GCSE Maths

Numbers
32 Topics | 2 Quizzes
Rounding to the Nearest 10, 100, 1000, and Beyond
Significant Figures
Significant Figures
Percentages
Simple Interest
Rounding and Estimation
Standard Form
Four Operational Rules
Conversion of Units to other Units
Compound Interest
Surds
Rational and Irrational Numbers
Greatest and Least Possible Values
2 of 2
Algebra
29 Topics | 1 Quiz
Algebraic Notation
Algebraic Vocabulary
Collecting Like Terms
Substituting Values into Formulae
Rearranging Equations
Algebraic Fractions
Expanding Brackets and Simplifying
Expanding Double Brackets
Factorisation
Factorising Quadratic Expressions
Factorising Quadratics with a Coefficient of x Squared Greater than 1
The Quadratic Formula
The Difference of Two Squares
Subject of Formula
Solving Linear Equations
Quadratic Equations
Completing the Square
Sequences and Nth Term
Arithmetic and Geometric Sequences
Quadratic Sequences
1 of 2
Ratio, Proportion and Rates of Change
7 Topics
Ratios
Scaling Ratios
Direct Proportion
Inverse Proportion
Reverse Percentages
Units and Conversions
Scale Factors
Geometry and Measures
16 Topics | 3 Quizzes
2D Shapes
2D Shapes Quiz
3D Shapes
3D Shapes Quiz
Types of angles
Angle Properties
Angle in Parallel Lines
Triangles
Triangles Quiz
Angles in Triangles
Polygons
Lines of Symmetry
Bearings
Loci and constructions
Constructing Triangles
Transformations
Enlargement
Plans and elevations
Areas and Volumes in Map Scale Problems
Probability
10 Topics
Introduction to Probability
Relative Frequency
Sample Space
Probability of Single Events
Probability of Combined Events
Probability Trees
Conditional probability
Set Notation
Venn diagrams
The Complement of a Set
Statistics
2 Topics
Population and samples
Representing data
Vectors
Mensuration and Calculation
7 Topics
Area
Volume
Congruence and similarity
Pythagoras’ theorem
Sine and cosine rule
Area of a triangle
Standard units of measure
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Rounding and Estimation

GCSE Maths Numbers Rounding and Estimation

The approximation symbol (≈) is used when you are indicating that a value is close to, but not exactly equal to, another value. It is typically used where:

  • You have rounded a number to a specific decimal place or a whole number. For example, if you calculate a value as 3.14159 and round it to two decimal places, you would write it as π ≈ 3.14.
  • You are using an estimated value or an average value to make a calculation. For example, if you estimate that the average speed of a car is around 55 mph, you would write it as v ≈ 55 mph.
  • The exact value is not necessary or practical for the given context. In some cases, an approximate value is sufficient for understanding or solving a problem, and using the exact value would be cumbersome or unnecessary.
  • You are working with irrational numbers, such as the square root of a non-perfect square, and you need to express the value as a decimal. For example, you might write √2 ≈ 1.414.

When faced with calculations involving awkward numbers, we can round these numbers to simplify the process. This is particularly useful for estimating results when exact calculations aren’t necessary. Keep in mind that the estimations we’ll look at are not perfect, but they provide a quick idea of the result without resorting to a calculator.

Let’s look at some examples involving estimation.

Examples

Example 1:

Estimate the value of \sqrt{36.012}.

We will round up 36.012 to a number, as the square root is easy to find. Instead of using 36.012, we will use 36 and \sqrt{36}=6. So, 6 will be used in place of \sqrt{36.012}.

Example 2:

Estimate the value of:

\dfrac{\sqrt{36.012}-2.967}{3.008}

We take \sqrt{36.012} as \sqrt{36}=6 and 2.967 as 3. We will also take 3.008 as 3. We then have:

\dfrac{\sqrt{36.012}-2.967}{3.008}\approx \dfrac{6-3}{3}

= \dfrac{3}{3}

= 1

A calculator gives 1.008644 to six decimal places. Our estimate of 1 is not bad, and all the calculations involved were much easier than if we used the original values.

Example 3:

Estimate, to two significant figures, the value of \left( \sqrt[3] {27.0413}-0.995\right) ^{1.97}

We round 27.0413 to 27, 0.995 to 1, and 1.97 to 2. We then have:

\left( \sqrt[3] {27.0413}-0.995\right) ^{1.97} \approx \left( \sqrt[3] {27}-1\right) ^{2}

= \left( 3-1\right) ^{2}

= 2^{2}

= 4

A calculator gives a value of 3.943, to 3 decimal places.

The percentage error involved is:

\dfrac{4-3.943}{3.943}\times 100\approx 1.45

The estimation in this example has a percentage error of 1.45%.

Example 4:

By writing each number correct to one significant figure, estimate the value of \dfrac{48.9\times 0.207^{2}}{3.94}

To one significant figure:

48.9=50, 0.207=0.2 and 3.94=4.

So, our calculation gives:

\dfrac{50\times 0.2^{2}}{4}=\dfrac{50\times 0.04}{4}

= 50 \times 0.01

= 0.5

Example 5:

Estimate, to one significant figure, the value of \frac { { 215.3 }^{ 2 }-{ 97.2 }^{ 2 } }{ \sqrt { 27.3 } }

We shall use 200^{(2)}, 100^{(2)}, and 25 in place of 215.3^{(2)}, 97.2^{(2)}, and 27.3 respectively. We then have

\frac { { 215.3 }^{ 2 }-{ 97.2 }^{ 2 } }{ \sqrt { 27.3 } } \approx \frac { { 200 }^{ 2 }-{ 100 }^{ 2 } }{ \sqrt { 25 } }

=\frac { (200+100)(200-100) }{ 5 }

=\frac { 300\times 100 }{ 5 }

=6000

Example 6:

Estimate the value of:

\sqrt{\dfrac{15.945\times 1.025}{7.005-2.915}}

We approximate the numbers to get:

\sqrt{\dfrac{15.945\times 1.025}{7.005-2.915}}\approx \sqrt{\dfrac{16}{4}}

= \sqrt{\dfrac{16}{4}}

= \sqrt{4}

= 2

Example 7:

Estimate 26.9135^{\dfrac{2}{3}}

We approximate the numbers and get:

26.9135^{\dfrac{2}{3}}\approx 27^{\dfrac{2}{3}}= \left( 3^{3}\right) ^{\dfrac{2}{3}}= 3^{2}

= 9

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