BIRKBECK MATHS SUPPORT
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Numbers 3
In this section we will look at
- improper fractions and mixed fractions
- multiplying and dividing fractions
- what decimals mean and exponents or indices can be used
- how ratios and percentages work
- exponents or indices with an introduction to algebra
- some examples of calculations involving all numbers
- some examples of word questions involving numbers
Helping you practice
At the end of the sheet there are some questions for you to practice.
Don’t worry if you can’t do these but do try to think about them. This
practice should help you improve. I find I often make mistakes the
first few times I practice, but after a while I understand better.
Videos
All the examples in this worksheet and all the answers to questions
are available as answer sheets or videos.
Good luck and enjoy!
Videos and more worksheets are available in other formats from
www.mathsupport.wordpress.com
www,mathsupport.wordpress.com Jackie Grant, Birkbeck College, 2010
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1. Improper and Mixed Fractions
Here we will look at different ways that fractions can be written but mean the
same thing.
So far we have considered equivalent fractions, e.g.
Now we will introduce Improper Fractions and Mixed Fractions.
PROPER FRACTIONS
Proper fractions are fractions where the numerator (number on the top) is
always smaller than the denominator (number on the bottom), e.g.
Any proper fraction is always less than one, and we can see that if we draw 2/3
as a diagram we don’t quite have one whole, e.g.
So all proper fractions are less than one.
IMPROPER FRACTIONS
Improper fractions are fractions where the numerator (number on the top) is
always larger than the denominator (number at the bottom) e.g.
Any improper fraction is always bigger than one whole and we can see this if
we draw a diagram of four thirds.
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So all improper fractions are bigger than the number one.
MIXED FRACTIONS
A mixed fraction is a fraction where we mix whole numbers and fractions. For
example one and a third is a mixed fraction and we could write this as
but by comparing the diagram above we can see that four thirds is the same
as one whole and a third. We will now show this using maths
Where we have used the fact that any number divided by itself is one, e.g.
CONVERTING MIXED FRACTIONS AND IMPROPER FRACTIONS
Here are some examples of converting between different types of fractions. To
see videos explaining these calculations visit www.youtube.com/jgrantbbk
Example 1:
is an improper fraction and can be written as a mixed fraction.
First we need to know how many times the denominator of the fraction goes
into the numerator of the fraction. Well 2 is the denominator and I can write the
numerator 5 as 2+2+1. So I can write
So five out of two is the same as two and a half.
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Example 2: One and a quarter is a mixed fraction and is written
but we can
write it as an improper fraction in the following way.
Example 3: It also doesn’t matter if the fraction is negative. If we have minus
two and a fifth (a mixed fraction) we can change this to an improper fraction
Notice that the whole expression is negative. This is because the whole of the
mixed fraction is negative.
2. Multiplying and Dividing Fractions
After the hard work on adding fractions and converting between improper and
proper fractions you’ll probably be pleased to know that multiplying and
dividing fractions is much easy. We’ll start with a simple example.
MULTIPLYING FRACTIONS
If I see a cake and eat a quarter of it and I do this three times in total, then I will
have eaten three quarters of a cake.
And another way to write this is as
And here we can see that when we multiply fractions we just multiply all the
numbers on the top (3 x 1) to get the total numerator (3), and all the numbers
on the bottom (1 x 4) to get the total denominator (4).
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Here is another example
But notice that we can simplify the answer, since the numerator and the
denominator can be written as multiples of 2:
Key Points:

When multiplying fractions make sure all the fractions are in the form of
proper or improper fractions. If any fractions are written as mixed
fractions you will need to change them to improper fractions.

When multiplying fractions just multiply the numerators together
(numbers on top) and the denominators together (numbers on bottom).

Remember you may be able to simplify your answer.
Example with mixed fractions
1) Calculate
Well we can’t multiply these at the moment as the first fraction is a mixed
fraction. So first we have to change the mixed fraction to an improper fraction
Now we can re-write the original equation and calculate the answer
More videos of fractions are available at www.mathsupport.wordpress.com
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DIVIDING FRACTIONS
Dividing one fraction by another is not so difficult. To help us think about this,
consider an example.
If I have half of a cake and I divide this between 2 people what do I get? To see
the answer we can draw a picture. Here is my half a cake:
I now divide this between 2 people; and we can see that they each get a
quarter of the original cake. Here is one of those pieces
So this means
one half divided by two is a quarter
half divided by two equals a quarter
Where each of these statements mean the same thing. Notice that we get the
answer by turning the dividing fraction upside down, so
To see examples of dividing fractions visit www.youtube.com/jgrantbbk
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4. Decimals
Once we have started to understand fractions, then decimals are easier to get
to grips with.
We start with the simplest decimals
Where 0.1 is pronounced as ‘zero point one’ or ‘nought point one’.
If we now consider some more general decimals
So generally we can write the following eight fractions as decimals
Any decimal that can be expressed exactly as a fraction is called a rational
number. If a decimal can’t be written exactly as a fraction it is called an
irrational number. Decimals can be converted to fractions as follows:
For more examples see the videos on www.mathsupport.wordpress.com
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5. Percentages and Ratios
If you can get confident with fractions then you will be much happier with ratios
and percentages, as calculating ratios or percentages involve the same ideas.
PERCENTAGES
The word percentage means ‘out of 100’ and is written as %. So 50% means ’50
out of 100’, which we can write as a fraction then as a decimal
But if we look at the fraction now we see that we can simplify the fraction
So 50% is the same as 0.50 or the same as half.
Here is another example. Seven percent is 7% which means 7 out of 100, and
we can write it as a fraction and as a decimal
We can’t simplify this fraction though as no number goes into both 7 and 100.
Here are some examples of questions that can be asked about percentage. We
try to answer these questions in more than one way as this helps to check we
have the right answer.
Example 1: A dress is in the sale. The price says £25, but there is a 20%
reduction. What is the actual price of the dress?
Answer 1: first we need to find out what 20% of £25 is, and we can write this as
20% of £25
Therefore if the dress is reduced by £5, it means it is £25 - £5 = £20
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Answer 2: Now we’ll do the same question but a different way. If the dress is
reduced by 20%, then 20% has been removed from the original price, since
100% - 20% = 80%. The new price is 80% of £25.
So the new price of the jumper is £20, which is the same as the answer above.
Example 2: A roofer quotes £400 to replace a roof. He adds VAT of 20%. What is
the full cost to replace the roof?
Answer 1: The total cost is £400 plus the extra 20%. So first we find out what
20% of £400 is
Then the total cost is £400 plus the extra £80 = £480
Answer 2: The total cost is 20% on top of the quote for £400. The £400 is 100%
and then there is an extra 20%. This makes 120% in total. So the full cost is
120% of £400
So 120% of £400 gives the total amount of £480.
RATIOS
Ratios are a simple way to compare amounts, usually when mixing things
together. For example to make orange juice you need 1 part orange juice to 3
parts water. This is a ratio of 1 part 3 parts, so the glass would look like this.
This glass has 3 parts water and
1 part orange juice way. It is in
the ratio 3 to 1. Written as 3:1
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We can now see that because there is 1 part of juice and 3 parts water, there
are four parts in total. So orange juice is one part out of four, and the juice is
three parts out of the four. We find we are back to fractions and percentages.
So if we want to make 6 litres in total, we need 1/4 (or 25%) orange juice and
3/4 (or 75%) water, which can be calculated as
Or we can carry out the same calculation using percentages
6. Exponents, Indices or Powers
Lots of jobs, such as nursing, business or engineering involve large and small
numbers and decimals. A very useful way to write very small and very large
numbers is to use exponents, indices or powers.
For example we could write 10x10 as ‘10 to the power 2’, which means multiply
10 by its self two times, so
In the case of 103 , the number 10 is the base and the 3 is the exponent, index or
power. We can use the same idea for any number, so
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This becomes useful though for writing very large numbers. For example
This way of writing numbers is called Scientific Notation or Standard Form,
and we will look at this more in the worksheet on Measurements and Units
But before we go any further we need to think about indices a bit more.
RULES OF INDICES
To make calculations and simplifications with numbers written with powers or
indices we need to learn and use the Rules of Indices. Below we show how
they are true, give examples and the general rule.
We have used a small bit of algebra in this section and if you feel lost try
reading the next section on algebra or watching the video on indices.
Notice that
This means that
Now notice how the indices on the left-hand side (which are 2 and 3) add up to
give the index (this is singular for indices) of the answer (which is 5).
We can write is as a general rule
For example
or we could also write using symbols, where x could be any number, for
example in this case 2 and y is any other number, in this example 3
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But x could be any number and y could any other number and the rule would
still be true. Let us look at
We can see that the rule still works, and further that if we change 4 for 6,
So if we use the symbol a to mean any number we can write the
First Rule of Indices is
Where n, x and y represents any numbers.
Notice now what happens if we write the following fraction
Well we know that if something multiplies the top of a fraction and the bottom
of a fraction then we can cancel, so this fraction can be simplified to
But if we re-write the top and bottom in terms of indices we have
We can see that the index on the right hand side can be got from the index of
the numerator minus the index of the denominator: 5-3 = 2.
The second rule of indices is
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We now consider what happens when we have 2x2x2 and multiply this by itself
four times, well lets write it out (we have left extra spaces in between each
group of 2x2x2)
But we can see on the left hand side that the total number of 2s multiplying
each other is twelve which is the same as 4x3, so we can write
(If you are not completely comfortable with using brackets there are lots of
examples explaining brackets in the first worksheet in the Calculations section.)
Writing this as a general rule for any number we have
The third rule of indices is
Finally: we notice three other things about indices:
Often it is only through practicing examples that things become clear. So try
these examples then read the theory again and it will become clearer.
Examples: simplifying using indices:
1)
6)
2)
7)
3)
4)
5)
8)
9)
10)
It is possible to have fractional indices and we will look at the meaning of these
in the Calculations section.
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6. Writing big and small numbers with indices
We already mentioned that it is possible to write large numbers using index
notation. For example
. But we will see now that it is also
possible to write small numbers in index form:
Consider the following decimals:
Continuing to follow this pattern we see that we could write
So for any small number, for example 0.002 we can write
If your job is writing small numbers regularly then it is easy to miss out the zeros
and so using indices is a much safer and easier way to write the numbers and
the index tells you how many zeros, including the one before the decimal point.
Here we will look at more examples:
1)
2)
3)
5)
6)
7)
8)
9)
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7. So what is Algebra?
Algebra uses the rules of numbers, such as addition, division, but instead of
using numbers it use variables. For example if I buy two arm chairs and a sofa,
then I can use the variable A to represent the armchairs and the variable S to
represent the sofa, then I have
If I now decide to buy another arm chair I have
But this is the same as three armchairs and a sofa, so we can simplify
SUBSTITUTION
Often the letters x and y are used to represent variables. For example we can
let x represent one burger and y represent one portion of fries. So if the first
customer orders two burgers and three portions of fries I can write this as
But now the second customer orders five burgers and two fries we write
Suppose we now find out that one place sells burgers for £1.50 and fries for
£1.00, we can substitute these in the following way
But if the burgers are £1.00 and the fries are £1.20, substitution gives
So we can see that it is cheaper for the first customer to go the first cafe but it is
cheaper for the second customer to go to the second cafe.
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8. Now your turn
Generally the more maths you practice the easier it gets. If you make mistakes
don’t worry. I generally find that if I make lots of mistakes I understand the
subject better when I have finished. If you want to see videos explaining these
ideas and showing the answers visit www.mathsupport.wordpress.com
A) Convert the following improper factions into mixed fractions
1)
4)
2)
5)
3)
6)
B) Multiplying and Dividing Fractions, try also to simplify your answer
1)
6)
2)
7)
3)
8)
4)
9)
5)
10)
C) Decimals, Percentages and Ratios
1)
2)
3)
4)
5)
6)
7) Water is mixed with glue in the ratio
3:1. If I use 1 litre of glue, how much
water do I need?
8) Orange juice is mixed in the ratio 1
parts juice to 4 parts water. How much
water is in 100ml of mixed juice?
9) The ratio of milk to dark chocolates
is 3:4. If there are 140 chocolates in a
bag, how many are milk chocolates?
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D) Working with indices, simplify the following
1)
6)
2)
7)
3)
8)
4)
9)
5)
10)
E) Word Questions
1) In a restaurant each slice of cake sold is one eight of a total cake. If there are
three and a half cakes left how can you show mathematically how many slices
are left for sale?
2) A suit has a price tag £100 and a shirt has a price tag £40. But there is a
special offer and everything is reduced by 15%. How much would the suit and
shirt cost with the reduction?
3) Humans produce 2 million red blood cells each second. If each blood cell
has mass of
. Show using indices the mass of blood cells produced
each second.
4) Two brothers decide to share the cost of a car. One of them pays £800 and
the other one £200. Write these amounts as the simplest ratio possible.
5) Jeans are sold for £30 each. The selling price is 50% more than the cost
price. What is the cost price of each pair of jeans?
6) The ratio of men to women working in a factory is 5:4. There are a total of 20
women. How many people work at the factory in total?
7) A survey reports that one half of families living in a particular block of flats
have a pet. Of these people half had a fish, a quarter had a hamster and a
quarter had a dog. What fraction of families had a fish?
8) A television has a mark up on the cost price of 40% and is then sold in the
sale for 10% discount. If the TV originally cost £100, how much was it sold for?
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