Algebra and number Revision Notes for AQA GCSE Further Maths

Algebra and number

Introduction

When working with algebra, you'll often need to combine your algebraic skills with number concepts like percentages and ratios. This combination is essential for solving real-world problems and creating mathematical expressions that represent practical situations.

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The ability to translate between percentage language, ratio relationships, and algebraic expressions is a fundamental skill that appears frequently in examinations and real-world applications.

Working with percentages in algebra

Converting percentages to algebraic expressions

One of the most important skills is translating percentage language into algebraic expressions. When we say a value is "increased by a percentage," we need to understand how to write this mathematically.

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Key principle: When a value $x$ is increased by a percentage, you add the original value to the percentage of that value.

Let's look at how this works step by step:

For " $x$ increased by 13%":

Start with the original value: $x$
Add 13% of $x$ : $x + \frac{13}{100} \times x$
Simplify: $x + 0.13x = 1.13x$

The coefficient 1.13 comes from combining the original value (represented by 1) with the increase (0.13).

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Worked Example: Converting Percentage Increases

To convert " $x$ increased by 25%" to an algebraic expression:

Step 1: Write the original value plus the percentage increase $x + \frac{25}{100} \times x$

Step 2: Convert the percentage to decimal form $x + 0.25x$

Step 3: Factor out $x$ $x(1 + 0.25) = 1.25x$

Therefore, " $x$ increased by 25%" becomes $1.25x$ .

General formula for percentage increases

When a value is increased by $n$ %, the algebraic expression becomes:

Original value + $n$ % of original value
$x + \frac{n}{100} \times x$
$x\left(1 + \frac{n}{100}\right)$
This simplifies to $x\left(1 + \frac{n}{100}\right)$ or using decimals: $x$ (1 + decimal form of percentage)

Working with ratios in algebra

The parts method

Ratios can be thought of in terms of parts of a whole. This makes complex ratio problems much easier to handle when combined with algebra.

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Key principle: If $p:q = 4:5$ , think of $p$ as representing 4 equal parts and $q$ as representing 5 equal parts.

Using this method:

$p$ represents 4 parts
$q$ represents 5 parts
Any expression involving $p$ and $q$ can be calculated by working out how many parts it represents

Solving complex ratio expressions

When you need to find ratios involving algebraic expressions like $p + 2q : 4q$ , use the parts method:

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Worked Example: Using the Parts Method

Given $p:q = 4:5$ , find the ratio $p + 2q : 4q$ .

Step 1: Identify the parts If $p:q = 4:5$ , then $p = 4$ parts and $q = 5$ parts

Step 2: Calculate each expression

$p + 2q = 4$ parts $+ 2(5$ parts $) = 4 + 10 = 14$ parts
$4q = 4(5$ parts $) = 20$ parts

Step 3: Form the new ratio $14$ parts $: 20$ parts $= 14:20$

Step 4: Simplify Divide both sides by their highest common factor (2) to get $7:10$

Alternative algebraic approach

You can also solve ratio problems by converting the ratio to algebraic expressions:

If $p:q = 4:5$ , then $\frac{p}{q} = \frac{4}{5}$
Therefore: $p = \frac{4}{5}q$
Substitute this into your expressions and simplify

Combining percentages and ratios

Sometimes you'll encounter problems that involve both percentages and ratios. The key is to convert percentages to fractions or decimals first, then use the relationship between the variables.

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Worked Example: Combining Percentages and Ratios

If $a$ is 75% of $b$ and $b:c = 3:2$ , find the relationship between $a$ and $c$ .

Step 1: Convert percentage to fraction $a = \frac{75}{100}b = \frac{3}{4}b$

Step 2: Use the ratio relationship $b:c = 3:2$ , so $\frac{b}{c} = \frac{3}{2}$ , therefore $b = \frac{3}{2}c$

Step 3: Substitute $a = \frac{3}{4} \times \frac{3}{2}c = \frac{9}{8}c$

Therefore, $a = \frac{9}{8}c$ or $a:c = 9:8$

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Common exam tips:

Always simplify fractions by looking for common factors to reduce ratios to their simplest form
Show your working by breaking down percentage calculations into clear steps
Check your logic - does your final answer make sense in the context?
Use substitution carefully to make sure you don't lose track of your variables

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Key Points to Remember:

When converting " $x$ increased by $n$ %", the result is $x\left(1 + \frac{n}{100}\right)$
The parts method makes ratio problems much more manageable - think of ratios as actual parts
Always simplify your final ratios by dividing by the highest common factor
Percentage problems often involve converting to fractions (like 75% = $\frac{3}{4}$ )
Show each step clearly when substituting values - this helps avoid errors and earns method marks

Algebra and number (AQA GCSE Further Maths): Revision Notes