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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Percentages

GCSE Maths Numbers Percentages

A percentage is a part of one hundred. When we say 30 out of 100, it is written as 30%, which is the same as \frac{30}{100} = \frac{3}{10}.

If we want to find x% of a number, we use the formula: \frac{10}{100} \times. For example, to find 10% of a number, we do \frac{10}{100} \times the number. Let’s look at some examples:

1. Find 10% of 120:

2. What percentage of £360 is £54?

3. What percentage of £12 is 18 pence?

1. 10% of 120 = \frac{10}{100} \times 120 = 12.

2. We calculate \frac{54}{360} \times 100\% = 15\%

3. We should convert the figures into the same unit. We can either say £12 = 1200 pence or 18 pence = £0.18. We’ll use £12 = 1200p.

\frac{18}{1200} \times 100\% = 1.5%

So, 18 pence is 1.5% of £12.

Converting Fractions, Decimals, and Percentages

Converting between fractions, decimals, and percentages is an important skill to have when working with percentages. Let’s look at how we can carry out these conversions:

Fraction to Percentage

To convert a fraction to a percentage, multiply the fraction by 100%. For example:

Convert \frac{3}{4} to a percentage.

\frac{3}{4} \times 100 = 75%

Decimal to Percentage

To convert a decimal to a percentage, multiply the decimal by 100%. For example:

Convert 0.25 to a percentage.

0.25 \times 100 = 25%

Percentage to Fraction

To convert a percentage to a fraction, divide the percentage by 100% and simplify the fraction if necessary. For example:

Convert 75% to a fraction.

\frac{75}{100} = \frac{3}{4}

Percentage to Decimal

To convert a percentage to a decimal, divide the percentage by 100%. For example:

Convert 25% to a decimal.

\frac{25}{100} = 0.25

Examples

1. Find 5\frac{1}{2}% of £200:

2. Calculate 3\frac{1}{7}% of £2100:

3. 0.2% of a population were over 70 years old. If the population of the country was 13.1 million, how many were over 70 years old?

1. 5\frac{1}{2}% = \frac{11}{200} \times £200 = £11.

2. 3\frac{1}{7}% = \frac{22}{700} \times £2100 = £66.

3. 0.2% of 13,100,000 = \frac{0.2}{100} \times 13,100,000 = 26,200.

Real-life Applications of Percentages

Discounts

When shopping, you might encounter discounts represented as percentages. To find the discounted price, subtract the discount percentage from 100%, then multiply the result by the original price. For example:

A dress is on sale with a 30% discount. If the original price is £50, what is the discounted price?

Discounted price = (100\% - 30\%) \times £50

= 70\% \times £50 = 0.7 \times £50 = £35

Tax Rates

Tax rates are often represented as percentages. To calculate the total amount to pay, including tax, multiply the original price by (1 + \text{tax rate}). For example:

A book costs £15, and the tax rate is 5%. What is the total price, including tax?

Total price = £15 \times (1 + 0.05) = £15 \times 1.05 = £15.75

Population Growth

Population growth is often represented as a percentage increase per year. To find the future population after a given number of years, use the formula:

Future population = \text{current population} \times (1 + \text{growth rate})^{\text{number of years}}

Let’s look at an example:

A town has a population of 5,000, and the population is growing at 2% per year. What will be the population after three years?

Future population = 5,000 \times (1 + 0.02)^3 = 5,000 \times 1.061208 = 5,306

After three years, the population will be approximately 5,306.

Examples

1. In a sale, all prices have been decreased by 15%. Find the new price of a toy which costs £20.

2. Prices of cars in a showroom have increased by 25%. David bought his new car for £30,000. What was the price of David’s car before the prices increased?

3. During a sale, all prices have decreased by 15%. Find the original price of a toy which now costs £17.

4. In 2020, Jane had an annual income of £60,000, and income tax was deducted from her income at a rate of 25%. What was the amount of money Jane received after the income tax deduction?

1. 15% of £20 = \frac{15}{100} \times £20 = £3.

So the new price of the toy is £20 – £3 = £17.

2. Here £30,000 represents 125%, because the price is 25% higher. So, \frac{125}{100} of the original price = £30,000

Original price = £30,000 \times \frac{100}{125} = £24,000

3. Since prices have gone down by 15%, this means that the £17 represents 85% (100 - 15).

\frac{85}{100} of the original price = £17

Original price = £17 \times \frac{100}{85} = £20.

4. To calculate the amount of money Jane received after the income tax deduction, we can use the following formula:

Income after tax = Income before tax – (Income before tax × Tax rate)

Plugging in the values, we have:

Income after tax = £60,000 - (£60,000 \times 0.25) Income after tax = £60,000 - £15,000 Income after tax = £45,000

Therefore, Jane received £45,000 after the income tax deduction.

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