Mastering Number Systems: Fractions, Decimals, and Percentages Revision Notes | GCSE AQA Higher Further Maths

1. Why Fractions, Decimals, and Percentages Matter

Every number question at GCSE — whether it involves money, probability, or measurement — uses fractions, decimals, or percentages in some form. Knowing how to move between them fluently is one of the highest-value skills on the paper.

Think of fractions, decimals, and percentages as three different languages for the same idea. The number $\frac{3}{4}$, the decimal $0.75$, and the percentage $75\%$ all describe exactly the same quantity. Switching between them quickly is what separates confident answers from hesitant ones.

2. Converting Between Fractions, Decimals, and Percentages

The conversion rules follow a clear pattern. To turn a fraction into a decimal, divide the numerator by the denominator. To turn a decimal into a percentage, multiply by 100. To reverse that, divide a percentage by 100 to get a decimal, then write it as a fraction and simplify.

Worked example: Converting a Fraction to a Decimal and Percentage

Write $\frac{7}{20}$ as a decimal and as a percentage.

1. $\frac{7}{20} = 7 \div 20$ — The fraction bar means divide. [M1] for recognising this.
2. $7 \div 20 = 0.35$ — Either use long division or note that $\frac{7}{20} = \frac{35}{100}$, so $0.35$. Suitable for non-calculator paper.
3. $0.35 \times 100 = 35\%$ — Multiply by 100 to shift to percentage. [A1] for correct percentage.

Answer: $\frac{7}{20} = 0.35 = 35\%$

3. Working with Fractions

Now that conversions are clear, the next step is calculating with fractions directly. The four operations — adding, subtracting, multiplying, and dividing — each follow a distinct method, and mixing them up is one of the most common sources of dropped marks.

• Adding and subtracting fractions: find a common denominator first, then add or subtract the numerators.
• Multiplying fractions: multiply numerators together and denominators together, then simplify.
• Dividing fractions: Keep the first fraction, Change the division to multiplication, Flip the second fraction (Keep-Change-Flip).

Notice that dividing by $\frac{2}{5}$ gave a result larger than the original $\frac{3}{4}$. Dividing by a number between 0 and 1 always makes the result larger — dividing by a fraction flips it and multiplies, so the value grows. This surprises many students who expect division to always reduce a number.

Worked example: Adding Fractions with Different Denominators

Calculate $\frac{2}{3} + \frac{3}{8}$. Give your answer as a fraction in its simplest form.

1. LCM of 3 and 8 is 24 — Both denominators must match before adding. [M1] for finding a common denominator.
2. $\frac{2}{3} = \frac{16}{24}$ and $\frac{3}{8} = \frac{9}{24}$ — $2 \times 8 = 16$ and $3 \times 3 = 9$. Multiply numerator and denominator by the same factor.
3. $\frac{16}{24} + \frac{9}{24} = \frac{25}{24}$ — Denominators stay the same; only numerators are added. [M1]
4. $\frac{25}{24}$ — HCF of 25 and 24 is 1, so already in simplest form. — As an improper fraction: $\frac{25}{24} = 1\frac{1}{24}$. [A1] for correct simplified answer.

Answer: $\frac{25}{24}$ or $1\frac{1}{24}$

4. Ratio and Proportion

Ratio and proportion build directly on fraction skills. A ratio compares quantities; proportion asks what fraction of the total each part represents. Both appear constantly in GCSE questions about sharing, scaling, and best-value problems.

Worked example: Sharing in a Given Ratio

Aisha and Ben share £360 in the ratio $5:4$. How much does each person receive?

1. $5 + 4 = 9$ parts in total — Add the ratio values to find how many equal shares exist. [M1]
2. $£360 \div 9 = £40$ per part — Divide the total amount by the number of parts. [M1]
3. Aisha: $5 \times £40 = £200$ Ben: $4 \times £40 = £160$ — Multiply each ratio value by the value of one part. [A1] for both correct values.
4. $£200 + £160 = £360$ ✓ — The shares must sum to the original total. If they do not, recheck step 2.

Answer: Aisha receives £200 and Ben receives £160.

5. Order of Operations

With fractions, decimals, and ratios covered, there is one more rule that governs every multi-step calculation: order of operations. Without it, the same expression can give different answers depending on the order you work through it.

The rule is BIDMAS: Brackets first, then Indices (powers and roots), then Division and Multiplication working left to right, and finally Addition and Subtraction working left to right. Division and multiplication have equal priority — work left to right when both appear.

Worked example: Applying Order of Operations

Calculate $3 + 4^2 \times (6 - 2) \div 8$

1. $6 - 2 = 4$, so expression becomes $3 + 4^2 \times 4 \div 8$ — Always resolve brackets before anything else. [M1]
2. $4^2 = 4 \times 4 = 16$, so expression becomes $3 + 16 \times 4 \div 8$ — Expand the power: $4 \times 4 = 16$. Never skip this multiplication step.
3. $16 \times 4 = 64$, then $64 \div 8 = 8$, so expression becomes $3 + 8$ — Work left to right: multiply first, then divide. [M1]
4. $3 + 8 = 11$ — Addition is the final step. [A1] for correct answer.

Answer: $11$

6. Percentages in Context

Percentage calculations appear in almost every GCSE paper, from finance questions to data interpretation. The three most tested types are: finding a percentage of an amount, finding what percentage one number is of another, and calculating percentage change.

Worked example: Calculating Percentage Change

A jacket costs £80. In a sale, it is reduced to £60. Calculate the percentage decrease.

1. $£80 - £60 = £20$ — Subtract the new price from the original price to find the decrease. [M1]
2. $\frac{20}{80}$ — Always divide by the original (starting) value, not the new value. This is a very common trap.
3. $\frac{20}{80} \times 100 = 0.25 \times 100 = 25\%$ — $20 \div 80 = 0.25$, then $0.25 \times 100 = 25$. [A1] for correct percentage.

Answer: $25\%$ decrease

7. Exam Technique and Common Mistakes

The methods above cover every core skill in this topic. What separates a Grade 4 from a Grade 7 is not knowing more content — it is avoiding the predictable errors that appear in every sitting. The following guidance targets the mistakes examiners see most often.

Mastering fractions, decimals, percentages, ratio, proportion, and order of operations satisfies the core number system requirement at GCSE — and these skills underpin every other topic from algebra to statistics, so the investment here pays off across the whole paper.