Number
Level 8
Reciprocals
2
b.
The reciprocal of 3/5 if 5/3; the reciprocal of a/b is b/a; the reciprocal of a is 1/a
Step 1 Write down the number as a fraction, if it isn’t already given as a fraction
c. 0.8
2
5
a.
The reciprocal of is
b.
Since 5 = , the reciprocal of 5 is
c.
0.8 =
d.
The reciprocal of is
e.
Since 3x =
8
10
5
5
1
4
3
1.
b. 5
d.
3
𝑥
4
e. 3x
1
3
3.
5
3
4
5
x
3x
1
4
x
3
, the reciprocal of 3x is
4
1
2
4
1
5
12
8
3
4
3
= ×
1
9
1
5
2.
6
1
6÷
3
=
4
1
6
1
2
1
2
3
1
5
6
5
3
5
= ×
3
1
= ×
=
5
1
2
5
We can add 10 pence + 14 pence to get an answer of 24 pence
Fractions
We can add 2a + 5a to get an answer of 7a
3
To add two quantities they must be the same type of quantity
23
To add (or subtract) fraction the fractions must be the same type
A fraction such as is called a proper fraction.
4
In a proper fraction the numerator (top) is smaller than the denominator (bottom)
A fraction such as is called an improper fraction.
4
In an improper fraction the numerator is larger than the denominator.
Improper fractions can be thought of as being “top heavy”.
Sometimes improper fractions are called vulgar fractions
1
A number such as 3 is called a mixed number
2
A mixed number consists of a whole number and a fraction
Each section of this diagram has been divided into
13
5
5
3
diagram that we can write this as 2 as a mixed number
5
Multiplying Fractions
Multiply the numerators to obtain the numerator of the answer and multiply the denominators to
obtain the denominators of the answer.
Examples
Answer
a.
2
=4
5
1
2
×
15
8
.
3
3
4
1
25
6
6
25
2
5
Fifths may be added to fifths, quarters may be added to quarters but to add fifths to quarters we must
rewrite as the same type.
That is we must find equivalent fractions which have the same denominator
Examples
2
5
5
1
3
8
8
4
+ =
5
3
− =
5
2
=
1
4
3
4
3
To find + first find fractions equivalent to and which have the same denominator
4
5
3
4
=
=
8
10
6
8
5
=
=
4
4
12
12
3
Then +
5
15
9
4
=
=
=
16
20
12
16
16
20
5
=
+
15
20
15
20
=
31
=1
44
+
21
20
4
11
20
To add (or subtract) mixed numbers we can begin by writing each mixed number as an improper
fraction
Examples
Work out
a.
=
12
This statement is also true for fractions
8
13
is an improper fraction, we can see from the
2
9
We cannot add 3x + 2a neither can we add 5cm + 6kg
Proper and Improper Fractions, Mixed Numbers
fifths. The shading represents 13 fifths that is
3
4
4
4. 1 ÷ 4 = ÷
5
3
For instance the reciprocal of 2 is found by keying 2 x-1 .

÷
=8
Adding and Subtracting Fractions

×
1
= ×
4
=1
6
The x-1 key on the calculator is used to find the reciprocal of a number

3
= ×
12
= ×
=
3x
1
1
= ×
÷
6
4
3
= ×
or . The reciprocal of 0.8 is or 1.25
5
3
÷6 = ÷
=
2
converting the mixed numbers into improper fractions gives
Examples
Find the reciprocals of
5
8
To divide by a fraction, multiply by the reciprocal.
Example
Answer
7
Dividing Fractions
Step 2 Invert the fraction. That is “tip the fraction upside down”
a.
×1
Simplifying the fractions by cancelling by 4 and 5 gives
To find the reciprocal of a number proceed as follows
2
2
5
4
9
4
9
3
×
10
×
10
3
b. 2
2
5
×1
7
2
2 is a factor of 4 and 10. 3 is a factor of 3 and 9. Simplifying the fractions,
cancelling by 2 and 3 gives
2
3
×
3
3 +1 =
8
1
5
=
2
15
3
1
4
1
3 −1 =
6
2
11
3
19
6
+
−
7
4
3
2
=
=
12
19
6
−
12
9
6
=
=
65
12
10
6
=5
=1
4
6
5
12
= 1
2
3
Another method of adding (or subtracting) mixed numbers is to add (or subtract) the whole numbers
then add or subtract the fractions
If more than one operation is involved in the same calculation then must follow BODMAS
Proportional Changes
Power and Roots
Calculating Original Quantities after Proportional Changes
Calculating with Powers and Roots
Example
In a factory, 870 workers are given a pay rise. This was 20% of all the workers. How many workers
are there altogether?
When more than one operation is to be done
Answer
Work out brackets
Then work out the powers and roots,
Then do multiply and divide
20% represents 70 workers.
Then do addition and subtraction
Divide by 20.
Examples
1% represents 70 ÷ 20 workers.
Calculate
Then multiply by 100
a.
100% represents all the workers
70 ÷ 20 x 100 = 350
So there are 350 workers altogether.
Example
The price of a car increased by 6% to £9116. Work out the price before the increase.
Answer
106% represents £9116.
Divide by 106.
1% represents £9116 ÷ 106
b.
1 - √9
c.
3√16 + 17 x (-2)
d.
4[2 + (-2)3]
Answer
a
5 + 72 = 5 + 49
=54
b
1 - √9 = 1 – 3
= –2
c
3√16 + 17 x (-2) = 3 x 4 + 17 x (-2)
= 12 – 34
= –22
d
4[2 + (-2)3] = 4[2 + (-8)]
= 4( -6 )
= -24
Then multiply by 100
100% represents the original price
9116 ÷ 106 x 100 = £8600
5 + 72
So the price before the increase was £8600
Indices
Repeated Proportional Changes
Rules of Indices
When more than one calculation involving percentages or fractions takes place, each must be done
separately.
am x an = am + n
For instance if a manufacturer sells an article for 25% profit and it is sold again for a further 40% profit
we cannot add the 25% and 40% to find the total percentage profit.
am ÷ an = am - n
(am)n = am x n
Examples
Example
Donna is a buyer for “Fashion Warehouse”. One line of dresses she buys cost £12.50. A 30% mark
up is put on these.
Use the rules of indices to write the following as single powers of 7
b. At the end of the summer these dresses are reduced by 25%. What profit or loss does the Fashion
Warehouse make on them?
a.
b.
c.
Answer
Answer
a. What price does the Fashion Warehouse sell these dresses for?
a. They sell for 130% of £12.50 =
130
100
b. Reduced price = 75% of £16.25 =
x 12.50 = £16.25
70
100
This reduced price is less than the cost.
Loss = £12.50 - £12.19 = 31p
74 x 79
78 ÷ 72
(75)3
a.
74 x 79 = 74 + 9
= 713
b.
78 ÷ 72 = 78 - 2
= 76
c.
(75)3 = 75 x 3
= 715
x 16.25 = £12.19 (to the nearest penny)
Standard Form
Adding and Subtracting Numbers written in Standard Form
The numbers 10, 100, 1000, 10000 ... can be rewritten as 101, 102, 103, 104 ...
4
4
The number 70000 can be rewritten as 7 x 10000 or as 7 x 10 or as 7.0 x 10 .
One way of calculating 3.6 x 106 + 2.3 x 104 is by rewriting 3.6 x 106 and 2.3 x 104 as ordinary
numbers then adding
The number 736 can be rewritten as 7.36 x 100 or as 7.36 x 102.
The numbers
1
1
10, 100
,
1
1000
.... can be rewritten as
The number 0.086 can be rewritten as
8.6
100,
or
8.6
102
1
101
,
1
102
,
1
103
.... or as 10-1, 10-2, 10-3 ....
or 8.6 x 10-2
The numbers 7 x 104, 7.36 x 102, 8.6 x 10-2 are written in a notation known as Standard Index
Notation. Standard Index Notation is usually called Standard Form.
Numbers written in standard form consist of two parts. They have a decimal number part in which
there is always just one digit (not Zero) before the decimal point and this part is multiplied by a power
of 10.
For instance the following numbers are in standard form 6.2 x 1014, 7.01 x 10-1, 8.3 x 100,
The following numbers are not in standard form 0.6 x 106, 78.2 x 10-3,
Another way of stating the standard form notation is
Standard form is
a x 10n where 1 ≤ a < 10 and n is an integer
Standard form is a very useful way of writing very large or very small numbers.
On a calculator screen the standard form notation is not always written in full the x and 10 are omitted.
For instance a calculator displays 7.3 x 105 as 7.3 05
Multiplying and Dividing numbers written in Standard Form
We use the laws of indices when we multiply or divide numbers written in standard form
We use as ax x ay = ax + y and ax ÷ ay = ax - y
Example
Calculate the following giving the answer in standard form
a.
(2.4 x 10-4) x (3.1 x 107)
b.
(3.4 x 108) ÷ (2.1 x 105)
Answers
a.
(2.4 x 10-4) x (3.1 x 107)
= 2.4 x 10-4 x 3.1 x 107
= 2.4 x 3.1 x 10-4 x 107
= 7.44 x 103
b.
(3.4 x 108) ÷ (2.1 x 105)
=
3.4 × 108
2.1×105
3.4
108
× 5
=
10
2.1
= 1.62 x 103
Using Scientific Mode on the Calculator (Calculators vary and you need to learn how to use your own)
To get the calculator operating in scientific mode
Key
MODE
Key
MODE three times followed by 2 followed by 4 if you want 4 significant figures
followed by 8
or
To get the calculator out of scientific mode
Key
MODE
Key
MODE three times followed by 3 followed by 1
followed by 9
or