Conversion of Unit of Measurement
Length
(mm = millimeter cm = centimeter m = metre)
1 cm = 10 mm, 1 m = 100 cm, 1km = 1000 m
1km = 1000m = 100000cm (1000 x 100)
Area
1 cm2 = 1cm x 1cm = 10mm x 10mm = 100mm2
1 m2 = 100 cm x 100 cm = 10000 cm2
1km2 = 1000m x 1000m = 1 000 000 m2
Mass
1 g = 1000mg, 1kg = 1000g
1 ton = 1000kg
TIME
1 min = 60 seconds (s)
1 hour (h) = 60 min (m)
1h = 60 x 60 = 3600s
1m = 1/60 h
1s = 1/60 m, 1/3600 h
1s = 1 x 109 nanosecond
0.5 or ½ hr = 30 mins
Volume
1 litre = 1000ml = 1000cm3
1cm3 = 1ml
Comparing and Ordering Numbers
Symbols
> greater than
< less than
≥ greater than or equal to
≤ less than or equal to
= equal
≠ not equal to
Tip to remember
4 > 3
greater = "open mouth" > “point” = less than
Similarly,
3 < 4
less than = “point” < greater = "open mouth"
Significant Digits
Rules
Number is Significant for
1. Every non-zero : 42549 (5 sig digits)
2. Zeros in between digits : 1001, 1.003 (4 sig digits)
3. Zeros at end of numbers with decimals : 300.100 (6 sig digits)
Number is not significant for
1. Zeros to left of numbers : 00003467 (4 sig digits)
2. Zeros at end of non-decimal numbers : 823000 (3 sig digits)
3. Zeros to right of decimal number < 1 : 0.0005 (1 sig digits)
[ For Decimal, remember to put a trailing zero for specified significant figure.
Eg: 0.200 is 3 significant figures]
Order Of Operations
1st Order : ( ) e Bracket Exponential BE / BO (e.O – Order)
2nd Order : ÷ X Multiplication And Division DM
3rd Order : + - Addition And Subtraction AS
[ B E D M A S] / [B O D M A S]
Direct and Indirect Proportion
Direct proportion: As one variable (Y) increases ↑, another variable (X) increases↑ at the same proportion/rate.
=> as y increase ↑, x also increases ↑ proportionally
y α x Symbol of proportion: α
The equation for direct proportion is
y = kx (where k is the constant of proportionality)
Inversely Proportion: As one variable (Y) increases↑, another variable (X) decreases ↓ at the same proportion/rate.
=> As y increases ↑, x decreases ↓ proportionally
The equation for inverse proportion:
y= k
Map Scale
Scale of drawing = Drawing length : Actual length
Map Scale = Map distance : Actual Distance
If the scale of a map is given as 1 : 200 000
=> 1cm represents 200 000 cm (200 000 cm = 2000m = 2km)
=> 1cm represents 2 km
1cm : 2km
1cm x 1cm = 1cm2: 2km x 2 km = 4km2
1cm2: 4 km2
- Area calculation: (first calculate 1cm2 represent how many km2) *
Standard Form
K x 10m where K is between 1 to 9 1 <= K < 10
Illustration of Common Order of Magnitude
Using 1 g:
1 gigagram (Gg) = 1 x 109 g
1 tonne (Mg) = 1 x 106 g
1 kilogram (kg) = 1 x 103 g
1 milligram(mg) = 1 x 10-3 g
1 microgram (µg) = 1 x 10-6 g
1 ng (nanogram) = 1 x 10-9 g
Indices Rules
Positive Indices
(1) am x an= am+n
(2) am ÷ an = am-n
(3) (am)n= am x n
Zero and Negative Indices
(4) a-m = 1/ am
(5) a0 = 1
Fractional Indices
(6) a1/2 = √a
(7) am/n = n√am
ALGEBRA
1. ab = a x b
2. a/b => a ÷ b => a x 1
b
3. a2= a x a, a3= a x a x a
4. a2b = a x a x b, ab2 = a x b x b
5. 3p = 3 x p = p + p + p
6. 3(p + q) = 3 x (p+ q) = 3p + 3q
7. 2(3 + y) = 2(3 + y) ÷ 5 = 2(3 + y)
5 5
8. a2 + a= a(a + 1) a2 = a x a,
Evaluating the "nth" Term
A formula can be given to a number sequence so that we can find the number in the sequence by substituting or "putting" the value into the formula.
The nth term => the number in the nth position in the number sequence.
Addition and Subtraction
An algebraic expression can be simplified by "grouping" of the same variables and number operations
Example
Simplify 2a + 4 + 2a + 1
= 2a + 4a + 4 + 1 (Step 1 : "Group" variables and numbers)
= 6a + 5 (Step 2 : Do the + - )
Expansion of Linear Expressions
Expansion usually involves removing the bracket.
=> a (b + c) = ab + ac
Expansion of
( a + b) ( c + d) = ac + ad + bc + bd
/\ /\
[ a x (c + d) + b x (c + d) ]
It is also called the "rainbow" arrow
Multiplication/Division OF ALGEBRAIC FRACTIONS
Simplify by 'reducing' the numbers and variables
(Crossing out same variables and numbers from numerator/denominator of the algebraic expressions)
Example
Simplify the algebraic Expression
a x 3b
9 a
Step 1 : Look for numbers that can be reduced => 3 and 1/9 can be reduced by 3
: a and 1/a can be reduced a/a = 1
(1a) a x 3b
3 9 a
(1b) a x b
3 a
Step 2 : Simplify
1 x b
3
= 1 b or b
3 3
Division of Algebraic fraction
First, to 'convert' the expression to multiplication before simplifying.
=> a ÷ c = a x d
b d b c
Example
Simplify 3b ÷ 2b
4 a
= 3b x a (Step 1 : Change the ÷ to multiply by 'flipping' the fraction)
4 2b (Step 2 : Look and "cross" same variables/numbers)
= 3a (Step 3 : Simplify)
8
Factorising Linear Algebraic Expression
Extracting Common Factor/s
Rule: ab + ac = a(b+ c)
Example
Factorise a2 + 3a
= a(a + 3). (Step 1 : Look for Common factors -> a, a2 = a x a)
To remember:
(a + b)2 = a2 + 2ab + b2 = (b + a)2
(a - b)2 = a2 - 2ab + b2
a2 – b2 = (a + b) x (a - b)
22 = 4 32 = 9 42 = 16 52 = 25 62 = 36
72 = 49 82 = 64 92 = 81 102 = 100 112 = 121
Solving Fractional Equations (linear equation)
Linear fractional Equation can usually be solved by simplifying the equation to having a single denominator.
Example
Solve y/2 + y/3 = 2
Step 1 : Find the lowest common factor (LCM) for the denominators
LCM of 2 and 3 = 6
Step 2 : Combine into single fraction
½y + ⅓y = 2
3 x y + 2 x y = 2
3 x 2 2 x 3
3y + 2y = 2
6
6 x 5y = 2 x 6 (multiply both side by 6 => "cross over : x 6 too)
6
5y = 12
y = 12/5 = 2 2/5
Multiplication property of Inequalities
x +ve number
if a < b and c > 0, then ac < bc eg: 2 < 3 , x 3 => 6 < 9
If a > b and c > 0, then ac > bc eg: 3 > 2, x 3 => 9 > 6
x –ve number
If a < b and c < 0, then ac > bc eg: 2 < 3 , x -3 => -6 > -9
=> the sign changed from < to >
If a > b and c < 0, then ac < bc eg: 3 > 2 , x -3 => -6 < -9
=> the sign changed from > to <
[[ x –ve : change sign from < to > or > to < ]]
Solving Fractional Quadratic Equations
Cross-Multiplying =>
a = c => x bd a = c x bd => ad = bc
b d b d
