GCSE Maths

Numbers
32 Topics | 1 Quiz
Integers
Prime Numbers and Composite Numbers
Positive and Negative Numbers
Order of Operations
Square Numbers and Cube Numbers
The Laws of Indices
Square Roots and Cube Roots
Triangular Numbers
The Fibonacci Sequence
Factors
Multiples
The Highest Common Factor
Fractions
Adding and Subtracting Fractions
Multiplication of Fractions
Division of Fractions
Fractional Indices
Decimal Places
Fractions and Decimals
Expressing Fractions as Decimal Numbers
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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
15 Topics
2D Shapes
Types of angles
Angle Properties
Angle in Parallel Lines
Triangles
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
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
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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Units and Conversions

GCSE Maths Ratio, Proportion and Rates of Change Units and Conversions

Units provide a standardised way to express quantities and measurements. We use conversions to make it easier to compare different values and when we’re trying to add and subtract values of different units of measure.

Ratios make it easier to convert between imperial and metric units. To do this, set up a proportion that links the imperial unit with its equivalent metric unit.

Let’s look at some common unit conversions and some imperial conversions.

Common Unit Conversions and Examples

Length

Some common metric length conversions include:

  • 1 km = 1,000 m
  • 1 m = 100 cm
  • 1 cm = 10 mm
  • 1 m = 1,000 mm

For example, to convert centimetres to metres, divide by 100:

150 cm = \frac{150}{100} = 1.5 m

An imperial conversion for length is:

1 inch ≈ 2.54 cm

Imperial conversion

Length (inches to centimetres):

1 inch = 2.54 cm

To convert inches to centimetres, use the ratio:

\frac{(x\:cm)}{(y\:inches)} = \frac{(2.54\:cm)}{(1 inch)}

Let’s look at an example:

Convert 8 inches to centimetres.

\frac{(8\:inches)}{(x\:cm)} = \frac{(1\:inch)}{(2.54 cm)}

Cross-multiply:

8 inches \times 2.54 cm = 1 inch × x cm

Solve for x:

x ≈ 20.32 cm

So, 8 inches = 20.32 cm.

Mass

Some common metric mass conversions include:

  • 1 kg = 1,000 g
  • 1 g = 1,000 mg
  • 1 tonne = 1,000 kg

To convert grams to kilograms, for example, divide by 1,000:

250\:g = \frac{250}{1,000} = 0.25\:kg

An imperial conversion for mass is:

1 pound ≈ 0.453592 kg

Imperial conversion

Mass (pounds to kilograms):

1 pound ≈ 0.453592 kg

To convert pounds to kilograms, use the ratio:

\frac{(x\:kg)}{(y\:pounds)} = \frac{(0.453592\:kg)}{(1\:pound)}

Let’s look at an example:

Convert 5 pounds to kilograms.

\frac{(5 pounds)}{(x kg)} = \frac{(1 pound)}{(0.453592 kg)}

Cross-multiply:

5 pounds × 0.453592 kg = 1 pound × x kg

Solve for x:

x ≈ 2.26796 kg

So, 5 pounds ≈ 2.26796 kg.

Volume

Common metric volume conversions include:

  • 1 l = 1000 ml
  • 1 ml = 1 cm³
  • 1 m³ = 1,000 l

For example, to convert millilitres to litres, divide by 1,000:

500\:ml = \frac{500}{1,000} = 0.5\:l

An imperial conversion for volume is:

1 imperial pint = 20 fluid ounces

Imperial conversion

Volume (fluid ounces to pints):

1 imperial pint = 20 fluid ounces

To convert fluid ounces to imperial pints, use the ratio:

*** QuickLaTeX cannot compile formula:
\frac{x\:pints}{y\:fluid\:ounces}} = \frac{1\:pint}{20\:fluid\:ounces}}

*** Error message:
Extra }, or forgotten $.
leading text: $\frac{x\:pints}{y\:fluid\:ounces}}

Let’s look at an example:

Convert 48 fluid ounces to imperial pints.

*** QuickLaTeX cannot compile formula:
\frac{x\:pints}}{48\:fluid\:ounces}} = \frac{1\:pint}}{20\:fluid\:ounces}}

*** Error message:
Argument of \frac  has an extra }.
leading text: $\frac{x\:pints}}

Cross-multiply:

*** QuickLaTeX cannot compile formula:
48\:fluid\:ounces} \times 1,\:pint = 20\:fluid\:ounces \times x,\:pints

*** Error message:
Extra }, or forgotten $.
leading text: $48\:fluid\:ounces}

Solve for x:

*** QuickLaTeX cannot compile formula:
x = \frac{48\:fluid\:ounces}}{20\:fluid\:ounces}} = 2.4\\:pints

*** Error message:
Argument of \frac  has an extra }.
leading text: $x = \frac{48\:fluid\:ounces}}

So, 48 fluid ounces = 2.4 imperial pints.

Time

Some common time conversions include:

  • 1 hour = 60 minutes
  • 1 minute = 60 seconds
  • 1 day = 24 hours

For example, to convert minutes to hours, divide by 60:

120 minutes = \frac{120}{60} = 2 hours

Unit Conversion Principles

Conversion Factors

Conversion factors are numerical values that allow us to convert from one unit to another. To change one unit of measure to another one, we need to know how much of the smaller one goes into the bigger one.

For example, to convert inches to centimetres, we can use the conversion factor 1 inch = 2.54 cm. To convert 10 inches to centimeters, we would multiply by the conversion factor: 10 inches × 2.54 cm/inch = 25.4 cm.

More than one conversion factor

You may need to multiply a given quantity by more than one conversion factor arranged in such a way that the original units cancel out, leaving only the desired units.

For example, to convert 3 hours to seconds, we can use: 3 hours × (60 minutes/hour) × (60 seconds/minute) = 10,800 seconds.

Conversion Tables

Conversion tables provide a convenient way to look up and compare conversion factors for different units. They can be helpful when working with less common units or when converting between Imperial system measurements to and metric system measurements.

Converting Units in Compound Measures

Compound measures are quantities that use a combination of two or more units, like speed (measured in metres per second), density (measured in kilograms per cubic metre), or pressure (measured in pascals).

To change the units for these types of measures, you’ll need to apply the correct conversion factors to each individual unit involved. This way, you can easily convert compound measures from one set of units to another.

For example, to convert a speed of 50 miles per hour to metres per second, use the conversion factors for miles to metres and hours to seconds:

50\text{miles/hour} \times (1609.34\text{metres/mile}) \times (\frac{1\text{hour}}{3600\text{seconds}}) \approx 22.35,\text{m/s}.

Conversion Problems and Strategies

Multi-step Conversions

Some problems require multiple conversion steps to obtain the desired units.

For example, if you need to convert a speed given in feet per second to kilometres per hour, you will need to perform two separate conversions: first, from feet to metres, and then from seconds to hours.

Let’s look at an example:

Convert 15 feet per second to kilometres per hour.

  • Conversion factor: 1 foot = 0.3048 metres
  • Conversion factor: 1 hour = 3600 seconds

Step 1: Convert feet to metres.

Conversion factor: 1 foot = 0.3048 metres

15 feet/second \times (0.3048 metres/foot) ≈ 4.57 metres/second

Step 2: Convert seconds to hours.

Conversion factor: 1 hour = 3600 seconds

4.57 metres/second × (3600 seconds/hour) ≈ 16,452 meters/hour

Step 3: Convert metres to kilometres

Conversion factor: 1 kilometre = 1000 metres

16,452 metres/hour × (1 kilometre/1000 metres) ≈ 16.46 km/h

Inverse Conversions

Inverse conversions are useful when you have a known conversion factor and need to reverse the process to obtain the original unit. This can be done by using the reciprocal of the conversion factor, essentially flipping the numerator and the denominator.

Let’s look at an example:

Convert 5 kilometres to miles. Conversion factor: 1 mile ≈ 1.609 kilometres.

Given conversion factor: 1 mile ≈ 1.609 kilometres

Reciprocal conversion factor: \frac {1}{1.609} ≈ 0.6214 (1 mile/1.609 kilometres)

5 kilometres × (0.6214 miles/kilometre) ≈ 3.11 miles

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