Download Fractions, Percentages, Ratios, Rates

Fractions, Percentages, Ratios, Rates
Fractions
3
Equivalent fractions
6
Simplifying fractions
11
A closer look at decimals
12
Changing between fractions and decimals
13
About percentages
17
Ratios
19
Rates
24
Using index notation
26
Very large and very small numbers
29
Suggested answers to activities
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Fractions
Consider this problem.
Divide 21 by 10.
The problem here is that 10 does not divide evenly into 21. It divides into it
twice, but there is a remainder left over. The answer is in between 2 and 3.
This could be illustrated on a number line, as follows.
How will you write down the answer?
You may have been taught to write something like ‘2 remainder 1’ or ‘2 r 1’
as your answer.
.
This gives the size of the remainder (1) relative to the number you were
dividing by (10).
Now use a calculator to divide 21 by 10.
Press 21 ÷ 10 =
Your calculator will express the answer as 2.1.
In everyday life, we often need to answer a question like the following.
10 cans of drink cost $21.00. What is the cost of one can?
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This time the question is about money, so you need to include a $ and two
figures after the decimal point in your answer. This time you need to write
the answer as $2.10.

There are many different ways of writing in-between numbers.

The different notations include: fractions, decimals, currency,
percentages, and ratios.

There is an infinite number of these numbers located between each
whole number.
A closer look at fractions
A fraction is simply a part of a whole.
When you write a fraction, you have a top number and a bottom number.
The top number tells us how many parts we are looking at; we call this the
numerator.
The bottom number tells us how many parts the whole is divided into.
In fractions we call the bottom number the denominator.
For each of the diagrams below:
(i) how many parts are shaded (the numerator)?
(ii) write the fraction that matches the shaded part.
(a)
(i)
_____
(ii)
_____
(i)
(ii)
_____
_____
(b)
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(c)
(i) _____
(ii) _____
(d)
(i) _____
(ii) _____
(a)
(i)
(ii)
2
(b)
(i)
(ii)
3
(c)
(i)
(ii)
5
(d)
(i)
(ii)
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Equivalent fractions
Equivalent fractions are fractions that have the same value, you can see this
by looking at the fraction diagrams.
Using multiplication to get an equivalent fraction
What happens when a fraction is changed to an equivalent fraction? Look at these examples.
Example 1
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Example 2
In these examples, both the numerator and denominator were multiplied by
the same number.
Find the missing number to complete the equivalent fraction:
(a)
(b)
(a)
(multiply numerator and denominator by 5)
(b)
(multiply numerator and denominator by 2)
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Using division to get an equivalent fraction
Sometimes we need to use division to form equivalent fractions.
Example 1
Example 2
In these examples, both the numerator and denominator are divided by the
same number.
Find the missing number to complete the equivalent fraction:
(a)
(b)
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(a)
(divide each by 3)
(b)
(divide each by 7)
Equivalent fractions are formed by multiplying or
dividing the numerator (top) and denominator (bottom)
by the same number.
Activity 4A
1
What fraction of the whole is shaded?
2
Divide this rectangle into quarters.
3
Divide this rectangle into eight
equal parts.
4
Compare your answers to Questions 2 & 3. What do you notice?
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5
Divide this shape (do you know its name?) into six equal parts.
6
Convert these fractions to their equivalent fractions by filling in the missing numbers:
(a)
(b)
(c)
(d)
(e)
7
Convert these fractions to their equivalent fractions by filling in the missing numbers:
(a)
(b)
(c)
(d)
(e)
Check your answers against those at the end of the topic.
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Simplifying fractions
Whenever you work with fractions, you are often expected to simplify them,
that is, reduce them to their lowest terms.
This means you have to find an equivalent fraction that has no common
factors in its numerator and denominator.
The easiest way to simplify a fraction is to divide the numerator and
denominator by the highest common factor.
Example
The factors of 20 are:
1, 2, 4, 5, 10, 20
The factors of 24 are:
1, 2, 3, 4, 6, 8, 12, 24
The highest common factor is 4.
So
However, if you are not sure of the highest common factor, you can keep
dividing the numerator and the denominator by any common factor until you
get the simplest terms. This is a little slower, but will give you the correct
answer. This is how you would do it.
Example
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Activity 4B
Write each of these fractions in their simplest terms.
(a)
(b)
(c)
(d)
(e)
(f)
Check your answers against those at the end of the topic.
A closer look at decimals
The use of decimals to denote ‘in-between’ numbers is really just an
extension of our whole number system.
In Topic 1, you learned about our place value system, so that when we write
‘four hundred and sixty two’ in numerals,

the 4 must be correctly placed in the hundreds column

the 6 must be correctly placed in the tens column, and

the 2 must be correctly placed in the units column
So the number is 462.
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The same is true with decimal parts of numbers. The position of each
numeral gives it a distinct value.
When we say a decimal number, each digit is read separately, so that 0.005
is read as ‘zero, point zero, zero, five’. To read 462.005 we have ‘four
hundred and sixty two, point zero, zero, five’.
Changing between fractions and
decimals
Fractions such as tenths, hundredths and so on are simple to change from
fraction notation to decimal notation.
Using the place value table given above:
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Changing from decimal notation to fraction notation is a similar process.
Other types of fractions are not as easy to change, if they do not have a
denominator of 10, 100, 1000 etc.
Example
To do this, we use the equivalent fraction method.
Another example
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Change the fraction to a decimal:
(a)
(b)
(c)
(a)
(b)
(c)
Activity 4C
1
Write these numbers using decimal notation:
(a) six tenths
(b) fifteen hundredths
(c) eight thousandths
(d) 34 thousandths
(e) three twentieths
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(f)
(g)
(h)
(i) three quarters
2
Write these decimals as fractions:
(a) 0.1
(b) 0.09
(c) 0.047
(d) 1.25
(e) 2.2
(f) 10.6
3
Which is bigger?
Example:
0.4 or 0.05?
Rewrite the decimals so they have the same
number of digits after the decimal point. Then
compare them like whole numbers.
0.4 = 0.40
0.05 = 0.05
0.4 is bigger
For each pair of numbers, decide which is bigger.
(a) 0.6 or 0.66?
(b) 4.4 or 4.404?
(c) 0.20 or 0.02?
(d) 5.818 or 5.88?
(e) 19.0505 or 19.505?
Check your answers against those at the end of the topic.
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About percentages
There is only one thing to remember when dealing with percentages, and
that is to keep thinking of what the word means. Simply,
The symbol for ‘per cent’, %, is a reminder of this fact. Think of it this way:

the slash / represents the horizontal fraction bar indicating ‘out of’

the noughts either side are the two noughts in 100.
So ‘per cent’ means
or ‘out of a hundred’.
Therefore:
This means that ‘percentage’ is really a ‘base 100’ idea, and that makes it a
simple extension of our base 10 (decimal) number system. Percentages are
used every day in shopping (15% off) and banking (interest rate 5.5%), as
well as in many other applications. You should be able to switch between
fraction, decimal and percentage notation for everyday numbers.
Example
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Activity 4D
Complete the missing cells in the following table.
Words
Fraction
Decimal
Percentage
(a) 3 tenths
(b)
(c) 6 tenths
(d)
60%
0.25
(e) 3 hundredths
(f)
(g)
5%
(h)
(i) 1 whole
Check your answers against those at the end of the topic.
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Ratios
A ratio compares two quantities of the same kind.
Order is important.
Here, the ratio of men to women is 2 to 1. We write this as 2 : 1.
This jug of lemon cordial was made by mixing 1 part syrup to 4 parts of
water.
The ratio of syrup to water is 1 : 4.
Adam has a collection of model planes.
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Or, looking at it another way, the length of each part on the real plane is
30 times the length of the corresponding part of the model.
As you can see, ratios can be written as fractions.
Example 1
In the main street of town, there were 6 sedans and 14 vans parked at
lunchtime. What is the ratio of sedans to vans?
Answer:
sedans : vans = 6 : 14 (take care to keep the correct order)
= 3 : 7 (reduce the ratio to its simplest form, by dividing
both parts by the same number, in this case 2)
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Simplify the ratio 24 : 36 _________________________________________________
2
There are 12 boys and 8 girls in a class. What is the ratio of girls to boys?
_____________________________________________________________________
1
2 : 3 (divide both parts by 12)
2
8 : 12 = 2 : 3 (divide both parts by 4)
Did you get the order correct?
Example 2
At 3 pm, the ratio of vans to trucks was 5 : 2. If there were 20 vans,
how many trucks were there?
Answer:
vans : trucks
=5:2
= 20 : ?
Now 5  4 = 20, so the number of trucks is 2  4 = 8.
You may also use the equivalent fraction method.
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1
Complete the ratios by filling in the blank squares.
(a) 2 : 5 = 4 :
(b) 8 : 1 = 24 :
(c) 3 : 2 =
:6
(d) 3 : 4 =
: 40
(e) 2 : 3 : 4 =
2
: 15 :
At a newsagency, the ratio of comics to novels is 8 : 25. How many novels are there if
there are 80 comics? ____________________________________________________
_____________________________________________________________________
3
The ratio of brown smarties to orange smarties is 2:3. Tim opens a packet and counted
21 orange smarties. How many brown smarties would he expect to find?
_____________________________________________________________________
1
(a)
(b)
(c)
(d)
(e)
4 : 10
24 : 3
9:6
30 : 40
10 : 15 : 20
2
Comics : novels = 8 : 25
= 80 : 250
There are 250 novels.
3
Brown : orange = 2 : 3
= 14 : 21
There are 14 brown smarties.
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Activity 4E
1
In a box of smarties, there are 10 yellow, 12 blue, 10 red and 8 green. Write the ratio of
(a) yellow to blue
(b) green to red
(c) red to yellow.
2
(a) The local Rotary club has 30 male members and 12 female members. What is the
ratio of male to female members in Rotary?
(b) The local garden club has four male members and 18 female members. What is
the ratio of male to total members in the Garden club?
3
Kay’s height is three times her son Charlie’s. What is the ratio of Kay’s height to
Charlie’s height?
4
Aziz mixes cement and sand together in the ratio 1 : 4. If he uses 25 kg of cement,
how much sand will he need?
5
Margaret’s special Christmas cake recipe uses cherries and almonds in the ratio 3 : 2.
If the recipe requires 100 g of almonds, what quantity of cherries is required?
6
Two towns are 1.7 cm apart on a map, which is drawn to a scale of 1 cm : 10 km.
How far apart are the towns in real life?
7
Liam’s model helicopter is manufactured to a scale of 1 : 20. If the rotor blades on a
full-sized helicopter are 480 cm long, how long would they be on Liam’s model?
Check your answers against those at the end of the topic.
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Rates
A rate compares two quantities of different kinds.
Examples
(a) 80 kilometres per hour (80 km/h) is a speed or rate of travel each hour.
(b) $12.50 per hour is a rate of pay each hour.
Rates are usually expressed in the simplest form.
Examples
(a) 150 km in 2 hours = 75 km in 1 hour (divide both numbers by 2)
= 75 km/h
(b) 500 sheep on 250 hectares = 2 sheep on 1 hectare
= 2 sheep per hectare
Write each of the following as a rate in its simplest form:
(a) 75 students for 3 teachers
_____________________________________________________________________
(b) 270 kilometres in 3 hours
_____________________________________________________________________
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(c) $12 for 6 litres
_____________________________________________________________________
(d) 20 degrees in 5 minutes
_____________________________________________________________________
(e) 240 passengers on 4 buses
_____________________________________________________________________
(a) 25 students per teacher
(b) 90 km/h
(c) $2/L
(d) 4 degrees/min
(e) 60 passengers per bus.
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Using index notation
In Topic 2, we had a brief look at the squaring operation, which you will
remember is the same as multiplying a number by itself.
The notation was to use the number 2 (because there are two identical
numbers multiplied) as an index, which is written raised (‘superscript’)
next to the number being squared.
Example
32 = 3  3 = 9
The same notation can be used for higher powers too.
Example
What does 34 mean?
If you guessed 3  3  3  3, then you were right.
This can be evaluated by working the multiplications from left to right,
giving 81.
We say, ‘three to the power of four equals eighty one’.
We write 34 = 81.
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Let’s take a closer look at the terminology used with index notation:
If the index is 2, this is the special case we have met before, called
squaring. We also have a special term for when the index is 3: it is called
cubing the number.
So 73 = 7  7  7 = ‘7 cubed’ = 343

Index notation is a convenient abbreviation for constant multiplication.

2  2  2  2  2  2 = 26 = 64

If the index is 2, this is ‘a square’.

If the index is 3, this is ‘a cube’.
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Activity 4F
1
Write in index notation:
(a) 2  2  2
(b) 3  3  3  3  3
(c) 10  10  10  10
(d) 6  6
(e) 5  5  5
2
Evaluate each of the powers in Question 1.
3
Evaluate:
(a) 27
(b) 52
(c) 10 cubed
(d) 44
(e) 63
4 Evaluate:
Check your answers against those at the end of the topic.
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Very large and very small numbers
In Topic 1, we saw how newspaper editors abbreviate large numbers like
1 500 000 000 to ‘1.5bn’.
Scientists also regularly work with very large (and very small) numbers and
also have a method of abbreviation, called scientific notation or standard
form. A scientist would write this number as:
1 500 000 000 = 1.5  109
This large number has now been written in a more compact form.
So rewriting 1 500 000 000 as 1.5  109 tells us straight away that ‘this
number is in the billions’ because a billion is 109.
Similarly,
2 700 000 = 2.7  106 says ‘this number is in the millions’.
125 = 1.25  102 says ‘this number is in the hundreds’.
Let’s look at how it’s done.
Did you notice the similarities in the 3 examples above?
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Writing numbers in scientific notation
Example 1
Write 108.5 in scientific notation
Step 1
Move the decimal point so that it is positioned between
the first and second digits. (This always produces
a number between 1 and 10.) In this case we get
1.085
Step 2
Put the multiplication sign in.
Step 3
Work out the power of 10 by counting the number of
hops it will take to move the decimal point back to
where it started.
1.085 
The answer is 108.5 = 1.085  102
Example 2
Write 45 000 in scientific notation.
Step 1
30
Put the decimal point between the first and second digits.
Note we do not have to write the following zeros.
4.5
4.5 
Step 2
Put the multiplication sign in.
Step 3
Work out the power of 10 by counting the number of hops
to get to the end of the number because there is no decimal
point.
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Write in scientific notation:
(a) 1287.4
(b) 1 320 000
(a) 1.2874  103
(b) 1.32  106
Example 3
Write 0.000 014 in scientific notation.
Step 1
Write your number between 1 & 10.
1.4
Step 2
Put the multiplication sign in.
1.4 
Step 3
Work out the power of 10 by counting the number of
hops it will take to move the decimal point back to
where it started.
This time it took 5 hops to the left.
It will be shown by making the
power of 10 a negative number.
This gives
1.4  10– 5
Did you notice the difference between this number and the other examples
we have looked at? This one is a very small number so we will have to move
the decimal point in the opposite direction. The power of 10 is negative.
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Write in scientific notation:
(a) 0.065
(b) 0·000 821.
(a) 6.5  10–2
(b) 8.21  10–4
To write a number in scientific notation (standard form), move the decimal
point to make a number between 1 and 10, then count the number of
decimal places back to its starting point.
Reversing the process
Changing from scientific notation to ordinary numerals. Changing back
from scientific notation to ordinary numerals is just a matter of reversing
the procedure.
Example 1
1.006  105 tells you to move the decimal point 5 places to the right.
Fill empty spaces with zeros.
This gives 100 600.
Example 2
4.76  10– 2 says to move the decimal point 2 places to the left.
Fill empty spaces with zeros. We should put a zero in front of the decimal
point. This gives 0.0476.
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Write as an ordinary number:
(a) 2.58  105
(b) 5.9  10–4
(a) 258 000
(b) 0.00059
Activity 4G
1
Write these numbers in scientific notation:
(a) 135 000
(b) 4 700 000
(c) 340.75
(d) 168 000
(e) 2 million
(f) 6543.2
(g) 0.006
(h) 0.15
(i) 0.000 072
(j) 0.95
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2
Change these from scientific notation into ordinary numerals:
(a) 4.25  104
(b) 1.17  106
(c) 8.85  103
(d) 5  102
(e) 9.99  109
(f) 1.0005  102
(g) 1.4  10–2
(h) 6.03  10–5
(i) 9  10–1
(j) 3.35  10–8
3
Write out the following sentences with the numbers changed into scientific notation:
(a) The speed of light is approximately 300 000 000 metres per second.
(b) The largest galaxy in the universe is 1070 million light years away.
(c) The wealthiest woman in the world is said to be Queen Elizabeth II, with a
personal fortune estimated at $15 billion.
(d) The smallest lizard in the world measures just 0.015 metres in length.
(e) The smallest of all free-living organisms is a type of bacteria which weighs
approximately 0.000 000 000 000 0001 grams.
Check your answers against those at the end of the topic.
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Suggested answers to activities
Activity 4A
4
The answers are the same because
5
Hexagon
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Activity 4B
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Activity 4C
(g) 0.01
(h) 101.5
(b) 4.404
(c) 0.20
(d) 5.88
(e) 19.505
(Rewrite as: 19.0505, 19.5050. So 19.505 is bigger.)
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Activity 4D
Activity 4E
1
(a) 10 : 12 (or 5 : 6)
(b) 8 : 10 (or 4 : 5)
(c) 10 : 10 (or 1 : 1)
2
(a) male : female = 30 : 12 (or 5 : 2)
(b) male : total = 4 : 22 (or 2 : 11)
3
Kaye : Charlie = 3 : 1
4
cement : sand
=1:4
= 25 : 100
 he needs 100 kg of sand.
5
cherries : almonds = 3 :2
= 150 : 100
 150 g of cherries are required.
6
1 cm : 10 km = 1.7 cm : 17 km
 they are 17 km apart.
7
38
24 cm
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Activity 4F
1
(a) 23
(b) 35
(c) 104
(d) 62
(e) 53
2
(a) 8
(b) 243
(c) 10 000
(d) 36
(e) 125
3
(a) 128
(b) 25
(c) 103 = 1000
(d) 256
(e) 216
4
(a) 5
(b) 10
(c) 6
(d) 9
(e) 8
(f) 4
(g) 5
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Activity 4G
1
(a) 1.35 × 105
(b) 4.7 × 106
(c) 3.4075 × 102
(d) 1.68 × 105
(e) 2 × 106
(f) 6.5432 × 103
(g) 6 × 10–3
(h) 1.5 × 10–1
(i) 7.2 × 10–5
(j) 9.5 × 10–1
2
(a) 42 500
(b) 1 170 000
(c) 8850
(d) 500
(e) 9 990 000 000
(f) 100.05
(g) 0.014
(h) 0.000 0603
(i) 0.9
(j) 0.000 000 033 5
3
(a) 3 × 108
(b) 1.07 × 109
(c) 1.5 × 1010
(d) 1.5 × 10–2
(e) 1 × 10–16
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