AQA GCSE Maths: Number and Ratio Revision Guide

Number is the foundation of GCSE Maths. It underpins every other topic on the specification -- if your number skills are unreliable, you will lose marks in algebra, geometry, statistics and probability as well as on the number questions themselves. On the AQA specification, Number contributes roughly 25% of the total marks across the three papers. Ratio, Proportion and Rates of Change adds another 20% or so. Together, these two content areas account for nearly half the exam.

That is a significant proportion of the available marks, and the good news is that much of the content is accessible, practical and highly learnable. The concepts build on each other logically, and the question types are predictable once you have seen enough past papers. This guide covers everything you need to know about the Number and Ratio topics on AQA GCSE Maths -- the key content, the techniques that earn marks, the common mistakes that lose them, and the exam strategies that make the difference between knowing the maths and actually scoring well.

How the AQA Papers Work

Before diving into topics, it is worth understanding the exam structure, because it shapes how you should revise.

AQA GCSE Maths consists of three equally weighted papers:

Paper 1: Non-Calculator

• 1 hour 30 minutes
• 80 marks
• No calculator allowed

Paper 2: Calculator

• 1 hour 30 minutes
• 80 marks
• Calculator allowed

Paper 3: Calculator

• 1 hour 30 minutes
• 80 marks
• Calculator allowed

The total raw mark is 240. Any topic from the specification can appear on any of the three papers, so you cannot afford to skip anything. Number and ratio questions appear across all three papers, but Paper 1 places particular demands on your arithmetic because you must perform every calculation by hand.

For a detailed breakdown of how to approach each paper tactically, see our AQA GCSE Maths exam technique guide.

Number Topics

Number covers a wide range of skills, from fundamental arithmetic to Higher-tier work with surds and bounds. Here is what you need to know for each key area.

Place Value, Ordering and Rounding

You must be able to order integers, decimals and negative numbers on a number line and compare their values. Rounding questions appear regularly and take two main forms: rounding to a given number of decimal places, and rounding to a given number of significant figures.

The rule for both is the same -- look at the digit immediately after the one you are rounding to. If it is 5 or above, round up; if it is below 5, round down. The most common error with significant figures is miscounting them, particularly when the number starts with zeros. In 0.00374, the first significant figure is 3, not 0. So 0.00374 rounded to 2 significant figures is 0.0037.

The Four Operations

You need fluency with addition, subtraction, multiplication and division of integers, decimals and negative numbers. On Paper 1, this means reliable written methods -- long multiplication, long division, column addition and subtraction. These skills are not glamorous, but they underpin almost every calculation you will do in the exam.

For negative numbers, remember the sign rules. Multiplying or dividing two numbers with the same sign gives a positive result. Two different signs give a negative result. Subtracting a negative is the same as adding: 7 - (-3) = 7 + 3 = 10. These rules come up constantly in algebra as well as in number questions.

Factors, Multiples, Primes, HCF and LCM

You should be able to find factors and multiples of any number, identify prime numbers, and express a number as a product of its prime factors using a factor tree or repeated division.

From the prime factorisation, you can find the Highest Common Factor (HCF) and the Lowest Common Multiple (LCM). The HCF is found by multiplying together the prime factors that appear in both numbers. The LCM is found by multiplying together the highest power of each prime factor that appears in either number.

For example, 60 = 2 squared x 3 x 5 and 90 = 2 x 3 squared x 5. The HCF is 2 x 3 x 5 = 30 (taking the lowest power of each common factor). The LCM is 2 squared x 3 squared x 5 = 180 (taking the highest power of each factor).

A very common mistake is confusing HCF and LCM. The HCF is always smaller than or equal to both numbers. The LCM is always larger than or equal to both numbers. If your answer does not fit that pattern, you have them the wrong way around.

Powers and Roots

Know the square numbers up to 15 squared (225) and the cube numbers up to 5 cubed (125). You need these for Paper 1 especially, where you cannot look them up on a calculator.

The laws of indices are essential at both tiers:

• a to the power m times a to the power n = a to the power (m + n)
• a to the power m divided by a to the power n = a to the power (m - n)
• (a to the power m) to the power n = a to the power (m x n)
• a to the power 0 = 1
• a to the power -n = 1 / (a to the power n)

At Higher tier, you also need fractional indices: a to the power 1/n means the nth root of a, and a to the power m/n means the nth root of a, raised to the power m. For example, 8 to the power 2/3 = (cube root of 8) squared = 2 squared = 4.

Standard Form

Standard form expresses very large or very small numbers as a value between 1 and 10, multiplied by a power of 10. For example, 4,500,000 = 4.5 x 10 to the power 6, and 0.00032 = 3.2 x 10 to the power -4.

You need to convert between ordinary numbers and standard form in both directions. You also need to perform calculations with standard form numbers -- adding, subtracting, multiplying and dividing. On calculator papers, make sure you know how to enter standard form using the EXP button (not the multiply key). On Paper 1, you will need to manipulate the powers of 10 by hand.

Fractions

Fraction arithmetic is one of the most heavily tested areas on Paper 1. You need all four operations:

• Adding and subtracting: Find a common denominator, convert both fractions, then add or subtract the numerators. Simplify the result.
• Multiplying: Multiply the numerators together and the denominators together. Simplify.
• Dividing: Flip the second fraction (take its reciprocal) and multiply.

You also need to convert fluently between fractions, decimals and percentages. Know the common equivalences -- 1/4 = 0.25 = 25%, 1/3 = 0.333... = 33.3%, 3/8 = 0.375 = 37.5% -- as these save time across the whole exam.

Mixed numbers must be converted to improper fractions before you can multiply or divide them. For example, 2 and 1/3 = 7/3.

Percentages

Percentage questions take many forms and appear on almost every paper:

• Percentage of an amount. Find 15% of 240: 10% is 24, 5% is 12, so 15% is 36. On calculator papers, multiply by the decimal: 240 x 0.15 = 36.
• Percentage increase and decrease. To increase 80 by 12%, either find 12% and add it (80 + 9.6 = 89.6) or multiply by 1.12 (80 x 1.12 = 89.6). To decrease, multiply by 0.88. The multiplier method is faster and less error-prone.
• Reverse percentages. If an item costs 69 pounds after a 15% reduction, the current price represents 85% of the original. So the original = 69 / 0.85 = approximately 81.18 pounds. A common mistake is finding 15% of 69 and adding it back -- this gives the wrong answer because the 15% was taken from the original price, not the reduced one.
• Compound interest and repeated percentage change. To find the value after 5 years of 3% annual growth, multiply by 1.03 to the power 5. This connects to the Ratio, Proportion and Rates of Change content area and appears at both tiers.

Surds -- Higher Tier Only

A surd is an irrational root that cannot be simplified to a whole number -- for example, the square root of 3 or the square root of 50. You need to be able to simplify surds by factoring out perfect squares: the square root of 50 = the square root of (25 x 2) = 5 times the square root of 2.

Rationalising the denominator means removing a surd from the bottom of a fraction. To rationalise 1 / (square root of 3), multiply top and bottom by the square root of 3 to get (square root of 3) / 3. For denominators of the form (a + square root of b), multiply by the conjugate (a - square root of b).

Surd questions appear in the final third of Higher papers and are a reliable source of 3-4 marks.

Estimation and Bounds

Estimation questions ask you to round each value to one significant figure and then perform the calculation. The purpose is to check whether an answer is reasonable. Always show your rounded values clearly before calculating -- this is where the method marks sit.

Bounds questions (Higher tier) involve upper and lower bounds. If a length is measured as 12 cm to the nearest centimetre, the lower bound is 11.5 cm and the upper bound is 12.5 cm. Error intervals are written as 11.5 less than or equal to length, which is less than 12.5. Note the less-than (not less-than-or-equal-to) on the upper bound -- the upper bound itself is not a possible value because it would round up to 13 cm.

When combining bounds in calculations, think about which combination gives the largest or smallest possible result. For a division, the maximum value comes from dividing the upper bound of the numerator by the lower bound of the denominator.

Ratio, Proportion and Rates of Change

This content area connects number skills to practical, real-world contexts. It is heavily tested because it maps naturally onto everyday problem-solving -- recipes, maps, speed, currency, and financial calculations.

Simplifying Ratios and Sharing in a Ratio

Simplify ratios the same way you simplify fractions -- divide all parts by their highest common factor. The ratio 12:18 simplifies to 2:3. Make sure both sides are in the same units before simplifying. If one quantity is in metres and the other in centimetres, convert first.

To share an amount in a given ratio, add the parts of the ratio to find the total number of shares, divide the amount by the total, and then multiply by each part. To share 200 pounds in the ratio 3:5: the total is 8 shares, one share = 25 pounds, so the split is 75 pounds and 125 pounds.

More demanding questions give you one part of the ratio and ask for the total, or give you the difference between two parts. For example: "The ratio of red to blue counters is 5:3. There are 12 more red counters than blue. How many counters are there in total?" The difference is 5 - 3 = 2 parts. If 2 parts = 12, then 1 part = 6, and the total = 8 parts = 48 counters.

Direct and Inverse Proportion

Two quantities are in direct proportion if, when one doubles, the other also doubles. The relationship has the form y = kx, where k is the constant of proportionality. To find k, substitute a known pair of values.

Two quantities are in inverse proportion if, when one doubles, the other halves. The relationship has the form y = k/x. Again, substitute a known pair to find k, then use the equation to find unknown values.

At Higher tier, you may also encounter proportional relationships involving powers or roots -- for example, y is proportional to x squared, or y is proportional to the square root of x. The method is the same: write the algebraic relationship, find k, and use the equation.

Rates of Change and Compound Measures

Speed, density and pressure are the three compound measures you must know:

• Speed = distance / time. Make sure you match units. If speed is in km/h and time is in minutes, convert minutes to hours first.
• Density = mass / volume. Density tells you how much mass is packed into a given volume. Units are typically g/cm cubed or kg/m cubed.
• Pressure = force / area. Units are typically N/m squared (pascals).

For each formula, you need to be able to rearrange to find any of the three quantities. The formula triangle method works, but understanding the algebra behind it is more reliable in multi-step problems.

A very common error is using the wrong units. If a question gives distance in metres and time in hours, your speed will be in m/h, which is unusual. Convert to consistent units before calculating, and always check that your answer has sensible units.

Growth and Decay

Compound interest, depreciation and population growth all use the same underlying model: repeated percentage change. The formula is:

Final value = initial value x (multiplier) to the power n

where n is the number of time periods. For 3% annual growth, the multiplier is 1.03. For 5% annual depreciation, the multiplier is 0.95.

On calculator papers, this is straightforward. On Paper 1, you may be asked to calculate two or three iterations by hand, so practise multiplying decimals without a calculator.

Gradient as a Rate of Change

On distance-time graphs, the gradient represents speed. On other real-life graphs, the gradient represents the rate at which one quantity changes relative to another -- for example, the rate at which water flows into a container or the rate of temperature change over time.

At Higher tier, you may need to estimate the gradient of a curve at a particular point by drawing a tangent. This links to the calculus ideas that underpin A-Level Maths, and examiners use it to assess whether students can interpret graphs as well as plot them.

Non-Calculator Techniques

Since Paper 1 is non-calculator and accounts for a third of the total marks, strong mental arithmetic and reliable written methods are not optional -- they are essential. Here are the key techniques to practise.

Long multiplication. Use a column method or grid method -- whichever you are most comfortable with. To multiply 347 by 26, break it into 347 x 20 and 347 x 6, calculate each, and add the results. Practise regularly so the method becomes automatic.

Long division. This is the written method students find most difficult, and the one most often avoided. Avoidance is a mistake -- long division appears on Paper 1, and if you cannot do it, you will lose marks. Use the bus stop method. If you are dividing 952 by 14, ask "how many 14s go into 95?" (6, remainder 11), then bring down the 2 to get 112, and "how many 14s go into 112?" (8). The answer is 68.

Fraction arithmetic without a calculator. Adding fractions with different denominators is a core Paper 1 skill. To add 2/3 and 3/5, the common denominator is 15: 10/15 + 9/15 = 19/15 = 1 and 4/15. Practise until this is fluent.

Percentage calculations. Build percentages from 10%. To find 37.5% of 160: 10% is 16, so 30% is 48, 5% is 8, 2.5% is 4, giving 48 + 8 + 4 = 60. This is reliable, easy to show as working, and earns method marks even if you make a small arithmetic slip.

Estimating answers. Always estimate before you calculate, especially on multi-step questions. If you are calculating 489 x 31, estimate 500 x 30 = 15,000. If your exact answer is 1,515 or 151,590, you know something has gone wrong. Estimation catches place-value errors that are otherwise easy to miss.

Common Mistakes

Examiners report the same errors year after year. Being aware of them gives you an advantage.

Confusing HCF and LCM. The HCF is always smaller than or equal to both numbers. The LCM is always larger than or equal to both. If your answer does not make sense, check which one the question is asking for.

Incorrectly applying percentage increase and decrease. Finding 20% of the new value instead of the original value in reverse percentage problems. Using the wrong multiplier -- for example, multiplying by 0.2 instead of 1.2 for a 20% increase, or multiplying by 1.15 instead of 0.85 for a 15% decrease.

Errors with negative numbers. Forgetting that subtracting a negative is the same as adding. Getting the sign wrong when multiplying or dividing. These errors cascade through multi-step problems and can cost several marks.

Not simplifying ratios fully. If the question asks you to give a ratio in its simplest form and you write 6:10 instead of 3:5, you will lose the final mark even if the rest of your working is correct.

Forgetting to use correct units in compound measures. Calculating speed in metres per minute when the question asks for km/h, or giving density without specifying g/cm cubed. Always check the units the question requires.

Rounding too early in multi-step problems. On calculator papers, use the ANS button to carry the full value through each step and round only at the end. On Paper 1, keep at least one extra digit in intermediate steps. Premature rounding is one of the most common causes of inaccurate final answers.

Exam Technique

Knowing the mathematics is only half the challenge. Converting that knowledge into marks requires good exam technique.

Show all working. On any question worth 2 or more marks, the mark scheme allocates separate method marks (M marks) and accuracy marks (A marks). If your final answer is wrong but your method is correct and visible, you can still earn most of the marks. If you write only the answer and it is wrong, you score zero. This is the single most important piece of exam technique for GCSE Maths.

Check that your answers make sense. If a question asks for the number of people at a concert and your answer is 3.7, something has gone wrong. If a percentage increase gives a result smaller than the original, check your method. A quick sense-check catches errors that cost marks.

Read "show that" questions carefully. In a "show that" question, the answer is given to you. Your job is to demonstrate how to reach it, step by step. Every intermediate calculation must be visible. You must not use the given answer as part of your working -- that is circular reasoning and earns no credit.

Check whether the question asks for exact values or rounded answers. If a question says "give your answer to 3 significant figures," give exactly that. If it says "give an exact answer," leave surds and fractions in your answer rather than converting to decimals. If the question does not specify, give your answer to a reasonable degree of accuracy -- typically 3 significant figures or 1 decimal place, depending on context.

Manage your time. Each paper is 90 minutes for 80 marks, giving you just over one minute per mark. Work through the paper from start to finish, spending roughly one minute per mark on each question. If a question is taking much longer than its marks justify, move on and come back to it. Leaving 5-10 minutes at the end to check your answers is always worthwhile.

Prepare with LearningBro

LearningBro's GCSE Maths courses are built around the AQA specification and designed to develop both your mathematical knowledge and your exam skills. The Number course covers every number topic on the specification, from place value and fractions through to Higher-tier surds and bounds. The Ratio, Proportion and Rates of Change course takes you through simplifying ratios, direct and inverse proportion, compound measures and growth and decay -- with practice questions that mirror real AQA exam questions in format and difficulty.

Once you are confident with individual topics, the AQA exam preparation course focuses on mixed-topic practice and exam technique under timed conditions. Combine topic-by-topic practice with regular past papers and the strategies in this guide, and you will be well prepared to maximise your marks.

For more on how to approach the three papers tactically, see our AQA GCSE Maths exam technique guide.

Good luck with your revision.