Subject: Mathematics | Level: GCSE | Exam Board: AQA
Master the fundamentals of Number, the bedrock of the GCSE Maths specification. From fraction conversions to index laws and prime factorisation, this topic is essential for securing marks across all three papers.
Revision Notes & Key Concepts
Revision Podcast Transcript
Welcome to your GCSE Maths revision podcast. I'm your tutor, and today we're diving deep into one of the most fundamental topics in the entire GCSE specification: Number. Whether you're sitting AQA, Edexcel, or OCR, Number underpins almost everything you'll be asked to do in your exam. So let's get started, and by the end of this episode, you'll feel genuinely confident walking into that exam hall. Number might sound basic, but don't be fooled. It accounts for a significant chunk of marks across all three papers, and the skills you build here — working with fractions, percentages, powers, and roots — will appear again and again throughout your exam. Examiners love to embed Number skills inside other topics, so mastering this now pays dividends across the whole paper. Let's begin with the core concepts. First up: Types of Numbers. You need to know the difference between natural numbers, integers, rational numbers, and irrational numbers. Natural numbers are your counting numbers — one, two, three, and so on. Integers include all whole numbers, both positive and negative, plus zero. Rational numbers are any number that can be written as a fraction — so that includes all integers, all terminating decimals like 0.25, and all recurring decimals like 0.333 repeating. Irrational numbers cannot be written as a fraction — the most famous examples are pi and the square root of two. Examiners will ask you to classify numbers, so make sure you can spot the difference instantly. Now let's talk about Place Value and Rounding. Place value is the foundation of all arithmetic. Every digit in a number has a value depending on its position — units, tens, hundreds, thousands, and so on. When rounding, you look at the digit immediately after the place you're rounding to. If it's five or more, you round up. If it's four or less, you round down. Candidates lose marks by rounding too early in a multi-step calculation — always keep full calculator precision until the very last step, then round your final answer. Significant figures are slightly different from decimal places, and this trips up a lot of students. Significant figures start counting from the first non-zero digit. So 0.00456 to two significant figures is 0.0046 — the leading zeros don't count. Decimal places, on the other hand, count from the decimal point itself. Read the question carefully — it will tell you which one to use. Moving on to Fractions. A fraction represents a part of a whole. The number on top is the numerator — think of it as the number of parts you have. The number on the bottom is the denominator — the total number of equal parts. To add or subtract fractions, you need a common denominator. To multiply fractions, simply multiply the numerators together and the denominators together. To divide fractions, flip the second fraction and multiply — this is called multiplying by the reciprocal. Here's a memory hook for dividing fractions: Keep, Change, Flip. Keep the first fraction as it is. Change the division sign to a multiplication sign. Flip the second fraction upside down. Then multiply across. So two-thirds divided by four-fifths becomes two-thirds multiplied by five-fourths, which gives ten-twelfths, simplified to five-sixths. Now, Percentages. Percentages are everywhere in real life — discounts, tax, interest rates — and examiners love to set them in context. To find a percentage of an amount, convert the percentage to a decimal and multiply. So 35 percent of 240 is 0.35 times 240, which equals 84. To find a percentage increase or decrease, use a multiplier. A 20 percent increase means multiplying by 1.20. A 15 percent decrease means multiplying by 0.85. This multiplier method is faster and less prone to errors than doing it in two steps. Reverse percentage questions are a Higher-tier favourite. These ask: after an increase or decrease, what was the original value? The key is to identify what the amount after the change represents as a decimal multiplier, then divide. If a price after a 30 percent increase is 260 pounds, the original price is 260 divided by 1.30, which gives 200 pounds. Never make the mistake of just finding 30 percent of the given amount and subtracting — that's the most common error on reverse percentage questions. Let's move to Powers, Roots, and Standard Form. A power or index tells you how many times to multiply a number by itself. Two to the power three is two times two times two, which equals eight. The laws of indices are essential: when multiplying powers with the same base, you add the indices. When dividing, you subtract. When raising a power to a power, you multiply. A negative index means the reciprocal — so two to the minus three is one over eight. A fractional index means a root — so 16 to the power of one-half is the square root of 16, which is 4. Standard form is used to write very large or very small numbers efficiently. A number in standard form is written as A times ten to the power n, where A is between 1 and 10, and n is an integer. So 3,400,000 in standard form is 3.4 times ten to the power 6. And 0.00056 is 5.6 times ten to the power minus 4. When multiplying numbers in standard form, multiply the A values and add the powers. When dividing, divide the A values and subtract the powers. Always check your answer is still in correct standard form — if your A value is outside the range 1 to 10, adjust it. Now let's talk about Factors, Multiples, and Primes. A factor of a number divides into it exactly with no remainder. A multiple is what you get when you multiply a number by an integer. A prime number has exactly two factors: one and itself. Note that one is not prime — this is a very common mistake. The prime numbers start: 2, 3, 5, 7, 11, 13, 17, 19, 23... Prime factor decomposition means breaking a number down into a product of its prime factors. Use a factor tree. For example, 60 equals 2 times 30, which is 2 times 2 times 15, which is 2 times 2 times 3 times 5. Written in index form: 2 squared times 3 times 5. This is the foundation for finding the Highest Common Factor and Lowest Common Multiple. For HCF, find the prime factors common to both numbers and multiply them. For LCM, find the prime factors of both numbers and take each prime to its highest power. Let's cover Ratio and Proportion. A ratio compares two or more quantities. To simplify a ratio, divide all parts by their HCF. To share an amount in a given ratio, find the total number of parts, divide the amount by the total parts to find one part, then multiply. For example, share 120 pounds in the ratio 3 to 5. Total parts: 8. One part: 120 divided by 8 equals 15. So the shares are 45 pounds and 75 pounds. Direct proportion means as one quantity increases, the other increases at the same rate. Inverse proportion means as one increases, the other decreases. In the exam, you'll often need to set up a proportionality equation. If y is directly proportional to x, write y equals kx, find k using given values, then use the equation to find unknowns. Now for some crucial exam technique. Let's talk about the most common mistakes and how to avoid them. Mistake one: not showing working. Even if you get the final answer wrong, method marks are available. Examiners are instructed to award marks for correct method even when arithmetic errors occur. Write every step clearly. Mistake two: premature rounding. In multi-step calculations, keep full precision throughout and only round at the very end. Rounding intermediate values introduces errors that compound. Mistake three: confusing percentage change with reverse percentage. If a question says 'the price was increased by 20 percent to reach 360 pounds, find the original price,' you divide by 1.20, not subtract 20 percent of 360. Mistake four: forgetting that one is not a prime number. If asked to list prime numbers, start from 2. Mistake five: index law errors. Remember — when multiplying powers, you add the indices. When dividing, you subtract. Do not multiply the indices when multiplying. Mistake six: not simplifying fractions fully. Always check your fraction is in its simplest form by dividing numerator and denominator by their HCF. Now for a quick-fire recall quiz. Pause after each question and try to answer before I give you the answer. Question one: What is 3/8 as a decimal? Answer: 0.375. Question two: Write 0.000047 in standard form. Answer: 4.7 times ten to the power minus 5. Question three: Find the HCF of 24 and 36. Answer: 12. Question four: A jacket costs 85 pounds after a 15 percent reduction. What was the original price? Answer: 85 divided by 0.85 equals 100 pounds. Question five: What is 2 to the power of negative 3? Answer: one-eighth, or 0.125. Question six: Express 72 as a product of its prime factors. Answer: 2 cubed times 3 squared. How did you do? If you struggled with any of those, go back and review that section of your notes. Let's wrap up with the key things to take away from today. Number is the bedrock of GCSE Maths. Master your fraction, decimal, and percentage conversions — they come up on every paper. Learn the index laws cold — they're tested directly and embedded in algebra, surds, and standard form questions. Always use the multiplier method for percentage problems, especially reverse percentages. Show every step of your working — method marks are your safety net. Use prime factor decomposition to find HCF and LCM efficiently. And remember: one is not prime, and irrational numbers cannot be written as fractions. You've got this. Keep practising, keep showing your working, and remember that every mark you earn in Number is a mark that builds your confidence across the whole paper. Good luck, and I'll see you in the next episode.
Key Terms & Definitions
- Integer
- A whole number that can be positive, negative, or zero.
- Prime Number
- A number with exactly two distinct factors: 1 and itself.
- Rational Number
- Any number that can be written as a fraction of two integers.
- Reciprocal
- The reciprocal of a number $x$ is $1/x$. Multiplying a number by its reciprocal gives 1.
- Standard Form
- A way of writing numbers as $A \times 10^n$, where $1 \leq A < 10$ and $n$ is an integer.
- Significant Figure
- The first non-zero digit in a number, reading from left to right, and subsequent digits.
Worked Examples
Worked Example
Question: Calculate the value of $3^{-2} \times 3^5$. Give your answer as an integer. (2 marks)
Solution: Step 1: Apply the multiplication law of indices by adding the powers: $-2 + 5 = 3$. Step 2: The expression simplifies to $3^3$. Step 3: Calculate the value: $3 \times 3 \times 3 = 27$. Final answer: 27
Worked Example
Question: A car loses 15% of its value each year. If it is bought for £12,000, calculate its value after 3 years. (3 marks)
Solution: Step 1: Identify the decimal multiplier for a 15% decrease: $100\% - 15\% = 85\% = 0.85$. Step 2: Set up the compound calculation: $12000 \times 0.85^3$. Step 3: Calculate the final value: $12000 \times 0.614125 = 7369.5$. Final answer: £7,369.50
Worked Example
Question: In a sale, the price of a television is reduced by 20%. The sale price is £360. Work out the original price of the television. (3 marks)
Solution: Step 1: Recognise this is a reverse percentage question. Let the original price be $x$. Step 2: Set up the equation: $x \times 0.80 = 360$. Step 3: Solve for $x$: $x = 360 \div 0.80$. Step 4: Calculate: $x = 450$. Final answer: £450
Practice Questions
Question: Work out $2 \frac{1}{3} + 1 \frac{4}{5}$. Give your answer as a mixed number in its simplest form.
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Question: Write 360 as a product of its prime factors in index form.
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Question: A population of bacteria increases by 8% every hour. The initial population is 5000. Calculate the population after 6 hours. Give your answer to the nearest whole number.
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Question: Estimate the value of $\frac{4.93 \times 18.2}{0.51}$.
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Question: Calculate $(3.2 \times 10^4) \times (4.5 \times 10^3)$. Give your answer in standard form.
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