under the bus stop (shown as the red number
4 ):

Step 6: We divide the number under the bus stop (4 0) by 8. We write the
number of times that 8 goes into 4 0 above the bus stop (5). If there was a
remainder, we would write the remainder before the next digit under the bus
stop, and continue onwards until we had as many decimal places as we
want, however in this case the remainder is zero, so we can stop – we have

an exact decimal answer.

Result: The answer is that is equivalent to the decimal 0.375.

If you try this for yourself with different fractions, you will soon come
across examples where the division never ends with a zero remainder – you
can keep going forever. In such cases, it is normal to calculate the division
to one more decimal place than needed, and then round up or down.

Let's do a couple of examples like that – let's calculate and to three

decimal places.

First – we need to calculate 3 ÷ 7:

Step 1: Draw the bus stop, with the number 7 in front of the bus stop, and the
number 3 underneath it.

Step 2: Since we are calculating a decimal number, we draw a decimal

point to the right of the number 3, and add some zeroes to the right of this.
We also add a decimal point above the bus stop, making sure it is lined up
with the decimal point below.

100
Step 3: We divide the first digit under the bus stop (3) by 7. We write the
number of times that 7 goes into 3 above the bus stop (0), and we write the
remainder before the next digit under the bus stop (shown as the red number
3 ):

Step 4: We divide the number under the bus stop (3 0) by 7. We write the
number of times that 7 goes into 3 0 above the bus stop (4), and we write the
remainder before the next digit under the bus stop (shown as the red number
2 ):

Step 5: We divide the number under the bus stop (2 0) by 7. We write the
number of times that 7 goes into 2 0 above the bus stop (2), and we write the
remainder before the next digit under the bus stop (shown as the red number
6 ):

Step 6: We divide the number under the bus stop (6 0) by 7. We write the
number of times that 7 goes into 6 0 above the bus stop (8) , and we write
the remainder before the next digit under the bus stop (shown as the red
number 4 ):

Step 7: We divide the number under the bus stop (4 0) by 7. We write the
number of times that 7 goes into 4 0 above the bus stop (5) , and we write
the remainder before the next digit under the bus stop (shown as the red
number 5 ):

101
Step 8: We could continue calculating more and more decimal places, but
since we only needed the answer to three decimal places and have already
calculated four decimal places (we always need to calculate one extra
decimal place to know whether to round-up or round-down), we can stop.

Step 9: If we were to truncate the result after three decimal places, this
would give 0.428. However we need to look at the fourth decimal place to
see whether to round-up or round-down when truncating. Since, the fourth
decimal place is a 5, we must round-up when truncating (we round-up if it
is 5 or more), so the answer is actually 0.429.

Result: The fraction is equivalent, to three decimal places, to the decimal

0.429.

Now let's try – we need to calculate 5 ÷ 7:

Step 1: Draw the bus stop, with the number 7 in front of the bus stop, and the
number 5 underneath it.

Step 2: Since we are calculating a decimal number, we draw a decimal

point to the right of the number 3, and add some zeroes to the right of this.
We also add a decimal point above the bus stop, making sure it is lined up
with the decimal point below.

Step 3: We divide the first digit under the bus stop (5) by 7. We write the
number of times that 7
goes into 5 above the bus stop (0), and we write the remainder before the
next digit under the bus stop (shown as the red number 5 ):

102
Step 4: We divide the number under the bus stop (5 0) by 7. We write the
number of times that 7 goes into 5 0 above the bus stop (7), and we write the
remainder before the next digit under the bus stop (shown as the red number
1 ):

Step 5: We divide the number under the bus stop (1 0) by 7. We write the
number of times that 7 goes into 1 0 above the bus stop (1), and we write the
remainder before the next digit under the bus stop (shown as the red number
3 ):

Step 6: We divide the number under the bus stop (3 0) by 7. We write the
number of times that 7 goes into 3 0 above the bus stop (4) , and we write
the remainder before the next digit under the bus stop (shown as the red
number 2 ):

Step 7: We divide the number under the bus stop (2 0) by 7. We write the
number of times that 7 goes into 2 0 above the bus stop (2) , and we write
the remainder before the next digit under the bus stop (shown as the red
number 6 ):

Step 8: We could continue calculating more and more decimal places, but
since we only needed the answer to three decimal places and have already
calculated four decimal places (we always need to calculate one extra
decimal place to know whether to round-up or round-down), we can stop.

103
Step 9: If we were to truncate the result after three decimal places, this
would give 0.714. However we need to look at the fourth decimal place to
see whether to round-up or round-down when truncating. Since, the fourth
decimal place is a 2, we must round-down when truncating (we round-up if
it is 0, 1, 2, 3, or 4), so the answer is 0.714.

Result: The fraction is equivalent, to three decimal places, to the decimal

0.714

What about mixed numbers and top heavy fractions?

In the case of a mixed number, we can simply convert the fraction

part into a decimal and add it on to the whole number part. For
example, if we wanted to convert into a decimal, we would
convert into a decimal (it comes to 0.375 – we did this one
previously), and then do 2 + 0.375 giving us a final answer of
2.375.
In the case of a top heavy fraction, we either could first convert it
into a mixed number and then go on from there, or we could do
the division directly on the top heavy fraction – dividing the
numerator (top number) by the denominator (bottom number)
until we have both the number in front of the decimal point and
the number of decimal places we need.

Questions:

3. Convert the following fractions to decimals. Where an exact answer

is not possible, give the answer to 3 decimal places.

Example Question: Convert to a decimal.

Answer: 3.286 (rounded to three decimal places)

Now you try:

3. (a) Convert to a decimal.

3. (b) Convert to a decimal.

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3. (c) Convert to a decimal.

3. (d) Convert to a decimal.

3. (e) Convert to a decimal.

3. (f) Convert to a decimal.

3. (g) Convert to a decimal.

3. (h) Convert to a decimal.

3. (i) Convert to a decimal.

3. (j) Convert to a decimal.

3. (k) Convert to a decimal.

3. (l) Convert to a decimal.

3. (m) Convert to a decimal.

3. (n) Convert to a decimal.

3. (o) Convert to a decimal.

D. Common Fraction Conversions to Remember

As you work with fractions and decimals more and more, you will soon
discover that the same conversions come up again and again – so it is very
helpful to try to remember the most common conversions.

Here are some conversions which you really should remember – they come
up all the time!

Notes:

105
* converts into a "repeating decimal " (also known as a "recurring
decimal ") – the 3s repeat on and on forever. Therefore, when converting
into a decimal we usually round to the required number of decimal places,
such as 0.33, 0.333, 0.3333, etc. You can learn more about repeating
decimals in my book, You Can Do Math: Repeating Decimals .

** also converts into a "repeating decimal " – the 6s repeat on and on

forever. Therefore, when converting into a decimal we usually round to
the required number of decimal places, such as 0.67, 0.667, 0.6667, etc.
Once again, you can learn more about repeating decimals in my book, You
Can Do Math: Repeating Decimals .

The conversions for tenths (and please note that some fractions involving
tenths have equivalent fifths fractions, and is of course also equivalent to
) are relatively easy to remember, so hopefully you should be remember
these too:

Finally, if you can, I would also recommend trying to remember the eighths,
as these come up often enough to be worth learning:

106
107
Answer to Questions in Chapter 8

1. (a) 0.0003
1. (b) 0.0431
1. (c) 56.2
1. (d) 7.098
1. (e) 3.33
1. (f) 0.333
1. (g) 3.3
1. (h) 3.3
1. (i) 3.33
1. (j) 5.42
1. (k) 5.02
1. (l) 5.002
1. (m) 5.021
1. (n) 5.321
1. (o) 10.10 (or 10.1)
2. (a) 0.28
2. (b) 0.15
2. (c) 0.14
2. (d) 2.25
2. (e) 6.6
2. (f) 0.4
2. (g) 1.6
2. (h) 4.15
2. (i) 3.28
2. (j) 0.525
2. (k) 0.18
2. (l) 1.06

108
2. (m) 0.98
2. (n) 1.4
2. (o) 4.96
3. (a) 0.833 (rounded to three decimal places)
3. (b) 1.875 (exact answer)
3. (c) 5.222 (rounded to three decimal places)
3. (d) 1.857 (rounded to three decimal places)
3. (e) 0.444 (rounded to three decimal places)
3. (f) 1.375 (exact answer)
3. (g) 0.556 (rounded to three decimal places)
3. (h) 1.833 (rounded to three decimal places)
3. (i) 0.429 (rounded to three decimal places)
3. (j) 0.778 (rounded to three decimal places)
3. (k) 0.222 (rounded to three decimal places)
3. (l) 0.625 (exact answer)
3. (m) 3.667 (rounded to three decimal places)
3. (n) 3.5 (exact answer)
3. (o) 1.75 (exact answer)

109
Chapter 9: Converting Decimals to Fractions

Sometimes it is necessary to convert decimals into fractions. In this chapter,

we explain how you can do this. If you are not yet familiar with decimal
numbers, we suggest you leave this chapter for now and return to it once you
have learned about them .

Please Note: As this book is intended simply to introduce fractions, and

build a basic level of knowledge - this book does not cover the more
advanced topic of how to convert "repeating decimals " (also known as
"recurring decimals ") into fractions. That topic is discussed, in
considerable detail, in another one of my books – You Can Do Math:
Repeating Decimals . Readers who wish to learn how to convert repeating
decimals are advised to first complete this book, and then move on to that
book .

Converting a decimal number into a fraction is a straightforward process,

and relies on four facts:

Fact 1: The first digit to the right of the decimal point is tenths, the second
digit is hundredths, the third digit is thousandths, and so on.

Let's try applying this knowledge on some decimal numbers:

First consider the decimal number 0.2, this is the same as 2

tenths, or the fraction .
Now consider the decimal number 0.03, this is the same as 3
hundredths, or the fraction .
What about the decimal umber 0.004? This is the same as 4
thousandths, or the fraction .

Fact 2: We can read all the digits to the right of the decimal point together
as one number.

Let's look at some examples of this:

If we encounter the 0.27, we should think of it as not as 2 tenths

and 7 hundredths, but rather as 27 hundredths, and thus as the

110
 fraction .
Likewise if encounter 0.036, we should think of it as not as 3
hundredths and 6 thousandths, but rather as 36 thousandths, and
thus as the fraction .
Likewise if encounter 0.408, we should think of it as not as 4
tenths and 8 thousandths, but rather as 408 thousandths, and thus
as the fraction .
Likewise if encounter 0.574, we should think of it as not as 5
tenths, 7 hundredths and 4 thousandths, but rather as 574
thousandths, and thus as the fraction .

Fact 3: When dealing with a decimal with digit(s) to the left of the decimal
point, these simply form the whole number part of a mixed number.

Let's look at some examples of this:

If we encounter the 43.27, we should think of the digits to the left

of decimal point as being the whole number part of a mixed
number. So 43.27 converts into .

Likewise if encounter 7.036, we convert it into .

Likewise if encounter 10.408, we convert it into .

Likewise if encounter 9.574, we convert it into .

Fact 4: Many fractions can be simplified. This applies whether the fraction
represents the entire number, or just the fraction part of a mixed number.

Let's look at some examples of this:

If we were to convert the decimal 0.35 to the fraction , we

should be aware that there may be another simpler equivalent
fraction (we look to see if we can divide the top and bottom
numbers by a common number and thus simplify the fraction). In
the case of , we would note that the top and bottom are both

111
 divisible by 5, and the faction be simplified to .

Likewise if we converted 10.4 into , we, we would

recognize the fraction part can be simplified (top and bottom are
both divisible by 2), giving .

Likewise if we converted 7.036 into , we would

recognize the fraction part can be simplified (top and bottom are
both divisible by 4), giving .

Questions:

1. Convert the following decimals to fractions. Where applicable,

express the answer as a whole number or as mixed number with the
fraction part in its lowest terms.

Example Question: Convert 0.425 to a decimal.

Answer: which simplifies to

Now you try:

1. (a) Convert 0.64 to a fraction.

1. (b) Convert 0.82 to a fraction.
1. (c) Convert 0.502 to a fraction.
1. (d) Convert 0.025 to a fraction.
1. (e) Convert 0.999 to a fraction.
1. (f) Convert 16.1 to a fraction. (Hint: Write it as a mixed number – the
digits to the right of the decimal point form the fraction part)
1. (g) Convert 5.2 to a fraction. (Hint: Write it as a mixed number – the
digits to the right of the decimal point form the fraction part – then look to
see if you can simplify the fraction part)
1. (h) Convert 3.03 to a fraction.
1. (i) Convert 3.05 to a fraction.
1. (j) Convert 0.14 to a fraction.

112
1. (k) Convert 2.18 to a fraction.
1. (l) Convert 3.001 to a fraction.
1. (m) Convert 4.01 to a fraction.
1. (n) Convert 9.09 to a fraction.
1. (o) Convert 5.5 to a fraction.
1. (p) Convert 2.75 to a fraction.
1. (q) Convert 3.1 to a fraction.
1. (r) Convert 3.10 to a fraction.
1. (s) Convert 10.10 to a fraction.
1. (t) Convert 0.4 to a fraction.

113
Answer to Questions in Chapter 9

1. (a) which simplifies to

1. (b) which simplifies to

1. (c) which simplifies to

1. (d) which simplifies to

1. (e)

1. (f)

1. (g) which simplifies to

1. (h)

1. (i) which simplifies to

1. (j) which simplifies to

1. (k) which simplifies to

1. (l)

1. (m)

1. (n)

1. (o) which simplifies to

1. (p) which simplifies to

1. (q)

1. (r) which simplifies to

1. (s) which simplifies to

1. (t) which simplifies to

114
115
Conclusion
Well done! If you have worked your way through this whole book, you now
should understand fractions, as well as mastered all the basic arithmetic
operations (addition, subtraction, multiplication, and division) using
fractions.

Keep up the good work, and keep practicing and learning! You are well on
the way to become a math wizard!

For more fractions fun (and to learn about advanced fractions), please go to
http://www.suniltanna.com/fractions

And if you enjoyed this book or it helped you, please post a positive
review on Amazon!

To find out about other educational books that I have written, please go to:

For math books: http://www.suniltanna.com/math

For science books: http://www.suniltanna.com/science

Remember: If you enjoyed this book or it helped you, please post a

positive review on Amazon!

Thanks!

116
‫اﻟﻤﺤﺘﻮﯾﺎت‬
Introduction 4
Chapter 1: What is a Fraction? 7
Chapter 2: Mixed Numbers and Top Heavy Fractions 31
Chapter 3: Simplifying and Comparing Fractions 45
Chapter 4: Adding fractions 59
Chapter 5: Subtracting Fractions 71
Chapter 6: Multiplying Fractions 76
Chapter 7: Dividing fractions 86
Chapter 8: Converting Fractions to Decimals 92
Chapter 9: Converting Decimals to Fractions 110
Conclusion 116

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