1.4.5
Limits of Accuracy
Rounding and Accuracy
Rounding and Accuracy
If we round numbers then the accuracy of our calculation can change. The weight of a cooking pot to the nearest kilogram is 7kg. What is the range of possible weights of the pan?
Upper bound is 7.5. Lowest number that won't round down to 7
Upper bound is 7.5. Lowest number that won't round down to 7
- The upper bound is 7.5
Upper and Lower Bounds
Upper and Lower Bounds
You need to know how to use bounds to work the higher and lower bounds of calculations with numbers that have been rounded.
Upper and lower bounds of calculations
Upper and lower bounds of calculations
- To work out the upper and lower bound of a calculation, you need to work out the upper and lower bound for each value used in a calculation.
- You can then use these individual bounds to work out the upper and lower bounds of the calculation.
Example - multiplication
Example - multiplication
- You are calculating the upper and lower bounds of the area of a rectangle.
- The height is 0.5 to 1 decimal place.
- The height's lower bound is 0.45 and the upper bound is 0.55.
- The length is 10 to 2 significant figures.
- The length's lower bound is 9.5 and upper bound is 10.5.
Example cont. - multiplication
Example cont. - multiplication
- The lower bound for the area is the smallest possible height × the smallest possible length.
- I.e. 0.45 × 9.5 = 4.275
- The upper bound for the area is the largest possible height × the largest possible length.
- I.e. 0.55 × 10.5 = 5.775
Example - division
Example - division
- You are calculating the upper and lower bounds of the speed = distance ÷ time.
- A car travels 30 km to 1 significant figure.
- The lower bound for the distance is 25 km and the upper bound is 35 km.
- The time is 7 hours to 1 significant figure.
- The lower bound of the time is 6.5 hours and the upper bound is 7.5 hours.
Example cont. - division
Example cont. - division
- The lower bound for the speed of the car is the smallest possible distance ÷ the largest possible time.
- I.e. 25 ÷ 7.5 = 3.33 km/h
- The upper bound for the speed of the car is the largest possible distance ÷ the smallest possible time.
- I.e. 35 ÷ 6.5 = 5.38 km/h
1Number
1.1Using Numbers
1.1.1Equal Symbols
1.1.2Negative Numbers
1.1.3BIDMAS
1.1.4Inverse Operations
1.1.5End of Topic Test- Ordering
1.1.6Factors
1.1.7Multiples
1.1.8Highest Common Factor
1.1.9Lowest Common Multiple
1.1.10Number of Ways to Do a Task
1.1.11Number of Combinations
1.1.12Number of Ways To Arrange Objects
1.1.13End of Topic Test - Factors, Multiples & Combos
1.1.14Standard Form
1.1.15Ordinary Numbers to Standard Form
1.1.16Addition in Standard Form
1.1.17Subtraction in Standard Form
1.1.18Multiplication in Standard Form
1.1.19Division in Standard Form
1.1.20Exam-Style Questions - Standard Form & Primes
1.1.21Long Division
1.1.22Prime Numbers
1.1.23End of Topic Test - Standard Form & Long Division
1.2Fractions, Decimals & Percentages
1.2.1Adding Fractions
1.2.2Subtracting Fractions
1.2.3Multiplying Fractions
1.2.4Dividing Fractions
1.2.5Mixed Fractions
1.2.6Calculating Percentages of Numbers
1.2.7Calculating Fractions of Numbers
1.2.8Multiply Decimals
1.2.9Place Value
1.2.10Recurring Decimals
1.2.11Exam-Style Questions - Recurring Decimals
1.2.12Convert Fractions & Terminating Decimals
1.2.13Convert Recurring Decimals
1.2.14Convert Decimals & Percentages
1.2.15Compare Decimals
1.2.16Compare Fractions
1.2.17Exam-Style Questions - Percentages
1.2.18End of Topic Test - Fractions & Decimals
1.3Powers & Roots
1.3.1Powers
1.3.2Roots
1.3.3Indices Advanced
1.3.4Estimating Powers
1.3.5Estimating Roots
1.3.6Multiplying Powers
1.3.7Dividing Powers
1.3.8Power Raised to Power
1.3.9Fractions Raised to a Power
1.3.10Numbers Raised to Negative Powers
1.3.11Roots of Numbers
1.3.12Numbers to the Power of 1 & 0
1.3.13Surds
1.3.14Simplify Surds
1.3.15Rationalise Denominator
1.3.16Exam-Style Questions - Indices
1.3.17End of Topic Test - Powers & Roots
2Algebra
2.1Introduction to Algebra
2.1.1Algebra Basics
2.1.2Substituting Values
2.1.3Algebraic Vocabulary
2.1.4Collecting Terms
2.1.5End of Topic Test - Algebra Basics & Power Rules
2.1.6Expanding Brackets I
2.1.7Common Factors
2.1.8Expanding Brackets II
2.1.9Rearrange For x
2.1.10Rearrange with Powers
2.1.11Exam-Style Questions - Basics of Algebra
2.1.12End of Topic Test - Expanding & Rearranging
2.2Manipulating Algebra
2.3Proofs & Functions
2.4Straight Line Graphs
2.5Common Graphs
2.6Transformations & Tangents
2.7Properties of Graphs
2.8Solving Equations
2.8.1Solve Linear Equations
2.8.2Linear Equations with Fractions
2.8.3Solve Linear Equations Graphically
2.8.4Quadratic ±
2.8.5Factorise Quadratics
2.8.6Quadratic Formula
2.8.7Complete the Square
2.8.8Solve Quadratics Graphically
2.8.9End of Topic Test - Solving Equations
2.8.10Simultaneous Equations
2.8.11Quadratic Simultaneous Equations
2.8.12Simultaneous Equations - Graphs
2.8.13Simultaneous Equations - Word Problems
2.8.14Approximate Solutions - Number of Iterations
2.8.15Approximate Solutions - Iterations to 1 DP
2.8.16Approximate Solutions - Iteration Formula
2.8.17Exam-Style Questions - Simultaneous Equations
2.8.18End of Topic Test - Simultaneous Equations
2.9Inequalities
2.10Sequences
2.10.1Nth Term Rules
2.10.2Recurrence Relationships
2.10.3Find Nth Term in Linear Sequence
2.10.4Check If a Number is In a Sequence
2.10.5Exam-Style Questions - Sequences
2.10.6Arithmetic & Geometric Progressions
2.10.7Key Sequences
2.10.8Quadratic Sequences
2.10.9Find Nth Term in Quadratic Sequence
2.10.10Exam-Style Questions - Geometric Series
2.10.11End of Topic Test - Sequences
3Ratio
3.1Ratios in Practice
3.2Manipulating Ratios
3.2.1Simplifying Ratios
3.2.2Turn Ratios into Equations
3.2.3Ratio Notation & Reduction
3.2.4End of Topic Test - Units & Ratios
3.2.5Part : Part Ratio
3.2.6Part : Whole Ratio
3.2.7Two Ratios
3.2.8Differences in Ratios
3.2.9Fractions in Ratios
3.2.10Numbers as Fractions
3.2.11Ratios to Fractions
3.2.12Exam-Style Questions - Ratios
3.2.13End of Topic Test - Fractions & Ratios
3.3Percentage & Interest
3.3.1Convert Percentages to Decimals
3.3.2Convert Percentages to Fractions
3.3.3Find the Percentage
3.3.4More/Less Percentage
3.3.5Price from Percentage Change
3.3.6Calculate Percentage Change from Price
3.3.7End of Topic Test - Percentages
3.3.8Profit & Interest & Discount
3.3.9Original Price after Discounts
3.3.10Simple Interest
3.3.11Compound Growth & Decay
3.3.12Compound Interest
3.3.13Compound Interest - After How Many Days?
3.3.14End of Topic Test - Percentages in Finance
3.4Proportion
4Geometry
4.1Introduction to Geometry
4.1.1Geometry Basics
4.1.2Geometry Basics 2
4.1.3Regular Polygons
4.1.4Triangles & Angles
4.1.5Midpoint
4.1.6Bisectors & Perpendiculars
4.1.7Locus
4.1.8Angle rules
4.1.9Parallel Angle Rules
4.1.10Angles in a Polygon
4.1.11Angles in a Triangle
4.1.12Exam-Style Questions - Angles in Shapes
4.1.13End of Topic Test - Geometry
4.2Triangles & Quadrilaterals
4.3Transformations
4.4Circle Basics
4.5Circle Theorems
4.6Measurements & Units
4.7Calculating Area
4.8Triangle Formulae
4.93D Shapes
4.9.1Types of 3D Shape
4.9.2Plans & Elevations
4.9.3Nets
4.9.4Volume of 3D Shapes
4.9.5Volume of a Cuboid
4.9.6Volume of a Cylinder
4.9.7Volume of a Sphere
4.9.8Volume of a Hemisphere
4.9.9Volume of a Cone
4.9.10Volume of a Pyramid
4.9.11Surface Area & Composite Shapes
4.9.12Surface Area of a Cuboid
4.9.13Surface Area of a Cylinder
4.9.14Surface Area of a Sphere/Hemisphere
4.9.15Surface Area of a Cone
4.9.16Exam-Style Questions - Volume
5Probability
5.1Introduction to Probability
5.2Representing Probabilities
5.3Probability Calculations
5.3.1Theoretical Probabilities
5.3.2Multiple Events
5.3.3Probability Calculations
5.3.4Dependent Events
5.3.5Independent Events
5.3.6Probability Tree Diagram
5.3.7Complex Tree Diagrams
5.3.8Conditional Probability
5.3.9Conditional Probability From a Two Way Table
5.3.10Conditional Probability Venn Diagrams
5.3.11Conditional Probability Tree Diagrams
5.3.12Exam-Style Questions - Probability
5.3.13End of Topic Test - Probability
6Statistics
6.1Introduction to Statistics
6.2Charts & Diagrams
6.2.1Bar Charts
6.2.2Pie Charts
6.2.3Stem & Leaf Diagrams
6.2.4Back to Back Stem & Leaf Diagrams
6.2.5End of Topic Test - Statistics
6.2.6Pictograms
6.2.7Time Series Graphs
6.2.8Histograms
6.2.9Histograms 2
6.2.10Frequency Polygons
6.2.11Cumulative Frequency
6.2.12Cumulative Frequency 2
6.2.13Scatter Graphs
6.2.14Scatter Graphs - Predictions
6.2.15Correlation & Causation
6.2.16Exam-Style Questions - Cumulative Frequency
6.3Comparing Datasets
Jump to other topics
1Number
1.1Using Numbers
1.1.1Equal Symbols
1.1.2Negative Numbers
1.1.3BIDMAS
1.1.4Inverse Operations
1.1.5End of Topic Test- Ordering
1.1.6Factors
1.1.7Multiples
1.1.8Highest Common Factor
1.1.9Lowest Common Multiple
1.1.10Number of Ways to Do a Task
1.1.11Number of Combinations
1.1.12Number of Ways To Arrange Objects
1.1.13End of Topic Test - Factors, Multiples & Combos
1.1.14Standard Form
1.1.15Ordinary Numbers to Standard Form
1.1.16Addition in Standard Form
1.1.17Subtraction in Standard Form
1.1.18Multiplication in Standard Form
1.1.19Division in Standard Form
1.1.20Exam-Style Questions - Standard Form & Primes
1.1.21Long Division
1.1.22Prime Numbers
1.1.23End of Topic Test - Standard Form & Long Division
1.2Fractions, Decimals & Percentages
1.2.1Adding Fractions
1.2.2Subtracting Fractions
1.2.3Multiplying Fractions
1.2.4Dividing Fractions
1.2.5Mixed Fractions
1.2.6Calculating Percentages of Numbers
1.2.7Calculating Fractions of Numbers
1.2.8Multiply Decimals
1.2.9Place Value
1.2.10Recurring Decimals
1.2.11Exam-Style Questions - Recurring Decimals
1.2.12Convert Fractions & Terminating Decimals
1.2.13Convert Recurring Decimals
1.2.14Convert Decimals & Percentages
1.2.15Compare Decimals
1.2.16Compare Fractions
1.2.17Exam-Style Questions - Percentages
1.2.18End of Topic Test - Fractions & Decimals
1.3Powers & Roots
1.3.1Powers
1.3.2Roots
1.3.3Indices Advanced
1.3.4Estimating Powers
1.3.5Estimating Roots
1.3.6Multiplying Powers
1.3.7Dividing Powers
1.3.8Power Raised to Power
1.3.9Fractions Raised to a Power
1.3.10Numbers Raised to Negative Powers
1.3.11Roots of Numbers
1.3.12Numbers to the Power of 1 & 0
1.3.13Surds
1.3.14Simplify Surds
1.3.15Rationalise Denominator
1.3.16Exam-Style Questions - Indices
1.3.17End of Topic Test - Powers & Roots
2Algebra
2.1Introduction to Algebra
2.1.1Algebra Basics
2.1.2Substituting Values
2.1.3Algebraic Vocabulary
2.1.4Collecting Terms
2.1.5End of Topic Test - Algebra Basics & Power Rules
2.1.6Expanding Brackets I
2.1.7Common Factors
2.1.8Expanding Brackets II
2.1.9Rearrange For x
2.1.10Rearrange with Powers
2.1.11Exam-Style Questions - Basics of Algebra
2.1.12End of Topic Test - Expanding & Rearranging
2.2Manipulating Algebra
2.3Proofs & Functions
2.4Straight Line Graphs
2.5Common Graphs
2.6Transformations & Tangents
2.7Properties of Graphs
2.8Solving Equations
2.8.1Solve Linear Equations
2.8.2Linear Equations with Fractions
2.8.3Solve Linear Equations Graphically
2.8.4Quadratic ±
2.8.5Factorise Quadratics
2.8.6Quadratic Formula
2.8.7Complete the Square
2.8.8Solve Quadratics Graphically
2.8.9End of Topic Test - Solving Equations
2.8.10Simultaneous Equations
2.8.11Quadratic Simultaneous Equations
2.8.12Simultaneous Equations - Graphs
2.8.13Simultaneous Equations - Word Problems
2.8.14Approximate Solutions - Number of Iterations
2.8.15Approximate Solutions - Iterations to 1 DP
2.8.16Approximate Solutions - Iteration Formula
2.8.17Exam-Style Questions - Simultaneous Equations
2.8.18End of Topic Test - Simultaneous Equations
2.9Inequalities
2.10Sequences
2.10.1Nth Term Rules
2.10.2Recurrence Relationships
2.10.3Find Nth Term in Linear Sequence
2.10.4Check If a Number is In a Sequence
2.10.5Exam-Style Questions - Sequences
2.10.6Arithmetic & Geometric Progressions
2.10.7Key Sequences
2.10.8Quadratic Sequences
2.10.9Find Nth Term in Quadratic Sequence
2.10.10Exam-Style Questions - Geometric Series
2.10.11End of Topic Test - Sequences
3Ratio
3.1Ratios in Practice
3.2Manipulating Ratios
3.2.1Simplifying Ratios
3.2.2Turn Ratios into Equations
3.2.3Ratio Notation & Reduction
3.2.4End of Topic Test - Units & Ratios
3.2.5Part : Part Ratio
3.2.6Part : Whole Ratio
3.2.7Two Ratios
3.2.8Differences in Ratios
3.2.9Fractions in Ratios
3.2.10Numbers as Fractions
3.2.11Ratios to Fractions
3.2.12Exam-Style Questions - Ratios
3.2.13End of Topic Test - Fractions & Ratios
3.3Percentage & Interest
3.3.1Convert Percentages to Decimals
3.3.2Convert Percentages to Fractions
3.3.3Find the Percentage
3.3.4More/Less Percentage
3.3.5Price from Percentage Change
3.3.6Calculate Percentage Change from Price
3.3.7End of Topic Test - Percentages
3.3.8Profit & Interest & Discount
3.3.9Original Price after Discounts
3.3.10Simple Interest
3.3.11Compound Growth & Decay
3.3.12Compound Interest
3.3.13Compound Interest - After How Many Days?
3.3.14End of Topic Test - Percentages in Finance
3.4Proportion
4Geometry
4.1Introduction to Geometry
4.1.1Geometry Basics
4.1.2Geometry Basics 2
4.1.3Regular Polygons
4.1.4Triangles & Angles
4.1.5Midpoint
4.1.6Bisectors & Perpendiculars
4.1.7Locus
4.1.8Angle rules
4.1.9Parallel Angle Rules
4.1.10Angles in a Polygon
4.1.11Angles in a Triangle
4.1.12Exam-Style Questions - Angles in Shapes
4.1.13End of Topic Test - Geometry
4.2Triangles & Quadrilaterals
4.3Transformations
4.4Circle Basics
4.5Circle Theorems
4.6Measurements & Units
4.7Calculating Area
4.8Triangle Formulae
4.93D Shapes
4.9.1Types of 3D Shape
4.9.2Plans & Elevations
4.9.3Nets
4.9.4Volume of 3D Shapes
4.9.5Volume of a Cuboid
4.9.6Volume of a Cylinder
4.9.7Volume of a Sphere
4.9.8Volume of a Hemisphere
4.9.9Volume of a Cone
4.9.10Volume of a Pyramid
4.9.11Surface Area & Composite Shapes
4.9.12Surface Area of a Cuboid
4.9.13Surface Area of a Cylinder
4.9.14Surface Area of a Sphere/Hemisphere
4.9.15Surface Area of a Cone
4.9.16Exam-Style Questions - Volume
5Probability
5.1Introduction to Probability
5.2Representing Probabilities
5.3Probability Calculations
5.3.1Theoretical Probabilities
5.3.2Multiple Events
5.3.3Probability Calculations
5.3.4Dependent Events
5.3.5Independent Events
5.3.6Probability Tree Diagram
5.3.7Complex Tree Diagrams
5.3.8Conditional Probability
5.3.9Conditional Probability From a Two Way Table
5.3.10Conditional Probability Venn Diagrams
5.3.11Conditional Probability Tree Diagrams
5.3.12Exam-Style Questions - Probability
5.3.13End of Topic Test - Probability
6Statistics
6.1Introduction to Statistics
6.2Charts & Diagrams
6.2.1Bar Charts
6.2.2Pie Charts
6.2.3Stem & Leaf Diagrams
6.2.4Back to Back Stem & Leaf Diagrams
6.2.5End of Topic Test - Statistics
6.2.6Pictograms
6.2.7Time Series Graphs
6.2.8Histograms
6.2.9Histograms 2
6.2.10Frequency Polygons
6.2.11Cumulative Frequency
6.2.12Cumulative Frequency 2
6.2.13Scatter Graphs
6.2.14Scatter Graphs - Predictions
6.2.15Correlation & Causation
6.2.16Exam-Style Questions - Cumulative Frequency
6.3Comparing Datasets
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