GCSE Maths

Numbers
32 Topics | 2 Quizzes
Integers
Integers Quiz
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
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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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Direct Proportion

GCSE Maths Ratio, Proportion and Rates of Change Direct Proportion

Direct proportion is a mathematical concept that describes the relationship between two variables when they increase or decrease at the same rate. In other words, when one variable increases, the other variable also increases, and vice versa.

Direct proportions are when two values increase or decrease together by the same factor. For example, there is a direct proportion between centimetres (cm) and metres (m), as 100 cm equals 1 m, so the multiplier is always 100.

The proportionality symbol is:

  • It is used to indicate that one expression changes in proportion to another

We can use it to represent a directly proportional relationship. For example, if t is directly proportional to p, we can write t \propto p.

Constant of Proportionality (k)

In a directly proportional relationship, there is a constant of proportionality, denoted as k. This constant represents the ratio between the two variables, expressed as y = kx.

The constant can be calculated from the information provided about x and y. If y is directly proportional to x, we write y ∝ x, which is equal to y = kx. If t is directly proportionate to p, then this means that t = kp, where k is a natural number showing that p is a multiple of t.

Solving Direct Proportion Problems

To solve direct proportion problems, follow these four steps:

Step 1: Find the proportional relationship between the variables: Identify the variables that are directly proportional and write the relationship using the proportionality symbol (∝).

Step 2: Form an equation with the constant of proportionality: Replace the proportionality symbol with an equal sign and include the constant of proportionality (k) in the equation, e.g., y = kx.

Step 3: Use the information provided to find the constant of proportionality: Using the values provided for x and y, calculate the constant of proportionality (k) by dividing one variable by the other, e.g., k = \frac{y}{x}.

Step 4: Substitute the constant of proportionality into the equation: Once you have calculated the constant of proportionality (k), substitute it back into the equation to find the unknown values.

Let’s look at an example:

If x \propto y and x = 6 when y = 18, find the value of y when x = 10.

First, write the proportion relationship: x \propto y.

Convert to an equation using the constant of proportionality: y = kx.

Use the given information to find the constant of proportionality:

k = \frac{y}{x}

= \frac{18}{6}

= 3

Substitute the constant of proportionality into the equation: y = 3 \times 10 = 30.

Direct Proportion in Graphs

Direct proportion graphs have specific characteristics. They are straight lines that pass through the origin (0,0) and have a constant slope equal to the constant of proportionality, k.

  • This graph has the equation y=kx

To plot a direct proportion graph, first calculate the constant of proportionality and create a table of values. Then, plot the points on the graph and draw a straight line through the points and the origin.

Let’s look at an example:

If y \propto x and y = 15 when x = 5, plot the graph of the direct proportion.

First, calculate the constant of proportionality: k = \frac{y}{x} = 1\frac{5}{5} = 3.

Create a table of values, such as:

xy
x = 1y = 3
x = 2y = 6
x = 3y = 9

Plot these points on the graph and draw a straight line through the points and the origin (0,0).

Example

Example 1:

Suppose the cost of apples is directly proportional to the number of apples bought. If 4 apples cost £12, how much would 10 apples cost?

1. Write the proportion relationship: Cost ∝ Number of apples

2. Convert to an equation using a constant of proportionality: Cost = k \times Number of apples

3. Use given information to find the constant of proportionality:

12 = k \times 4

k = 3

4. Substitute the constant of proportionality into the equation: Cost = 3 × Number of apples

5. Calculate the cost for 10 apples: Cost = 3 \times 10 = £30

10 apples would cost £30.

Example 2:

Suppose that y varies directly as x^2 and y = 12 when x = 2.

Form an equation connecting x and y, then use it to find the value of y when x=5. Also, find the value of x when y=108.

We are given that y is directly proportional to the square of x, which can be expressed as y \propto x^2. To form an equation connecting x and y, we introduce a constant of proportionality, k, resulting in the equation y = kx^2.

To find the value of k, we use the given data point: when x = 2, y = 12.

Substitute the values into the equation: 12 = k(2^2)

12 = 4k

k = 3

So, the equation connecting x and y is y = 3x^2.

Now, we can use this equation to find the value of y when x = 5:

y = 3(5^2)

= 3(25)

= 75

When x = 5, y = 75.

Next, we’ll find the value of x when y = 108:

3x^2 = 108

x^2 = \frac{108}{3}

x^2 = 36

x = \pm\sqrt{36}

x = \pm6

When y = 108, x can be either 6 or -6.

Example 3:

The speed of a car is directly proportional to the force applied to its accelerator pedal. If applying a force of 20 N results in a speed of 40 km/h, what would be the speed when a force of 50 N is applied?

1. Write the proportion relationship: Speed ∝ Force

2. Convert to an equation using a constant of proportionality: Speed = k \times Force

3. Use given information to find the constant of proportionality:

40 = k \times 20

k = 2

4. Substitute the constant of proportionality into the equation: Speed = 2 \times Force

5. Calculate the speed for a force of 50 N: Speed = 2 \times 50 = 100 km/h

The speed would be 100 km/h when a force of 50 N is applied.

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