Ratio, Proportion, and Order of Operations Revision Notes | GCSE AQA Higher Further Maths
To simplify a ratio, divide all parts by their highest common factor; BIDMAS sets the order of operations. Notes, worked examples and 10 questions answered.
Ratio, Proportion, and Order of Operations is part of Knowledge and use of numbers and the number system in GCSE AQA Higher Further Maths.
4 min
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4
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GCSE AQA Higher
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Reviewed by Revise Right
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What this lesson covers
To simplify a ratio, divide all parts by their highest common factor; BIDMAS sets the order of operations. Notes, worked examples and 10 questions answered.
Apply the correct order of operations (BIDMAS) to evaluate numerical expressions involving brackets, indices, division, multiplication, addition, and subtraction
Express and simplify ratios, including reducing to simplest form and dividing quantities in a given ratio
Solve proportion problems using the unitary method and by identifying multiplicative relationships between quantities
Convert fluently between fractions, decimals, and percentages and use these representations to solve problems
Key ideas to keep in view
Use this lesson to stay inside Knowledge and use of numbers and the number system while connecting the detail back to the wider Number topic.
Order of Operations (BIDMAS): The agreed sequence for evaluating expressions: Brackets, Indices, Division and Multiplication (left to right), Addition and Subtraction (left to right). Used whenever an expression contains more than one operation.
Ratio: A comparison of two or more quantities of the same type, written as $a: b$. Used when sharing or comparing amounts in a fixed relationship.
Proportion: A statement that two ratios are equal, or a relationship where quantities scale together at a constant rate. Used when finding unknown values in direct or inverse scaling problems.
Unitary Method: A proportion technique that finds the value of one unit first, then scales to the required quantity. Used in best-value, currency, and recipe problems.
Revision notes
Read the core explanations from Ratio, Proportion, and Order of Operations before testing yourself from memory.
1. Why These Skills Are Exam-Critical
Ratio, proportion, and order of operations appear in almost every GCSE paper — often hidden inside geometry, algebra, or statistics questions. Mastering them means you collect marks that other students drop through careless arithmetic or misread calculations.
These topics connect tightly: a proportion problem might ask you to work with fractions, a ratio question might require percentage conversion, and every multi-step calculation demands correct order of operations. Think of BIDMAS as the grammar rule that makes all other maths make sense.
2. Order of Operations — BIDMAS
Here is the key insight: Division and Multiplication share the same priority level — you work through them left to right, not multiplication before division. The same applies to Addition and Subtraction. Getting this wrong is one of the most common sources of lost marks at GCSE.
Worked example: Applying BIDMAS to a Multi-Operation Expression
Calculate $3 + 4^2 \times (10 - 4) \div 8$
$(10 - 4) = 6$ — Brackets have the highest priority. Replace the bracket with its value. [M1]
$4^2 = 4 \times 4 = 16$ — Indices come next. $4^2$ means $4 \times 4$, which equals 16. Never skip this expansion.
$3 + 16 \times 6 \div 8$ — Substitute the results from steps 1 and 2 back in.
$16 \times 6 = 96$, then $96 \div 8 = 12$ — Multiplication and division share equal priority — work left to right. [M1]
$3 + 12 = 15$ — Addition is last. Final answer confirmed. [A1]
Answer: $15$
Order of operations underpins every calculation in this lesson. Now that the priority rules are clear, the next step is applying them inside ratio and proportion problems — where the arithmetic is often multi-step.
3. Ratio — Comparing and Sharing Quantities
A ratio tells you how two or more quantities relate to each other. The ratio $3: 5$ means for every 3 parts of one quantity, there are 5 parts of another. Ratios do not carry units — they are pure comparisons.
To simplify a ratio, divide all parts by their highest common factor (HCF). For example, $12: 18$ simplifies to $2: 3$ because the HCF of 12 and 18 is 6. The simplified ratio is equivalent — it describes the same relationship.
Worked example: Dividing a Quantity in a Given Ratio
Share £40 between Ahmed and Priya in the ratio $3: 5$.
$3 + 5 = 8$ parts in total — Add the ratio values to find how many equal parts the whole is divided into. [M1]
$£40 \div 8 = £5$ per part — Divide the total amount by the number of parts.
Ahmed: $3 \times £5 = £15$
Priya: $5 \times £5 = £25$ — Multiply the value of one part by each person's ratio number. [A1 each]
$£15 + £25 = £40$ ✓ — The shares must sum to the original total. This is your sanity check — always do it.
Answer: Ahmed receives £15 and Priya receives £25.
Sharing in a ratio is the foundation. The next level is working backwards — finding the original total or one quantity when you know the other. This is where proportion comes in.
4. Proportion — Scaling with the Unitary Method
Proportion problems involve quantities that scale together. In direct proportion, when one quantity doubles, the other doubles too. The most reliable exam method is the unitary method: find the value of one unit first, then scale to the required amount.
Worked example: Using the Unitary Method
5 kg of apples cost £6.00. How much do 8 kg cost?
$£6.00 \div 5 = £1.20$ per kg — Divide by 5 to reach the unit value. This is the 'unitary' step. [M1]
$£1.20 \times 8 = £9.60$ — Multiply the unit value by the required quantity. [A1]
$£9.60$ — Always include the currency or unit in your final answer — missing units can cost a mark.
Answer: 8 kg of apples cost £9.60.
The unitary method works for any direct proportion scenario: recipes, currency exchange, speed problems, and best-value comparisons. The structure is always the same — divide to find one, multiply to find many.
5. Fractions, Decimals, and Percentages — Fluent Conversion
Fractions, decimals, and percentages are three ways of expressing the same value. Being able to move between them quickly is essential — exam questions often give information in one form and ask for an answer in another.
Dividing by a fraction is a common stumbling block. When you divide by a fraction, you multiply by its reciprocal — the fraction flipped upside down. For example, $6 \div \frac{1}{3} = 6 \times 3 = 18$. Dividing by a number less than 1 always gives a result larger than the original, not smaller.
Worked example: Converting Between Forms and Using Percentages
A jacket costs £80. It is reduced by $\frac{3}{8}$ of its price. What is the sale price?
$\frac{3}{8} = 3 \div 8 = 0.375$ — Divide numerator by denominator. This is the proportion being removed. [M1]
$0.375 \times £80 = £30$ — $0.375 \times 80 = 30$. The discount is £30.
$£80 - £30 = £50$ — Subtract the discount from the original price. [A1]
$1 - \frac{3}{8} = \frac{5}{8}$, so $\frac{5}{8} \times £80 = £50$ — A quicker route: find what fraction remains and multiply directly. Both methods give the same answer.
Answer: The sale price is £50.
6. Combining Skills — Multi-Step Problems
Exam questions rarely test one skill in isolation. A typical higher-demand question might give you a ratio, ask you to find a percentage of one share, then compare it to a fraction of another. The method is always: identify what you know, plan the steps, apply BIDMAS at each stage.
Worked example: Multi-Step: Ratio, Percentage, and Fraction Combined
Mohammed and Aisha share £120 in the ratio $2: 3$. Mohammed spends $40\%$ of his share and Aisha spends $\frac{1}{5}$ of hers. How much do they spend in total?
$2 + 3 = 5$ parts. $£120 \div 5 = £24$ per part. — Unitary method applied to the ratio. [M1]
$£19.20 + £14.40 = £33.60$ — Add both amounts for the final answer. [A1]
Answer: Mohammed and Aisha spend £33.60 in total.
7. Exam Technique and Common Mistakes
These topics are reliable mark-earners when approached methodically, but they are also reliable mark-losers when students rush. The mistakes below appear repeatedly in mark schemes — knowing them in advance puts you ahead.
Every topic in this lesson — BIDMAS, ratio, proportion, and fraction-decimal-percentage conversion — feeds into the same core skill: structured, step-by-step arithmetic with a clear check at the end. Practise each method until the sequence is automatic, and you will find these marks reliable on any paper.
Sources and content standards
Published by Revise Right. Use the official specification reference to check the requirements for your course.
To simplify a ratio, divide all parts by their highest common factor (HCF). For example, 12:18 simplifies to 2:3 because the HCF of 12 and 18 is 6. The simplified ratio is equivalent and describes the same relationship.
What is BIDMAS?
BIDMAS is the priority order for operations: Brackets first, then Indices (powers and roots), then Division and Multiplication together, working left to right, then Addition and Subtraction together, working left to right. Division and multiplication share the same priority, so they are worked left to right rather than multiplication before division.
How do you share an amount in a given ratio?
To share an amount in a ratio, add the ratio parts, divide the total by the number of parts to find one part, then multiply by each ratio number. Sharing £40 in the ratio 3:5 gives 8 parts of £5, so the shares are £15 and £25, which add back to £40.
What is the unitary method?
The unitary method solves direct proportion problems by finding the value of one unit first, then scaling to the required amount. If 5 kg of apples cost £6.00, then 1 kg costs £6.00 ÷ 5 = £1.20, so 8 kg cost £1.20 × 8 = £9.60.
Do you multiply before you divide in BIDMAS?
Multiplication does not come before division in BIDMAS: division and multiplication share the same priority level and are worked left to right. The same applies to addition and subtraction, and getting this wrong is one of the most common sources of lost marks.
What does a ratio mean?
A ratio tells how two or more quantities relate to each other. The ratio 3:5 means for every 3 parts of one quantity there are 5 parts of another, and ratios do not carry units because they are pure comparisons.
How do you find the sale price after a fraction reduction?
The sale price after a fraction discount is found by working out the discount and subtracting it, or by multiplying by the fraction that remains. A £80 jacket reduced by 3/8 has a discount of 0.375 × £80 = £30, so the sale price is £50, the same as 5/8 × £80.
How do you check an answer when sharing in a ratio?
An answer for sharing in a ratio is checked by adding the shares, which must sum to the original total. Sharing £40 in the ratio 3:5 gives £15 and £25, and £15 + £25 = £40, confirming the answer.
How do you solve a multi-step ratio and percentage problem?
A multi-step ratio and percentage problem is solved by sharing first, then applying each percentage or fraction. Sharing £120 in the ratio 2:3 gives £48 and £72; 40% of £48 is £19.20 and 1/5 of £72 is £14.40, so the total spent is £33.60.
What is the difference between BIDMAS and PEMDAS?
BIDMAS and PEMDAS are names for the same order of operations: Brackets first, then Indices (powers and roots), then Division and Multiplication left to right, then Addition and Subtraction left to right. For example, 3 + 4² × (10 − 4) ÷ 8 = 15.