Numbers and the number system Revision Notes for AQA GCSE Further Maths

Numbers and the number system

Numbers form the foundation of all mathematical work you'll encounter in your GCSE studies. This topic covers essential skills in working with ratios, percentages, and fractions that will appear throughout your mathematics course. These fundamental number skills are tested both directly and as part of more complex problems across all areas of mathematics.

Working with ratios

A ratio compares quantities by showing how many times one amount contains another. When working with ratios, it's crucial to ensure all quantities are expressed in the same units before simplifying.

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Always convert to the same units first! This is the most common source of errors in ratio problems. Never attempt to simplify ratios with mixed units.

Simplifying ratios with different units

When ratios involve different units, your first step is always to convert everything to the same unit. Then you can simplify by dividing both parts by their common factors.

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Worked Example: Simplifying 3 kilometres to 840 metres

Step 1: Convert to the same units

3 kilometres = 3 × 1000 = 3000 metres
Ratio becomes 3000:840

Step 2: Simplify by finding common factors

3000:840 ÷ 10 = 300:84
300:84 ÷ 3 = 100:28
100:28 ÷ 4 = 25:7

Therefore, 3 km : 840 m = 25:7 in its simplest form.

Percentage calculations

Percentages represent parts of 100, making them useful for comparing different quantities. Converting between percentages and decimals is a key skill you'll use constantly in GCSE mathematics.

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Remember that "percent" literally means "per hundred" - this helps you understand why we divide by 100 to convert to decimals.

Finding a percentage of a number

To find a percentage of any number, convert the percentage to a decimal by dividing by 100, then multiply by your number.

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Worked Example: Finding 43% of 5680

Step 1: Convert percentage to decimal

43% ÷ 100 = 0.43

Step 2: Multiply by the number

0.43 × 5680 = 2442.4

Therefore, 43% of 5680 = 2442.4

This method works for any percentage calculation and is often easier than trying to work with fractions.

Percentage increases

When increasing a number by a percentage, you can use a shortcut method that combines the original amount with the increase in one calculation.

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Worked Example: Increasing 540 by 17.5%

Method: Find (100% + 17.5%) of the original number

Step 1: Calculate the total percentage

100% + 17.5% = 117.5%

Step 2: Convert to decimal and multiply

117.5% = 1.175
540 × 1.175 = 634.5

Therefore, 540 increased by 17.5% = 634.5

This method is particularly useful because it reduces the chance of calculation errors and is faster than calculating the increase separately and adding it on.

Fraction operations without a calculator

Working with fractions requires understanding the fundamental rules of arithmetic, especially when calculators aren't allowed. The key is to follow the order of operations carefully and remember the basic rules for each operation.

Division with fractions

The key rule for dividing fractions is to multiply by the reciprocal (flip the second fraction upside down).

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Worked Example: Calculating $\frac{9}{10} - \frac{2}{5} ÷ \frac{6}{7}$

Step 1: Follow order of operations (division before subtraction)

First calculate: $\frac{2}{5} ÷ \frac{6}{7}$

Step 2: Divide by multiplying by the reciprocal

$\frac{2}{5} ÷ \frac{6}{7} = \frac{2}{5} × \frac{7}{6}$

Step 3: Multiply the fractions

$\frac{2}{5} × \frac{7}{6} = \frac{14}{30} = \frac{7}{15}$

Step 4: Subtract from the original fraction

$\frac{9}{10} - \frac{7}{15}$ (requires common denominator to complete)

chatImportant

Remember "flip and multiply" for fraction division - this is one of the most frequently tested skills in non-calculator papers.

Complex ratio problems

Sometimes you'll encounter problems involving multiple connected ratios that need to be combined or manipulated. These problems require systematic thinking and careful attention to common terms.

Connecting ratios through common terms

When you have two ratios that share a common term, you can connect them by making that term the same in both ratios.

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Worked Example: Finding x : z when x : y = 5 : 3 and y : z = 4 : 7

Step 1: Identify the common term

y appears in both ratios with different values (3 and 4)

Step 2: Find the least common multiple of the y values

LCM of 3 and 4 = 12

Step 3: Scale both ratios to make y = 12

x : y = 5 : 3 becomes 20 : 12 (multiply by 4)
y : z = 4 : 7 becomes 12 : 21 (multiply by 3)

Step 4: Combine the ratios

x : y : z = 20 : 12 : 21
Therefore x : z = 20 : 21

The key insight is that the common term must have the same value in both ratios before you can connect them effectively.

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Key Points to Remember:

Always convert to the same units before simplifying ratios - this prevents errors and ensures accurate results
Convert percentages to decimals by dividing by 100, then multiply to find percentage values
For percentage increases, multiply by (100 + increase)% converted to decimal form - this is faster and more reliable
When dividing fractions, flip the second fraction and multiply instead - remember "flip and multiply"
Connect multiple ratios by finding equivalent ratios with matching common terms - use the least common multiple for efficiency

Numbers and the number system (AQA GCSE Further Maths): Revision Notes