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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Ratios

GCSE Maths Ratio, Proportion and Rates of Change Ratios

A ratio is a comparison of quantities with the same units, typically represented by two or more numbers separated by a colon (e.g., 3:5:7).

Ratios can be used to describe relationships in everyday life and in various mixtures. For example, if you have two red sweets and four blue sweets, the ratio of red sweets to blue sweets can be expressed as 2:4.

In a mixture of cement, sand, and gravel, if there are 3 parts cement, 5 parts sand, and 7 parts gravel, the ratio representing this mixture would be 3:5:7.

Simplifying Ratios

Ratios can be simplified by finding common factors and reducing them to their lowest form. For example, the ratio 12:18 can be simplified to 2:3 by dividing both numbers by their common factor, 6. This process is similar to simplifying fractions.

Let’s look at an example:

Simplify the ratio 16:24.

Both numbers have a common factor of 8. Dividing each number by 8, we get:

\frac{16}{8} : \frac{24}{8} = 2:3

So, the simplified ratio is 2:3.

Converting Ratios

Ratios can be converted to fractions and percentages to better understand the relative quantities in a given situation.

To convert a ratio to fractions, first sum the numbers in the ratio. If the ratio is a:b:c, calculate a+b+c. The fractions for each component would be \frac{a}{a+b+c}, \frac{b}{a+b+c}, and \frac{c}{a+b+c}.

To convert a fraction to a percentage, multiply the fraction by 100.

Let’s look at an example:

Convert the ratio 3:5:7 to fractions and percentages.

First, add the numbers in the ratio: 3+5+7=15. Then, calculate the fractions for each component: \frac{3}{15}, \frac{5}{15}, and \frac{7}{15}.

Next, convert each fraction to a percentage:

  • \frac{3}{15}\times 100 = 20%
  • \frac{5}{15}\times 100 = 33.33%
  • \frac{7}{15}\times 100 = 46.67%

Equivalent Ratios

Equivalent ratios are ratios that represent the same relationship between two quantities, even though the actual numbers in the ratios may be different. If two ratios have the same value, then they are equivalent.

For example, the ratios 2:3 and 4:6 are equivalent because the relationship between the numbers in each ratio is the same (multiplying the first number by 2 gives the second number).

How to Find Equivalent Ratios

To find equivalent ratios, you can either multiply or divide both terms of a given ratio by the same non-zero number. This will maintain the proportion and create a new ratio with the same relationship between the quantities.

Let’s look at an example:

Find three equivalent ratios for the ratio 6:10.

We can multiply or divide both terms of the ratio by different non-zero numbers to find equivalent ratios.

  • Multiply by 2: (6 \times 2):(10 \times 2) = 12:20
  • Multiply by 3: (6 \times 3):(10 \times 3) = 18:30
  • Divide by 2: (6 \div 2):(10 \div 2) = 3:5

So, three equivalent ratios for the ratio 6:10 are 12:20, 18:30, and 3:5.

Examples

Example 1:

Sam and Ram share £200 in the ratio 3 : 2. How much does Sam get?

To find Sam’s share, we first determine the total number of parts in the ratio and then find the value of each part. Finally, we calculate Sam’s share based on the number of parts he gets.

Total parts in the ratio: 3 + 2 = 5 Each part’s value: \frac{200}{5} =£40

Sam’s share (3 parts): 3 \times £40 = £120

Sam gets £120.

Example 2:

Adam has £50 to share among his three children: Bill, Claire, and Diana, in the ratio 4:5:1. How much will each child receive?

First, add the numbers in the ratio: 4+5+1=10. Then, calculate the share for each child:

  • Bill: \dfrac{4}{10}\times £50=£20
  • Claire: \dfrac{5}{10}\times £50=£25
  • Diana: \dfrac{1}{10}\times £50=£5

Example 3:

Alan and Bernard share £230 in the ratio 1\frac{1}{2} : 2\frac{1}{3}. How much does Bernard get?

First, we need to simplify the ratio 1\frac{1}{2} : 2\frac{1}{3}. Convert it into improper fractions: this is \frac{3}{2} : \frac{7}{3}.

Now, we need to find a common denominator for the two fractions. The least common multiple of 2 and 3 is 6. So, we multiply both improper fractions by 6 to eliminate the denominators:

\frac{3}{2} : \frac{7}{3} =\frac{3}{2} \times 6 : \frac{7}{3} \times 6 = 9 : 14

The simplified ratio is 9 : 14. Now, we can find the share of Bernard:

Total parts in the ratio: 9 + 14 = 23 Each part’s value: \frac{230}{23} = £10

Bernard’s share (14 parts): 14 \times £10 = £140

Bernard gets £140.

Example 4:

Alan and Bernard share a certain amount of money in the ratio 2 : 3. Given that Bernard receives £15 more than Alan. How much does Alan receive? And how much does Bernard receive?

Let’s denote the total amount of money shared as T, Alan’s share as A, and Bernard’s share as B. We have the following relationships:

1. \frac{A}{T} = \frac{2}{5} and \frac{B}{T} = \frac{3}{5}

2. B = A +£15

First, we need to find the total amount of money (T) by using the information that Bernard receives £15 more than Alan. Since the difference between the two shares is \frac{1}{5} of the total amount, we can write:

\frac{1}{5} \times T = £15

Now we can solve for T:

T = 15 \times 5 = £75

Now that we know the total amount of money shared, we can find Alan’s and Bernard’s shares:

Alan: A = \frac{2}{5} \times £75 = £30 Bernard: B = \frac{3}{5} \times £75 = £45

So, Alan receives £30, and Bernard receives £45.

Example 5:

Given that Ahmad’s age : Brown’s age = 3 : 4 and Brown’s age : Caulas’ age = 4 : 7. Find the ratio of Ahmad’s age : Brown’s age : Caulus’ age.

Denoting Ahmad, Brown, and Caulus by A, B, and C respectively, we have: A : B = 3 : 4 and B : C = 4 : 7.

To find the combined ratio, we need to ensure that the value for Brown’s age in both ratios is equal. To do this, we can cross-multiply the ratios:

(A : B) \times (B : C) = (3 : 4) \times (4 : 7)

Now, multiply the respective terms: A : B : C = (3 \times 4) : (4 × 4) : (4 \times 7)

= 12 : 16 : 28

We can simplify this ratio further by dividing each term by the greatest common divisor, which is 4:

A : B : C = \frac{12}{4} : \frac{16}{4} : \frac{28}{4} = 3 : 4 : 7

The ratio of Ahmad’s age : Brown’s age : Caulus’ age is 3 : 4 : 7.

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