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Approximation and Estimation

- Calculus
- Absolute Maxima and Minima
- Absolute and Conditional Convergence
- Accumulation Function
- Accumulation Problems
- Algebraic Functions
- Alternating Series
- Antiderivatives
- Application of Derivatives
- Approximating Areas
- Arc Length of a Curve
- Arithmetic Series
- Average Value of a Function
- Calculus of Parametric Curves
- Candidate Test
- Combining Differentiation Rules
- Combining Functions
- Continuity
- Continuity Over an Interval
- Convergence Tests
- Cost and Revenue
- Density and Center of Mass
- Derivative Functions
- Derivative of Exponential Function
- Derivative of Inverse Function
- Derivative of Logarithmic Functions
- Derivative of Trigonometric Functions
- Derivatives
- Derivatives and Continuity
- Derivatives and the Shape of a Graph
- Derivatives of Inverse Trigonometric Functions
- Derivatives of Polar Functions
- Derivatives of Sec, Csc and Cot
- Derivatives of Sin, Cos and Tan
- Determining Volumes by Slicing
- Direction Fields
- Disk Method
- Divergence Test
- Eliminating the Parameter
- Euler's Method
- Evaluating a Definite Integral
- Evaluation Theorem
- Exponential Functions
- Finding Limits
- Finding Limits of Specific Functions
- First Derivative Test
- Function Transformations
- General Solution of Differential Equation
- Geometric Series
- Growth Rate of Functions
- Higher-Order Derivatives
- Hydrostatic Pressure
- Hyperbolic Functions
- Implicit Differentiation Tangent Line
- Implicit Relations
- Improper Integrals
- Indefinite Integral
- Indeterminate Forms
- Initial Value Problem Differential Equations
- Integral Test
- Integrals of Exponential Functions
- Integrals of Motion
- Integrating Even and Odd Functions
- Integration Formula
- Integration Tables
- Integration Using Long Division
- Integration of Logarithmic Functions
- Integration using Inverse Trigonometric Functions
- Intermediate Value Theorem
- Inverse Trigonometric Functions
- Jump Discontinuity
- Lagrange Error Bound
- Limit Laws
- Limit of Vector Valued Function
- Limit of a Sequence
- Limits
- Limits at Infinity
- Limits of a Function
- Linear Approximations and Differentials
- Linear Differential Equation
- Linear Functions
- Logarithmic Differentiation
- Logarithmic Functions
- Logistic Differential Equation
- Maclaurin Series
- Manipulating Functions
- Maxima and Minima
- Maxima and Minima Problems
- Mean Value Theorem for Integrals
- Models for Population Growth
- Motion Along a Line
- Motion in Space
- Natural Logarithmic Function
- Net Change Theorem
- Newton's Method
- Nonhomogeneous Differential Equation
- One-Sided Limits
- Optimization Problems
- P Series
- Particle Model Motion
- Particular Solutions to Differential Equations
- Polar Coordinates
- Polar Coordinates Functions
- Polar Curves
- Population Change
- Power Series
- Ratio Test
- Removable Discontinuity
- Riemann Sum
- Rolle's Theorem
- Root Test
- Second Derivative Test
- Separable Equations
- Simpson's Rule
- Solid of Revolution
- Solutions to Differential Equations
- Surface Area of Revolution
- Symmetry of Functions
- Tangent Lines
- Taylor Polynomials
- Taylor Series
- Techniques of Integration
- The Fundamental Theorem of Calculus
- The Mean Value Theorem
- The Power Rule
- The Squeeze Theorem
- The Trapezoidal Rule
- Theorems of Continuity
- Trigonometric Substitution
- Vector Valued Function
- Vectors in Calculus
- Vectors in Space
- Washer Method
- Decision Maths
- Geometry
- 2 Dimensional Figures
- 3 Dimensional Vectors
- 3-Dimensional Figures
- Altitude
- Angles in Circles
- Arc Measures
- Area and Volume
- Area of Circles
- Area of Circular Sector
- Area of Parallelograms
- Area of Plane Figures
- Area of Rectangles
- Area of Regular Polygons
- Area of Rhombus
- Area of Trapezoid
- Area of a Kite
- Composition
- Congruence Transformations
- Congruent Triangles
- Convexity in Polygons
- Coordinate Systems
- Dilations
- Distance and Midpoints
- Equation of Circles
- Equilateral Triangles
- Figures
- Fundamentals of Geometry
- Geometric Inequalities
- Geometric Mean
- Geometric Probability
- Glide Reflections
- HL ASA and AAS
- Identity Map
- Inscribed Angles
- Isometry
- Isosceles Triangles
- Law of Cosines
- Law of Sines
- Linear Measure and Precision
- Median
- Parallel Lines Theorem
- Parallelograms
- Perpendicular Bisector
- Plane Geometry
- Polygons
- Projections
- Properties of Chords
- Proportionality Theorems
- Pythagoras Theorem
- Rectangle
- Reflection in Geometry
- Regular Polygon
- Rhombuses
- Right Triangles
- Rotations
- SSS and SAS
- Segment Length
- Similarity
- Similarity Transformations
- Special quadrilaterals
- Squares
- Surface Area of Cone
- Surface Area of Cylinder
- Surface Area of Prism
- Surface Area of Sphere
- Surface Area of a Solid
- Surface of Pyramids
- Symmetry
- Translations
- Trapezoids
- Triangle Inequalities
- Triangles
- Using Similar Polygons
- Vector Addition
- Vector Product
- Volume of Cone
- Volume of Cylinder
- Volume of Pyramid
- Volume of Solid
- Volume of Sphere
- Volume of prisms
- Mechanics Maths
- Acceleration and Time
- Acceleration and Velocity
- Angular Speed
- Assumptions
- Calculus Kinematics
- Coefficient of Friction
- Connected Particles
- Constant Acceleration
- Constant Acceleration Equations
- Converting Units
- Force as a Vector
- Kinematics
- Newton's First Law
- Newton's Law of Gravitation
- Newton's Second Law
- Newton's Third Law
- Projectiles
- Pulleys
- Resolving Forces
- Statics and Dynamics
- Tension in Strings
- Variable Acceleration
- Probability and Statistics
- Bar Graphs
- Basic Probability
- Charts and Diagrams
- Conditional Probabilities
- Continuous and Discrete Data
- Frequency, Frequency Tables and Levels of Measurement
- Independent Events Probability
- Line Graphs
- Mean Median and Mode
- Mutually Exclusive Probabilities
- Probability Rules
- Probability of Combined Events
- Quartiles and Interquartile Range
- Systematic Listing
- Pure Maths
- ASA Theorem
- Absolute Value Equations and Inequalities
- Addition and Subtraction of Rational Expressions
- Addition, Subtraction, Multiplication and Division
- Algebra
- Algebraic Fractions
- Algebraic Notation
- Algebraic Representation
- Analyzing Graphs of Polynomials
- Angle Measure
- Angles
- Angles in Polygons
- Approximation and Estimation
- Area and Circumference of a Circle
- Area and Perimeter of Quadrilaterals
- Area of Triangles
- Arithmetic Sequences
- Average Rate of Change
- Bijective Functions
- Binomial Expansion
- Binomial Theorem
- Chain Rule
- Circle Theorems
- Circles
- Circles Maths
- Combination of Functions
- Combinatorics
- Common Factors
- Common Multiples
- Completing the Square
- Completing the Squares
- Complex Numbers
- Composite Functions
- Composition of Functions
- Compound Interest
- Compound Units
- Conic Sections
- Construction and Loci
- Converting Metrics
- Convexity and Concavity
- Coordinate Geometry
- Coordinates in Four Quadrants
- Cubic Function Graph
- Cubic Polynomial Graphs
- Data transformations
- Deductive Reasoning
- Definite Integrals
- Deriving Equations
- Determinant of Inverse Matrix
- Determinants
- Differential Equations
- Differentiation
- Differentiation Rules
- Differentiation from First Principles
- Differentiation of Hyperbolic Functions
- Direct and Inverse proportions
- Disjoint and Overlapping Events
- Disproof by Counterexample
- Distance from a Point to a Line
- Divisibility Tests
- Double Angle and Half Angle Formulas
- Drawing Conclusions from Examples
- Ellipse
- Equation of Line in 3D
- Equation of a Perpendicular Bisector
- Equation of a circle
- Equations
- Equations and Identities
- Equations and Inequalities
- Estimation in Real Life
- Euclidean Algorithm
- Evaluating and Graphing Polynomials
- Even Functions
- Exponential Form of Complex Numbers
- Exponential Rules
- Exponentials and Logarithms
- Expression Math
- Expressions and Formulas
- Faces Edges and Vertices
- Factorials
- Factoring Polynomials
- Factoring Quadratic Equations
- Factorising expressions
- Factors
- Finding Maxima and Minima Using Derivatives
- Finding Rational Zeros
- Finding the Area
- Forms of Quadratic Functions
- Fractional Powers
- Fractional Ratio
- Fractions
- Fractions and Decimals
- Fractions and Factors
- Fractions in Expressions and Equations
- Fractions, Decimals and Percentages
- Function Basics
- Functional Analysis
- Functions
- Fundamental Counting Principle
- Fundamental Theorem of Algebra
- Generating Terms of a Sequence
- Geometric Sequence
- Gradient and Intercept
- Graphical Representation
- Graphing Rational Functions
- Graphing Trigonometric Functions
- Graphs
- Graphs and Differentiation
- Graphs of Common Functions
- Graphs of Exponents and Logarithms
- Graphs of Trigonometric Functions
- Greatest Common Divisor
- Growth and Decay
- Growth of Functions
- Highest Common Factor
- Hyperbolas
- Imaginary Unit and Polar Bijection
- Implicit differentiation
- Inductive Reasoning
- Inequalities Maths
- Infinite geometric series
- Injective functions
- Instantaneous Rate of Change
- Integers
- Integrating Polynomials
- Integrating Trig Functions
- Integrating e^x and 1/x
- Integration
- Integration Using Partial Fractions
- Integration by Parts
- Integration by Substitution
- Integration of Hyperbolic Functions
- Interest
- Inverse Hyperbolic Functions
- Inverse Matrices
- Inverse and Joint Variation
- Inverse functions
- Iterative Methods
- Law of Cosines in Algebra
- Law of Sines in Algebra
- Laws of Logs
- Limits of Accuracy
- Linear Expressions
- Linear Systems
- Linear Transformations of Matrices
- Location of Roots
- Logarithm Base
- Logic
- Lower and Upper Bounds
- Lowest Common Denominator
- Lowest Common Multiple
- Math formula
- Matrices
- Matrix Addition and Subtraction
- Matrix Determinant
- Matrix Multiplication
- Metric and Imperial Units
- Misleading Graphs
- Mixed Expressions
- Modulus Functions
- Modulus and Phase
- Multiples of Pi
- Multiplication and Division of Fractions
- Multiplicative Relationship
- Multiplying and Dividing Rational Expressions
- Natural Logarithm
- Natural Numbers
- Notation
- Number
- Number Line
- Number Systems
- Numerical Methods
- Odd functions
- Open Sentences and Identities
- Operation with Complex Numbers
- Operations with Decimals
- Operations with Matrices
- Operations with Polynomials
- Order of Operations
- Parabola
- Parallel Lines
- Parametric Differentiation
- Parametric Equations
- Parametric Integration
- Partial Fractions
- Pascal's Triangle
- Percentage
- Percentage Increase and Decrease
- Percentage as fraction or decimals
- Perimeter of a Triangle
- Permutations and Combinations
- Perpendicular Lines
- Points Lines and Planes
- Polynomial Graphs
- Polynomials
- Powers Roots And Radicals
- Powers and Exponents
- Powers and Roots
- Prime Factorization
- Prime Numbers
- Problem-solving Models and Strategies
- Product Rule
- Proof
- Proof and Mathematical Induction
- Proof by Contradiction
- Proof by Deduction
- Proof by Exhaustion
- Proof by Induction
- Properties of Exponents
- Proportion
- Proving an Identity
- Pythagorean Identities
- Quadratic Equations
- Quadratic Function Graphs
- Quadratic Graphs
- Quadratic functions
- Quadrilaterals
- Quotient Rule
- Radians
- Radical Functions
- Rates of Change
- Ratio
- Ratio Fractions
- Rational Exponents
- Rational Expressions
- Rational Functions
- Rational Numbers and Fractions
- Ratios as Fractions
- Real Numbers
- Reciprocal Graphs
- Recurrence Relation
- Recursion and Special Sequences
- Remainder and Factor Theorems
- Representation of Complex Numbers
- Rewriting Formulas and Equations
- Roots of Complex Numbers
- Roots of Polynomials
- Rounding
- SAS Theorem
- SSS Theorem
- Scalar Triple Product
- Scale Drawings and Maps
- Scale Factors
- Scientific Notation
- Second Order Recurrence Relation
- Sector of a Circle
- Segment of a Circle
- Sequences
- Sequences and Series
- Series Maths
- Sets Math
- Similar Triangles
- Similar and Congruent Shapes
- Simple Interest
- Simplifying Fractions
- Simplifying Radicals
- Simultaneous Equations
- Sine and Cosine Rules
- Small Angle Approximation
- Solving Linear Equations
- Solving Linear Systems
- Solving Quadratic Equations
- Solving Radical Inequalities
- Solving Rational Equations
- Solving Simultaneous Equations Using Matrices
- Solving Systems of Inequalities
- Solving Trigonometric Equations
- Solving and Graphing Quadratic Equations
- Solving and Graphing Quadratic Inequalities
- Special Products
- Standard Form
- Standard Integrals
- Standard Unit
- Straight Line Graphs
- Substraction and addition of fractions
- Sum and Difference of Angles Formulas
- Sum of Natural Numbers
- Surds
- Surjective functions
- Tables and Graphs
- Tangent of a Circle
- The Quadratic Formula and the Discriminant
- Transformations
- Transformations of Graphs
- Translations of Trigonometric Functions
- Triangle Rules
- Triangle trigonometry
- Trigonometric Functions
- Trigonometric Functions of General Angles
- Trigonometric Identities
- Trigonometric Ratios
- Trigonometry
- Turning Points
- Types of Functions
- Types of Numbers
- Types of Triangles
- Unit Circle
- Units
- Variables in Algebra
- Vectors
- Verifying Trigonometric Identities
- Writing Equations
- Writing Linear Equations
- Statistics
- Bias in Experiments
- Binomial Distribution
- Binomial Hypothesis Test
- Bivariate Data
- Box Plots
- Categorical Data
- Categorical Variables
- Central Limit Theorem
- Chi Square Test for Goodness of Fit
- Chi Square Test for Homogeneity
- Chi Square Test for Independence
- Chi-Square Distribution
- Combining Random Variables
- Comparing Data
- Comparing Two Means Hypothesis Testing
- Conditional Probability
- Conducting a Study
- Conducting a Survey
- Conducting an Experiment
- Confidence Interval for Population Mean
- Confidence Interval for Population Proportion
- Confidence Interval for Slope of Regression Line
- Confidence Interval for the Difference of Two Means
- Confidence Intervals
- Correlation Math
- Cumulative Frequency
- Data Analysis
- Data Interpretation
- Discrete Random Variable
- Distributions
- Dot Plot
- Empirical Rule
- Errors in Hypothesis Testing
- Events (Probability)
- Frequency Polygons
- Generalization and Conclusions
- Geometric Distribution
- Histograms
- Hypothesis Test for Correlation
- Hypothesis Test of Two Population Proportions
- Hypothesis Testing
- Inference for Distributions of Categorical Data
- Inferences in Statistics
- Large Data Set
- Least Squares Linear Regression
- Linear Interpolation
- Linear Regression
- Measures of Central Tendency
- Methods of Data Collection
- Normal Distribution
- Normal Distribution Hypothesis Test
- Normal Distribution Percentile
- Point Estimation
- Probability
- Probability Calculations
- Probability Distribution
- Probability Generating Function
- Quantitative Variables
- Quartiles
- Random Variables
- Randomized Block Design
- Residuals
- Sample Mean
- Sample Proportion
- Sampling
- Sampling Distribution
- Scatter Graphs
- Single Variable Data
- Standard Deviation
- Standard Normal Distribution
- Statistical Graphs
- Statistical Measures
- Stem and Leaf Graph
- Survey Bias
- Transforming Random Variables
- Tree Diagram
- Two Categorical Variables
- Two Quantitative Variables
- Type I Error
- Type II Error
- Types of Data in Statistics
- Venn Diagrams

In maths, there are times when we have something long and tedious to work out and need to get to an answer quickly. You may be in a non-calculator exam and need to know what seventy-three multiplied by seven is. Or, you may be in a restaurant trying to **guess** how much the bill is going to come to. At times like these, it is useful to use approximation and estimation. Luckily, this article is all about **approximation and estimation**. Let's start with some definitions:

An **approximation** is a value that is **close** to the **true** **value** but **not** **quite** **equal** to it. Note that the symbol for "approximately" is.

We could approximate pi by saying that. In actual fact, pi is an irrational number that never ends, so is certainly not an exact value. However, it is a very good approximation.

**Estimation** is a process where we either guess or roughly calculate something. Our objective is to obtain a value that is as close to the true value as possible.

The definition of pi is the ratio of a circle's circumference and diameter. Thus, if we get a piece of string to measure the circumference of a circle, and divide it by its diameter, we can estimate pi. Suppose, we measure a diameter of a circle as 10 cm, and a circumference of 31.4 cm, we could say. Therefore, we have just estimated pi to be 3.14.

For this topic, it is also important that we know how to round numbers. Let's quickly recap before going any further...

**Round the number 3728 to the nearest 10, 100, and 1000. **

**Solution:**

When rounding to the nearest 10, we need to look at the digits from the 10s column onwards. In this case, we have 28. Now, we must ask ourselves, is 28 closer to 20 or 30? The answer is 30, since 28 is only 2 away from 30, whereas it is 8 away from 20. Thus, 28 rounds to 30 and so 3728 rounds up to 2730.

When rounding to the nearest 100, we look at the digits from the 100s column onwards. In this case, we have 728. Now, we must ask ourselves, is 728 closer to 700 or 800? Clearly, it is closer to 700 and so in this case, 3728 rounds down to 3700.

When rounding to the nearest 1000, we look at the digits from the 1000s column onwards. This is 3728. Now, we must ask ourselves, is 3728 closer to 3000 or 4000. In this case, it is closer to 4000 so we round up to 4000 when rounding to the nearest 1000.

Now that we have defined some key terms, we will now look at some examples using approximation and estimation.

To estimate a calculation, first **round** all the numbers to something that is "easy" to work with. For example, it is hard to multiply 72 by 91, however, it is much easier to work out and so we would round each of the numbers to the nearest 10 to form an estimate. This **rounding** process is an example of **approximating**. In other words, 72 is approximately 70 and 91 is approximately 90 so we use those numbers to work out our estimation. Sometimes, it is easier to round to the nearest whole number, hundred, or even decimal place. Choose something sensible that enables you to work out the estimation in your head.

**Work out an estimate for **

**Solution:**

This calculation is quite difficult to work out without using a calculator. However, if we round each number to the nearest 10, we obtain. Thus, we can say that the estimation for the calculation is 18.

We could go one step further and work out the **percentage error** between the estimated value and the real value. Using a calculator, we can find that. If we subtract the estimated value from the real value, we get what is called the absolute error. In this case, the absolute error is 0.0256881 (which is promising as it shows that our estimated value is close to the real value due to such a small absolute error).

Now, if we divide the absolute error by the actual value, and then multiply by 100, we get the percentage error. If we do this, we get the percentage error as. Since this number is small, we can see that we have a good estimation as we have such a small percentage error.

**I buy 32 packets of crisps for a party. Each packet costs 21p. Estimate the total cost of the crisps.**

**Solution:**

The total cost is 32 lots of 21p. Thus, we need to calculate to work out the cost (in pence). We can round both numbers to the nearest 10 to get which is far easier to work out. We get. Thus, we can say that the total cost of the 32 packets of crisps is about £6. The true value ofand so we can see that our estimated value is close to the actual value.

If we were to go one step further and work out the percentage error, we would need to subtract the estimated value from the true value , divide by the actual value and then multiply by the true value as follows:

. Thus, the percentage error of our estimate is 10.7%.

**Estimate the cost of 123 paper plates that cost 11p each and 157 napkins that cost 9p each **

**Solution:**

The total cost (in pence) of the paper plates is going to beand the total cost of the napkins is. Thus the total cost of both the paper plates and napkins is. We can approximate the numbers in this calculation and instead work out . Thus, an estimate for the total cost is £28.

We could work out the percentage error by working out the true value of the cost. In this case, it is . Thus, the percentage error is.

**Estimate the value of **

**Solution:**

If we round 301 to the nearest 10, we get 300. If we round 9.01 to the nearest 10, we get 10. Now, 0.499 is approximately 0.5, so let's round it to that. Thus, we have . Thus, 6000 is an estimate.

The true value of and so we can see that our estimated value is relatively close. The absolute error of our estimate is and our percentage error is . Thus, we can say that our estimation is out by approximately 10.4%.

**Estimate the value of **

**Solution: **

4.98 is quite close to 5, and it is easy to square 5, so let's approximate 4.98 as 5. 0.482 is close to 0.5, and it is fairly easy to divide by one half. Thus, we have the estimate . Thus, the estimation for this calculation is 50.

We could work out the percentage error by working out the true value of.

Thus, the percentage error is .

**Estimate the value of **

**Solution:**

51.3 is approximately 50, and 0.53 is approximately 0.5. Thus, we have the estimation . Thus, an estimate for the value is 10.

We could work out the percentage error by working out the true value of .

Thus, the percentage error is .

You may be questioning what the actual difference between estimation and approximation actually is. They are both very similar concepts, so how do we determine whether something is an estimation or an approximation?

Estimation is the process of **roughly** obtaining a solution to something that we don't already know. For example, you may estimate that the number of sweets in a jar is around 30, but you do not know how many there are **exactly**. You may also be in a shop, and want to estimate how much everything is going to come to. By rounding the price of everything in your basket to the nearest pound, you can get an estimation, but you do not know the true value until you go through the bills to pay. By rounding, you are approximating the price of each item; you know the true price but you want to make the calculation simpler. Going back to the first example, if it turns out there are actually 32 sweets in the jar, and each of them cost 23p, you can estimate the cost of all 32 by approximating 32 to 30 and 23 to 25.

So, the main difference between estimation and approximation is that with estimation, you do not know the true value. On the contrary, with approximation, you know the true value but want to alter it slightly to turn it into something that is easier to work with.

Being able to estimate and approximate is actually a very handy tool to have in everyday life. It enables us to quickly make approximate calculations in our heads rather than relying heavily on a calculator. It also makes us really good at that game of "guess the bill", where we guess the cost of a restaurant bill before it has come.

Mathematicians often use approximations, for example, when working out solutions to equations that are hard to solve. Later in the GCSE course, you may come across iterative techniques to approximate solutions of higher-order equations (I know, we have some interesting stuff coming).

Some other uses of estimation include trying to work out the value of something. For example, property evaluators estimate the value of a property by looking at various factors including the house size, access to local travel networks, access to schools, number of bedrooms, socio-economic status of the area, and more.

Estimations enable us to make predictions, and approximations make numbers easier to work with. Of course, there are times when our estimations are rubbish and deviate massively from the true value. There are also times when approximations are not sensible. However, in general, they provide us with a really useful tool way of working things out.

- An
**approximation**is a value that is**close**to the**true value**but**not quite equal**to it. **Estimation**is a process where we either guess or roughly calculate something.- To estimate a calculation, first round (approximately) all the numbers involved to something that is "easy" to work with. Then compute the calculation in your head.
- The
**difference**between estimation and approximation is that estimation is where we are trying to work out the true value by either guessing or using rounding techniques. An approximation is where we already know the true value, but take a value close to the true value so that it is easier to work with.

Choose a value that is close to the true value.

Where we try to work out the value of something we don't already know using approximations.

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