GCSE Maths

Numbers
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Expressing Fractions as Decimal Numbers
Nearest 10, 100, 1000, …
Nearest 10, 100, 1000, 10000, …, million and so on
Percentages
More on Percentages: Applications to Interest Calculations
Significant Figures
Significant Figures
Rounding Up and Estimation
Standard Form
Order of Operations
Four Operational Rules
Square Roots and Cube Roots
Conversion of Units to other Units
Compound Interest
Surds
Rational and Irrational Numbers
Errors and Uncertainties in Measurements
Greatest and Least Possible Values
2 of 2
Vectors
1 Topic
Vectors
Probability
6 Topics
Introduction to probability
Finding the sum of probabilities
Conditional probability
Set Notation
Venn diagrams
The Complement of a Set
Statistics
2 Topics
Population and samples
Representing data
Algebra
16 Topics | 1 Quiz
Algebraic Notation
Algebraic Vocabulary
Collecting Like Terms
Expanding Brackets and Simplifying
Expanding Double Brackets
The Difference of Two Squares
Substituting into Formulae
Subject of Formula
Factorisation
Factorising Quadratic Expressions
More on Factorisation
Equations
Quadratic Equations
Mathematical Formulae
Algebraic Equivalence and Proof
Functions
Algebra
Sequences
2 Topics | 1 Quiz
Generating sequences
Nth Term Sequences
Sequences
Ratio, Proportion and Rates of Change
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Inverse Proportion
Conversion
Scale Factors
Percentages
Direct Proportion
Rate, Ratio and Proportion Examples
Ratios
Ratios
Ratio, Proportion and Rates of Change
Geometry and Measures
9 Topics
Loci and constructions
Properties of angles
Properties of polygons
Transformations
Enlargement
Circle theorems
Plane figure properties
Plans and elevations
Areas and Volumes in Map Scale Problems
Mensuration and Calculation
10 Topics
Map scales
Bearings
Area
Volume
Circles: arcs and sectors
Congruence and similarity
Pythagoras’ theorem
Sine and cosine rule
Area of a triangle
Standard units of measure
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Order of Operations

GCSE Maths Numbers Order of Operations

Suppose we wish to evaluate
2+3\times 4+3^{ 2 }+25\div 5+({ 2 }^{ 2 }-1)\times 2
To evaluate the above, we need to perform the operations in the correct order.
We do evaluate Brackets first, then we evaluate Orders which includes powers, indices, square roots, etc. Then we evaluate Division, followed by Multiplication, Addition and finally Subtraction.


This is easily remembered from the ‘word’ BODMAS, where
B stands for Brackets
O for Orders
D for Division
M for Multiplication
A for Addition and
S for Subtraction


So, for our problems we should first perform ({ 2 }^{ 2 }-1), that gives us 4-1=3
So that, we now have
2+3\times 4-{ 3 }^{ 2 }+25\div 5+3\times 2
Next, we take down the orders, we have { 3 }^{ 2 } which is 9.
So, now we have
2+3\times 4-9+5+3\times 2
After Division, comes Multiplication. So, evaluate 3\times 4=12 and 3\times 2=6
2+12-9+5+6
Following the rule of BODMAS, we now perform the addition.
2+12+5+6=25
Finally, we have 25-9=16
Therefore 2+3\times 4-{ 3 }^{ 2 }+25\div 5+({ 2 }^{ 2 }-1)\times 2=16


We notice that Addition and Subtraction are given the same priority in the order of operation. So we could have done 12-9=3in the above first and then add 2+3+5+6=16

Examples

Find the value of:
  1. 36-(5+({ 3 }^{ 2 }-2)\times 3)\div 13
  2. { (54\div 9\times 3-10) }^{ 2 }+15
  1. 36-(5+({ 3 }^{ 2 }-2)\times 3)\div 13=36-(5+7\times 3)\div 13
    • =36-26\div 13
    • =36-26\div 13
    • =36-2
    • =34
  2. { (54\div 9\times 3-10) }^{ 2 }+15={ (6\times 3-10) }^{ 2 }
    • ={ (18-10) }^{ 2 }
    • ={ 8 }^{ 2 }
    • =64

Find the value of 6+8\times 9\div 3
6+8\times 9\div 3=6+8\times 3 since 9\div 3=3 is done first.
=6+24
=30

Find the value of \frac { 1 }{ 1+\frac { 2 }{ 3 } }
First 1+\frac { 2 }{ 3 } =\frac { 3 }{ 3 } +\frac { 2 }{ 3 } =\frac { 5 }{ 3 }
3+\frac { 1 }{ (\cfrac { 3 }{ 5 } ) }
=\frac { 15 }{ 5 } +\frac { 3 }{ 5 }
=\frac { 18 }{ 5 }
=3\frac { 3 }{ 5 }

Evaluate, giving your answer as simplest as possible.

  1. 1+{ 2 }^{ 2 }-1\times 3+({ 4 }^{ 2 }\div 2\times 3)
    • Bracket first { 4 }^{ 2 }\div 2\times 3=16\div 2\times 3
    • =8\times 3
    • =24
  2. 1+{ 2 }^{ 2 }-1\times 3+24=1+4-1\times 3+24
    • =1+4-3+24
    • =26

Evaluate { { (2 }^{ 2 }-3) }^{ 100 }-({ { 2 }^{ 3 }-7) }^{ 200 }
{ { (2 }^{ 2 }-3) }^{ 100 }-({ { 2 }^{ 3 }-7) }^{ 200 }
{ 1 }^{ 100 }-{ 1 }^{ 100 }
=1-1
=0

If we have more brackets within brackets, then we perform the innermost brackets first and move outwards. For example, to evaluate
2+5-({ 2 }^{ 2 }+({ 6 }^{ 3 }\div 12))^{ 2 }\times 3
We do { 6 }^{ 3 }\div 12 first to have 216\div 12=18.
This brings us to
2+5-{ 2 }^{ 2 }+{ 18 }^{ 2 }\times 3
Next we take { 2 }^{ 2 }+18=4+18=22, so that now we have
2+5-{ 22 }^{ 2 }\times 3
Now { 2 }^{ 2 }=48450
2+5-484\times 3=7-1452
=-1445

Another example


Evaluate 3\times { 5 }^{ 2 }-12\div 4.
3\times { 5 }^{ 2 }-12\div 4=3\times 25-12\div 4 (Power first); { 5 }^{ 2 }=25
=3\times 25-3 (Division, 12\div 4=3)
=75-3 (Multiplication)
=72

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