IGC SE
MATHEMATICS z
Fractions, percentages and
basic number work
z
SATURN IS 1.2 X 109KM FROM EARTH!
This week’s work is
required to fulfill these
topics on the syllabus:
Topic 1: number, set notation
and language (core)
Topic 2: squares and cubes
(core)
Topic 3: vulgar and decimal
fractions
Topic 6: standard form (core)
Topic 7: the four rules (core)
Topic 8: estimation (core)
Topic 11: percentages (core)
(supplement)
Topic 12: use of a calculator
(core)
When you have completed this
week’s work, you will be able to:
This week’s work covers a few
basic number topics, but one
of the most important for
scientists is
that of
standard
form. Really
large
numbers (like
1.2 billion
km) need to
be written in
a shortened
format, to
make them manageable for
calculations. If you don’t
understand the way the
number in the title is written,
don’t worry, this lesson will
teach you how to write
numbers in this format. All
the topics covered this week
• simplify fractions
• use directed numbers
and order operations
• increase or decrease an
amount by a given
percentage
need to be covered by all
students. You won’t know if
you need to go into the core or
supplement
exam until
you have
finished at
least one
term of
work. In the
box below,
the topics
covered are
either in
orange or purple. Orange
topics need to be covered by
all students, but purple topics
will only appear on the
supplement exam. If you find
these topics increasingly hard,
then speak to your supervisor
about which exam to take.
• express numbers in
standard form
• round using significant
figures and decimal
places
Pace yourself....
Each week, you will be expected to complete a chapter of your text book. Some chapters are bigger
than others. Have a look through the lesson that is posted and the chapter, work out how you will
divide up the work over the week. You may wish to do it in 3 one and a half hour sessions, or 5 one
hour sessions. The work is normally divided up into different topics and it will be less confusing if
you cover one topic a day and answer the exercises on that topic. For this week only, I have put a day
next to each topic to give you an idea of how to pace your work.
WHEN TO HAND IN ASSIGNMENTS
Have a look at the text book and you will see it is divided up into 5 units, or modules. Each unit has 5
sections to it. Number and algebra always come first and these will take two weeks. After the algebra
section has been completed, you will need to sit an online multichoice assignment. It will be posted
the same week as your algebra lesson. Click on each question, use paper and pen to work out the
answer and then click on your choice of answer. Number and algebra are being tested this way as
they take the longest to type up into a word processed document. You still need to do the same
amount of work as if you were submitting a typed document. Keep records of your working out. If
you find that you haven’t done well with these assignments, email me for more help. Look in your
text book and find a question out of an exercise which is the type you need more help with, email me
with the question and exercise number and I will set it out for you.
You then have 3 weeks to complete the graphs, shape and space and handling data sections. After the
handling data lesson, there will be a written assignment to complete. You can either type out your
answers (with full working out shown!) Or, you can scan your handwritten work in (this needs to be in
a word processing document so that I can add my comments to it). It is normal for me to give you
further guidance after the first assignment about how it needs to be set out and this is training for
what is expected of you in the exam.
Fractions, percentages and decimals
(Monday)
Are you confident with decimals? Start with the number 5 and keep adding 0.4 until you get to 20.
Write down the answer each time. Now check with your calculator - were you correct?
Now look at these vulgar fractions:
a) 1/2 b) 3/4 c) 2/5 d) 4/10
Can you change these to decimal fractions (this is the correct name for what we usually call
‘decimals’)? Answers are: 0.5, 0.75, 0.4, 0.4
Egyptian number system
Even the Ancient Egyptians used a decimal number system. You can see in the picture, the images
they used for the various powers of ten. If you get confused with decimals, then you need to rewind
back to the place value tables that you used when first learning about what a number means. There
is no shame in needing to revise this table and you should keep a copy in front of you for as long as
you need it until you can do the work in your head without looking. Here’s what it looks like:
H
T
U
.
1/10
1/100
0
.
4
0
.
0
3
0
.
5
5
1/1000
1/10000
5
Using the table makes it easy to change from decimal fractions to vulgar fractions. You can see that
I’ve put the number 0.4 in the table. As the last digit, 4, is in the tenths column, then this is 4 tenths,
or 4/10
The second number in the table is 0.03 Here, the last digit is in the hundredths column, so as a
vulgar fraction, this should be written as 3/100
The last number in the table is 0.555 and this is written as 555/1000 as the last digit is in the
thousandths column. After writing a number as a vulgar fraction, you need to cancel it down to it’s
simplest term, so 555/1000 simplifies to 111/200
Going the other way, from fractions to decimals can be harder. If a fraction has a denominator
(bottom number) which is a power of ten, then it’s easy. You can use the table. If you had 3/10 for
example, this means 3 tenths, so you put 3 in the tenths column and you have 0.3
If you had a denominator that could be easily changed to a power of 10, for example 6/50 can be
changed to 12/100, then the problem is not too hard. The last digit, the 2, needs to be in the
hundredths column and you have 0.12
Vulgar fractions that can’t be changed to decimals in this easy way can be changed using a calculator.
For example, 3/8 As 8 does not divide exactly into 100, then you can use the method of dividing 3
by 8. After all, a vulgar fraction such as 3/8 does mean 3 divided by 8.
If you put this into your calculator, you will get the answer 0.375. To check if you are correct, write
0.375 as a fraction and you get 375/1000 now cancel this down to it’s simplest form and the answer is
3/8
Changing a fraction to a percentage is easy if it already has a denominator of 100. ‘Per’ cent means
‘per’ hundred. So 23/100 is 23%
Again, if you have a denominator which easily changes to 100, like ten, then this is also simple.
6/10 = 60/100 = 60%
Using our example above,
3/8 = 0.375
Now we need to recall how to change a decimal to a percentage. You simply multiply by 100. Going
back to our place value table, this is the same as shifting the number two places to the left, so
0.375 x 100 = 37.5
Many people learn this as moving the decimal point two places to the right. If this helps you, then
don’t forget it, but it is actually the number which is being moved to the left as it is multiplied by each
power of ten.
3/8 = 0.375 = 37.5%
Go...
This week, you have a choice, you should now be able to complete Exercise 1 in your text book.
Read the examples first, take notes in your maths book and then work through the exercise. If you
feel confident with the work, you can complete exercise 1* instead (this is the slightly harder version of
exercise 1). However, if you have not received your text book yet, you can work through the
worksheets posted this week instead, or you can do the worksheets instead of the book exercise!
REMEMBERTO MARK YOUR WORK AS YOU GO. THERE IS NO POINT DOING
QUESTION AFTER QUESTION IF YOU HAVE USED THE WRONG METHOD WITH
THE FIRST ONE! ANSWERS CAN BE FOUND IN THE PARENTS’ SUPPORT AREA.
Order of operation (Tuesday)
Hopefully, ‘order of operation’ should bring to mind BODMAS or BIDMAS. This is another topic
that should have been covered pre-IGCSE, but is reviewed here because of it’s importance. Have a
look at this problem:
4-3x2
What is the answer? Well, depending on what order you did this problem, you could come up with 2
(by subtracting 3 from 4 to get one then multiplying by 2) or you could come up with -2 (by
multiplying 3 by 2 to get 6, then subtracting 6 from 4). Actually, the second answer, -2, is correct.
That is because multiplications should be carried out before subtractions.
You should be able to see from this very simple example, that the success of your future maths career
rests on your ability to solve mathematical problems in the correct order!
This topic is easy to follow. The acronym BIDMAS stands for:
Brackets
Indices
Division
Multiplication
Addition
Subtraction
If there are any of these operations in your calculation, then they need to be carried out in the order
shown. If you are used to using the word BODMAS instead, there is no difference in order, just that
the ‘O’ stands for ‘of ’ as in ‘to the power of ’ instead of being called an index number. There are
good examples given in the exercise book on this topic and if you don’t have your book yet, then you
can read through these when it arrives.
Directed number is also covered with this topic. You have been using directed numbers since starting
maths. 2 + 2 means ‘positive 2 added to positive 2’. We assume all numbers are positive unless
otherwise stated. But calling a number a directed number, means to give it direction. Is it negative or
positive?
In your text book, you will see calculations such as:
(-2) x 4
which means ‘negative 2 multiplied by positive 4’ The brackets are used to clarify the direction of the
number, rather than what operation you should perform.
Go...
You need to read the example on page 2 of your text book. If you are a little unsure of the topic still,
work through exercise 2 in order. The questions will get gradually harder. If you are fairly confident,
work through the even numbers of exercise 2 then the even numbers of exercise 2*. If you are super
confident, work through the whole of exercise 2* only. (This is the guidance for all your work - you
should start to get an idea of how difficult or easy you are finding your maths course).
If you haven’t received your textbook yet, click on the directed numbers weblink posted this week,
read through the examples and have a go at the questions.
Percentages (Wednesday)
You can see from the pictures above, just a few examples of the uses of percentages. The ability to
work with and understand percentages is not only important in many jobs,where you need to
calculate with them, but also in your daily life, for shopping, working out your taxes, taking out a loan
and much more.
In the introduction, we looked at changing fractions and
decimals to percentages. We also need to look at finding a
percentage of an amount and working out what percentage one
number is of another number.
Firstly, finding the percentage of an amount. Every time you
need to reduce a number by a given percentage, such as in a
sale, you will need to use this technique. As percent means per
hundred, it is easiest to start out by looking percentages as an
amount of 100.
Example
What is 12% of 100?
This means ‘what is 12 out of every hundred’. As we only have
one hundred here, the answer is 12.
What is 12% of 300?
As we have three hundreds here, to find 12%, we have 3 lots of 12,
which is 36
These are easy examples as the number is a multiple of 100. How
do we find 12% of 150?
Well, this is also not too hard, we can find 12% of 100 (12) and add this to 12% of 50 (6) so the
answer is 18. But, as the numbers get harder, you will need to get out your calculator!
12 out of 100 can be written as 12/100
so 12% of 150 can be written as:
12/100 of 150
In maths, the ‘of ’ means multiply, so this can be shown as:
12/100 x 150
Now, get your calculator out and input 12 divided by 100, then multiply by 150. You should get the
answer 18.
This method can now be used for the harder calculations.
Example
What is 21% of $256?
Put into the correct form:
21/100 x 256 = $53.76
Don’t forget to answer all money questions to 2 decimal places.
Many questions involving percentages ask you to find a reduced or increased
price. What is the new price of a coat that originally cost $256 and is now reduced by 21% in a sale?
We worked out 21% of $256 above, it’s $53.76
256 - 53.76 = 202.24
There is a quicker way to do this calculation. If we reduce a coat by 21%, we really need to find
79% of the original cost (this is the original 100% minus 21%)
Putting this into your calculator:
79/100 x 256 = 202.24
Now we have reduced a two stage calculation into one stage. You can also put this straight in as a
decimal if you are confident:
0.79 x 256 = 202.24
Increases work just the same. If a man buys a house for £90000 and sell it three years later at an
increase of 8%, how much does he sell it for?
The old method would be to find 8% and add it on, but you need to find the original 100% plus 8%,
so this is 108% of the original value (yes, you can have more than 100%!)
As a fraction:
108/100 x 90000 = £97200
As a decimal:
1.08 x 90000 = £97200
Much quicker!
Go...
You need to read the example on page 3 of your text book. If you are a little unsure of the topic still,
work through exercise 3 in order. The questions will get gradually harder. If you are fairly confident,
work through the even numbers of exercise 3 then the even numbers of exercise 3*. If you are super
confident, work through the whole of exercise 3* only. (Same method as before!)
If you haven’t received your textbook yet, you will need to work through the additional pdf posted
this week on fractions and percentages to answer questions on this topic.
Exodus 12:37
The Israelites journeyed from
Rameses to Sukkoth. There
were about six hundred
thousand men on foot, besides
women and children.
Standard form positive indices
(Thursday)
Linking back to the introduction, using standard form is a method of writing large numbers in a
more manageable, compact form. Once they are written in this form, it will be much easier to use
them in calculations later on.
When you add 300 to 500, you don’t begin at the number one and start counting. You ignore the
hundreds and add 3 to 5 and get 8. Using this method, you know that 300 plus 500 is 800. Standard
form takes a number between 1 and 10 and then looks at how many times it needs to be multiplied by
10. It is easier seen by example:
300 is the number 3, multiplied by 100 (that is multiplied by 10 then 10 again)
This can be written as:
300 = 3 x 100 = 3 x 102
As 102 = 100
Using this method, 6000 is the number 6 multiplied by 1000
(which is 10 x 10 x 10 or 103) This can be written as:
6000 = 6 x 1000 = 6 x 103
According to the quote from Exodus, the number of men
leaving Rameses was 600 000 men. Here’s a picture of the
Obama rally in 2008. There were approximately 100 000
people there - this gives you some idea of what a group of 600
000 people looks like!
Here’s 600 000 in standard form:
600 000 = 6 x 100 000 = 6 x 105
Now that you know how to write numbers in standard form, you are ready to calculate with them.
Here is the wonderful thing with standard form, you can do really big calculations really quickly.
Take the number 10000 and multiply it by 1000000. Can you do this in your head? Probably not!
But, when you multiply powers of 10 by each other, you simply add the zeros. Here’s some examples:
10 x 10 = 100
100 x 10 = 1000
100 x 100 = 10000
You can see that the answer has the number of zeros found in the two first numbers added together.
Using this method,
10000 x 1000000 = 10000000000
In standard form, this can be written as:
104 x 106 = 1010
The rule here is that when you multiply numbers with the same base (here the base is 10) then you
add the indices. This is known as the first rule of indices. In algebra, this is stated as:
an x am = an+m
The second rule of indices is about dividing numbers with the same base.
100 ÷ 10 = 10
1000 ÷ 10 = 100
Each time, you can see that if you subtract the number of zeros of the second number, from the first
number, you have the correct number of zeros in the answer. So, for a larger number:
100000 ÷ 100 = 1000
In standard form:
105 ÷ 102 = 103
The rule here is that if you divide numbers of the same base, you subtract the second index from the
first. In algebra, this is stated as:
an ÷ am = an-m
The third rule is about using brackets. Here is a problem in standard form:
(104)2
in long-hand, this can be written as:
(10 x 10 x 10 x 10) x (10 x 10 x 10 x 10) = 100000000 = 108
The rule here is that the indices are multiplied. In algebra, this is written as (an)m = anm
Go...
Further examples of these rules can be found on page 5 and after reading them, work through
exercise 4 (or 4* if you feel confident) Mark your work. If you haven’t received the book yet, you can
work through the section on indices posted this week along with the lesson.
S.F and D.P (Friday)
Here, I have written the abbreviation for significant figures (s.f) and decimal places (d.p) You need to
get used to using these as often as possible. From now on, each time you round an answer, you should
be stating the degree to which you have rounded your answer. Most students have covered these
rounding methods before starting IGCSE, but it is necessary to make sure that it is fully understood
before continuing with the course. Research work covered by one scientist is often continued by
another. If the first scientist didn’t make good notes of what he was doing, then the future scientists
would find it very difficult to continue on with his work. One of the
most important technical notes by a scientist will be the degree of
rounding he has used. For example, when designing a road bridge did he round the mass of vehicles it can take to the nearest tonne or
kg?
Decimal places
This is probably the easiest of the two. If you are asked to round a
number to 3 d.p then you count 3 numbers after the decimal point
and chop the rest of the digits off. If the digit after the third
decimal place (the forth) is 5 or over, you round the third digit up. Here’s some examples:
Round 3.12345 to 3d.p
Answer:
3.123
Round 3.214567 to 3d.p
Answer:
3.215
The third decimal place is rounded up to 5 as the number after was 5 or over.
Significant figures
Some students find this technique harder, but it follows set rules. If you are asked to round a number
to 3s.f, you count 3 non zero digits from the left and change the rest of the digits to zero. The most
important point to note here is that the magnitude (size) of the number can not be changed. For
example, if you were asked to round 935 to the nearest hundred, you would round to 900, not 9.
What you have actually done here is round to 1s.f since you only have
one significant figure (the 9) and the rest are zeros.
(Our picture shows the alternative definition of a significant figure!)
Here’s another example. Round 23456 to the nearest thousand. The
answer is 23000. This is the same as rounding to 2s.f.
Round 80290 to 3s.f and the answer is 80300 (we have 3 significant
figures, but after the third, there is a number over 5, so we need to round
up the third digit) after that, the rest of the number turns to zeros.
When rounding decimals, you can ignore all zeros after the decimal point unless they are place
holders. For example,
Round 0.12345678 to 3s.f
counting from the left, the first three non zero numbers are 123, so the number rounds to
0.12300000
But the zeros that follow after 123 do not change the size of the number and can be ignored.
However, if you are asked to round 0.1023456 to 3s.f the answer would be
0.102
Here, the zero in between the 1 and the 2 is a place holder and can not be ignored. It is there to
make the number the size that it is.
The best way to practice these is just to answer a lot of them. I would suggest that all students work
through exercises 5 and 5*. Marking as you go to make sure that you have fully understood this
important work.
If you find you are getting most answers wrong to this exercise, or any of the others that you have
done, then email me soon, before the work begins to accumulate!
Again, if you haven’t received the book, you can work through this work in the worksheets posted this
week.