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NUMBER
INTEGERS
are whole numbers (0,1,2 …) with a sign (either + or -)
HOW TO add and subtract integers
 Change any double signs to a single sign
using these rules …..
If signs are the same, change to +
If signs are different, change to eg.
+3 + -5
eg.
-3 - -4
 On a number line start at the first
number and ….
move right if you are adding,
move left if you are subtracting.
signs different, change to –
+3 + -5
=
Start move right 4 
+3-5
-4
-3
-2
signs same, change to +
-3 - -4
=
-1
0
1
2
-4
-3
-2
-1
0
1
4
Start
 move left 5
-3 + 4
3
2
+3
4
HOW TO multiply and divide integers
 If signs are the same the answer is +
If signs are different the answer is eg.
eg
-3 x -5
 Multiply or divide the numbers
-3 x -5 signs same so answer is +
+ 15
. -10  +2 -10  +2 signs different so answer is -
POWERS
- 5
are used to show repeated multiplication. They are also know as
indices or exponents.
eg.
2³
means 2 multiplied 3 times
2 x 2 x 2 = 8
eg.
(-3)4
means -3 multiplied 4 times
-3 x -3 x -3 x -3 = 81
eg.
45
Means 4 multiplied 5 times
using calculator
ROOTS 
are the opposite of powers.
SQUARE ROOT
eg.
9
CUBE ROOT
eg.
³8
OTHER ROOTS
eg. 5
1024
= 3 (because 3 x 3 = 9)
= 2 (because 2 x 2 x 2 = 8)
using calculator
4 xy 5 = 1024
using calculator
9 =
using calculator
³8 =
5 x 1024 = 4
OPERATIONS
B
E
work out anything
inside the brackets first.
eg.
eg.
If different operations are used in a calculation then follow
BEDMAS order.
then work out and
D M
Exponents (powers)
Division or Multiplication
3² + 5 x 4
9+5 x 4
5 x 10²
5 x 100
eg.
5 x (6 + 4)²
12 –16 (5 +3) + 7
A S
then do any
Addition or Subtraction
9 + 20
= 29
= 500
12 – 2 + 7 = 17
12 –16  8 + 7
STANDARD FORM
finally do any
is a more compact way to write large and small
numbers
HOW TO write numbers in standard form
 Put a decimal point after the
first digit that is not a zero,
making a number between 1
and 10.
 Work out how many places the decimal point needs to
move to get to its original position. This is the power
of 10 to use. If the original number is less than 0 the
power is negative.
eg.
256 000
2.56
2.56000.
dp moved 5 places
2.56 x 105
eg.
0.007
7.0
0.007.
dp moved 3 places
7.0 x 10-3 (Number ‹ 0)
eg.
1.365
1.365
1.365
dp moved 0 places
1.365 x 100
HOW TO take numbers out of standard form
Move the decimal point the same number of times as the power of 10. Move left if the power
is negative. Zeros may need to be used as place holders.
eg.
8.01 x 104
move decimal point right 4 places.
8.0100.
eg.
1.923 x 10-2
move decimal point left 2 places.
0.01.923
DECIMALS
80100
0.01923
are numbers that have a decimal point in them
HOW TO write a DECIMAL as a FRACTION
Put decimal digits over 10, 100, 1000 ..etc (depending on the number of digits) then simplify
the fraction.
eg.
0.8
8
10
2
2
4
5
eg.
0.35
35
100
5
5
7
20
eg.
0.025
25
1000
25
25
1
40
HOW TO write a DECIMAL as a PERCENTAGE
Multiply decimal by 100 (same as moving
decimal point 2 places to the right)
eg.
0.35
0.35 x 100 = 35
35 %
FRACTIONS
are parts of a whole
PROPER FRACTION
eg.
2
3
top is smaller than
bottom
IMPROPER FRACTION
eg.
5
3
MIXED NUMBER
eg.
top is bigger than
bottom
2 a counting number
3 and a fraction
1
HOW TO simplify a fraction
Find a number that divides into the top and bottom.
Divide that number into the top and bottom. Repeat
until the fraction is in the simplest form.
eg.
15
18
3 divides
into both
15  3
18  3
5
6
HOW TO change an improper fraction into a mixed number
eg.
Work out how many times the bottom number goes
into the top number, and what remainder is left.
5
3
3 goes into 5 once
with two remaining
2
3
1
HOW TO change a MIXED number into an IMPROPER fraction
eg.
Multiply the top number by the bottom number and
then add the top number. This gives the new top
number for the fraction.
1
2
3
5
3
1 x 3 + 2 = 5
HOW TO write a FRACTION as a DECIMAL
eg.
Divide top by bottom on a calculator
4
5
4  5 =
0.8
HOW TO add or subtract fractions
 Check that the
denominator is the
same.
eg.
½+⅛
 If not, then find
 Change the
the lowest common
fraction/s to an
denominator.
equivalent fraction
LCD of 2 and 8 is
8
1 becomes 4
2
8
 Add or subtract
the numerators
and simplify.
4 + 1
8
5
8
HOW TO multiply fractions
Simplify the top numbers with the bottom numbers.
Multiply top by top then multiply bottom by bottom.
eg.
3
4
x
8
5
2
3
41 3 x 1
x
8
5 2 x 5
3
10
HOW TO divide fractions
Change the  to x and flip over the second
fraction (the reciprocal). Multiply as above.
eg.
2

3
3
4
2
3
3
 4
2
4
x
3
3
8
9
PERCENTAGES %
means out of 100.
HOW TO write a PERCENTAGE as a DECIMAL
Divide percentage by 100 (same as moving decimal point
2 places left)
eg.
40 %
40  100 = .40
0.4
55 %
55  5
100  5
11
20
HOW TO write a PERCENTAGE as a FRACTION
Put the percentage over 100 and then simplify the
fraction.
eg.
HOW TO find a percentage of an amount
Write percentage as a decimal and multiply by the
amount.
eg.
8 % of 25 kg
0.8 x 25
2 kg
HOW TO increase or decrease an amount by a percentage
 Work out the % after the increase (by adding % to 100%) or
after decrease (by subtracting the % from 100%)
 Work out % of this amount.
(as above)
eg.
increase 25kg by 15 %
100% + 15 % = 115%
115% of 25kg ..(as above)
eg.
decrease $320 by 20 %
100% - 20% = 80%
80% of $320 .. (as above)
HOW TO find the percentage increase or decrease
% Increase or % Decrease =
eg.
Actual Increase/Decrease
Original Amount
Population increases from 2400 to 3800.
What is the % increase?
actual increase
3800 – 2400 = 1400
x 100
1400
2400
x 100 = 58%
HOW TO find the original amount after a % increase or decrease
 Work out the % after the increase or
decrease.
eg.
 Write an equation using X
for the original amount
100%
+ 10 %
110%
Item costs $264 after a 10% increase.
What was the price before the increase?
RATIO
110% of X is $264
1.1 x X = 264
 Solve the
equation.
X = 264  1.1
X = $240
compares the size of two or more quantities measured in the same units.
HOW TO work out the ratio of quantities
 Write quantities down in same order as
asked for and in same units. Separate with a
colon :
eg.
2 litres of juice costs $1.60 and 2 litres
of milk costs $1.20. What is the ratio of
cost of milk to juice.
 Change any decimals into whole numbers by
multiplying by 10, 100 etc. Simplify the ratio like
a fraction.
milk : juice
1.2 : 1.6
multiplying by
10 gives.
12 : 16
dividing by 4 gives
3 : 4
HOW TO split up an amount into a given ratio
 Work out total number of
parts by adding ratio numbers.
eg.
Divide $350 into
the ratio of 2:5
eg.
Divide 18m into
the ratio of 1:3:6
7 parts
in total
10 parts
in total
 Divide the amount by total
of parts
 Multiply this by each number
in ratio
350
7
= $50
2 x $50 and 5 x $50
$100 and $250
18
10
= 1.8
1 x 1.8, 3 x 1.8 and
6 x 1.8
1.8m, 5.4m and
10.8m
PROPORTION
problems of this type “If 5 apples cost $3.50 how much will 7
apples cost?”
HOW TO work out proportion questions
 Decide if 1 unit (item, person) would be more
or less. If more then multiply. If less then
divide.
 Work out the
cost or time
for 1 unit.
eg.
8 pens cost $3.60. How much will 1 pen would cost
less so divide
5 pens cost?
 Use cost or time for 1 unit to
work out cost or time for
other amount.
1 pen costs $3.60  8
= $0.45
eg.
3 people take 4 days to fold some 1 person would
take more time
letters. How long would it take 8
than 3 so multiply
people?
1 person would take
4 days x 3
= 12 days
5 pens cost 5 x $0.45 =
$2.25
8 people would take
12 days  8
= 1.5 days
SPACE AND SHAPE
METRIC UNITS
HOW TO change one unit to another
Either multiply or divide by the number which connects units as shown below.
LENGTH
AREA
VOLUME
mm

10
x
cm

100
x
mm²
mm³
m

1000
x
km

10²
x
cm²

100²
x
m²

1000²
x
km²

10³
x
cm³

100³
x
m³

1000³
x
km³
CAPACITY
mL

1000
x
L
MASS
mg

1000
x
g
1 Hectare = 10 000m²
1 mL = 1 cm³
1 L = 1000 cm³

1000
x
kg

1000
x
tonne
PERIMETER
the total distance around the outside of a shape
units: mm, cm, m, km etc
CIRCLE
POLYGONS
C =  x d
or
C = 2r
Perimeter = side lengths added together
AREA
The Perimeter of a circle is called the Circumference
is the number of squares
units: mm², cm², m² etc
RECTANGLE
PARALLELOGRAM
that fit into a shape.
TRIANGLES
AREA = b x h
AREA = ½ x b x h
TRAPEZIUMS
CIRCLES
a & b are
the parallel sides
AREA =  x r²
AREA = ½ x (a + b) x h
ADDING AREAS
SUBTRACTING AREAS
eg:
eg:
=
A1
+
=
AREA = A1 + A2
VOLUME
ODD SHAPES
Draw a grid over shape and
estimate the number of squares.
AREA = A1 - A2
is the number of cubes
units: mm³, cm³, m³ etc
PRISMS have two identical ends and the same cross
section along its length
PYRAMIDS
Height is
from base to
top point
VOLUME = 1/3 x BASE AREA x height
VOLUME = END AREA x length
PYTHAGORAS
that fit into a shape.
when working with right-angle triangles with all three sides
a² + b² = c²
LONGEST² = SHORTER² + SHORTER²
side
side
side
HOW TO find the missing side
 Square the sides
eg.
4
6
4²+ 6²
 Add or Subtract
16 + 36
52
x = √52
7.2 (1dp)
eg.
 Square Root
3
7
7²- 3²
49 – 9
40
x = √40
6.3 (1dp)
TRIGONOMETRY
POLYGONS
Perimeter = side lengths added together
Sin, Cos and Tan used for right-angled triangles when
working with 2 sides and 1 angle.
CIRCLE
C =  x d
or
C = 2r
The Perimeter of a circle is called the Circumference