Basic Geometry Revision Notes
Subject: Mathematics | Level: GCSE | Exam Board: OCR
Master the fundamental connections between fractions, decimals, and percentages. These core skills are essential for both Foundation and Higher tier exams, unlocking marks across multiple topics from probability to compound interest.
Revision Notes & Key Concepts
Revision Podcast Transcript
GCSE Mathematics Podcast: Fractions, Decimals and Percentages Running time: approximately 10 minutes Voice: Female, warm, conversational, enthusiastic tutor tone --- [INTRO — 1 minute] Hello and welcome to your GCSE Maths revision podcast. I'm really glad you've pressed play today, because the topic we're covering — Fractions, Decimals and Percentages — is one of those golden topics that comes up in virtually every single GCSE Maths paper. Whether you're sitting Foundation or Higher, you will meet these ideas again and again, so getting them absolutely solid is one of the best investments you can make in your revision time. In this episode, we're going to cover all the core concepts clearly and confidently, then I'll walk you through the exam tips and common mistakes that catch students out every year, followed by a quick-fire recall quiz to test yourself, and then a summary to lock it all in. So grab a pen, maybe a piece of paper, and let's get started. --- [CORE CONCEPTS — 5 minutes] Let's begin with the big picture. Fractions, decimals, and percentages are simply three different ways of writing the same thing — a part of a whole. Think of a pizza cut into four equal slices. If you eat one slice, you've eaten one quarter of the pizza. That's a fraction: one over four. As a decimal, that's zero point two five. As a percentage, that's twenty-five percent. Same amount of pizza — three different ways to say it. So the first skill you need is conversion — moving fluently between these three forms. Let's start with fractions to decimals. The rule is beautifully simple: divide the top number — the numerator — by the bottom number — the denominator. So three-quarters means three divided by four, which gives you zero point seven five. Easy. Three divided by four. Always divide top by bottom. Now, decimals to percentages. Even simpler: multiply by one hundred. Zero point seven five multiplied by one hundred equals seventy-five percent. You're essentially shifting the decimal point two places to the right. And percentages back to fractions? Write the percentage over one hundred, then simplify. Seventy-five percent becomes seventy-five over one hundred. Both are divisible by twenty-five, so that simplifies to three-quarters. Perfect — we've gone all the way around the triangle. Now here's where candidates lose marks: converting percentages to decimals incorrectly. This is one of the most common errors in GCSE papers. Five percent is NOT zero point five. Five percent is zero point zero five. You divide by one hundred — so move the decimal point two places to the LEFT. Five becomes zero point zero five. Fifteen percent becomes zero point one five. Thirty percent becomes zero point three. Say it with me: divide by one hundred, move two places left. Next up: fractions of quantities. If a question says "find three-fifths of two hundred and forty," you have two methods. Method one: divide by the denominator first, then multiply by the numerator. Two hundred and forty divided by five is forty-eight. Forty-eight multiplied by three is one hundred and forty-four. Method two: multiply by the fraction directly — two hundred and forty multiplied by three-fifths. Either way, you get one hundred and forty-four. Always show your working — examiners award method marks even if your final answer is wrong. Now let's talk about percentage multipliers, because this is where the real efficiency lies in exam technique. Instead of finding ten percent and then adding it on, use a single multiplier. For a percentage increase, your multiplier is one plus the decimal form of the percentage. So a twenty percent increase means you multiply by one point two. A fifteen percent increase means multiply by one point one five. Simple. For a percentage decrease, your multiplier is one minus the decimal. So a twenty percent decrease means multiply by zero point eight. A thirty percent decrease means multiply by zero point seven. Here's the golden rule: INCREASE — one PLUS the decimal. DECREASE — one MINUS the decimal. Let me give you an example. A jacket costs eighty pounds. It's reduced by thirty-five percent in a sale. What's the sale price? The multiplier for a thirty-five percent decrease is one minus zero point three five, which equals zero point six five. Eighty multiplied by zero point six five equals fifty-two pounds. That's your answer in one step. No messing around finding thirty-five percent and subtracting — one multiplication and you're done. Now let's cover ordering mixed types, because exam questions love asking you to put fractions, decimals, and percentages in order. The trick is to convert everything to the same form first — decimals are usually easiest. Convert all values to decimals, order them, then write them back in their original form. Don't try to compare a fraction directly with a percentage — convert first, compare second. For Higher tier students, there's one extra concept: recurring decimals as fractions. A recurring decimal is one where digits repeat forever, like zero point three recurring — that's zero point three three three three going on forever. The fraction equivalent is one-third. Zero point six recurring is two-thirds. For more complex ones, you use an algebraic method. Let x equal zero point one two recurring. Then one hundred x equals twelve point one two recurring. Subtract: ninety-nine x equals twelve. So x equals twelve over ninety-nine, which simplifies to four over thirty-three. This is a Higher-only technique, but it's very learnable and examiners reward it well. --- [EXAM TIPS AND COMMON MISTAKES — 2 minutes] Right, let's talk exam strategy, because knowing the maths is only half the battle. Tip one: always show your working. For any question worth two or more marks, there are method marks available. Even if you make an arithmetic slip, you can still earn marks for the correct method. Write down every step. Tip two: read the question carefully for what form the answer should be in. If it says "give your answer as a fraction in its simplest form," don't leave it as a decimal. If it says "give your answer to two decimal places," don't leave it as a fraction. Candidates lose marks every year by giving the right value in the wrong form. Tip three: use estimation to check your answers. If you're calculating fifteen percent of eighty, you know ten percent is eight, so fifteen percent should be twelve. If your calculator gives you something wildly different, you've made an error somewhere. Tip four: the most common mistake on percentage multipliers is using zero point one for a ten percent increase instead of one point one. Remember — for an increase, you keep the original AND add the percentage. One point one means you have one hundred percent of the original plus ten percent more. Zero point one would only give you ten percent of the original — that's a completely different thing. Tip five: when simplifying fractions, always check for common factors. Divide both numerator and denominator by their highest common factor. If you're not sure of the HCF, divide by any common factor and keep going until you can't simplify further. --- [QUICK-FIRE RECALL QUIZ — 1 minute] Right, quick-fire quiz time! I'll ask a question, give you a few seconds to think, then I'll give the answer. Ready? Question one: What is three-quarters as a decimal? ... Answer: zero point seven five. Question two: What is the percentage multiplier for a twenty-five percent increase? ... Answer: one point two five. Question three: Convert forty percent to a fraction in its simplest form. ... Answer: two-fifths. Question four: What is the percentage multiplier for a fifteen percent decrease? ... Answer: zero point eight five. Question five: What is zero point three recurring as a fraction? ... Answer: one-third. How did you do? If you got all five, brilliant — you're in great shape. If you stumbled on any, go back and re-read that section of your notes. --- [SUMMARY AND SIGN-OFF — 1 minute] Let's wrap up with the key things to lock in before your exam. One: Fractions, decimals, and percentages are three forms of the same value — convert fluently between them. Two: Fraction to decimal — divide top by bottom. Decimal to percentage — multiply by one hundred. Three: Percentage to decimal — divide by one hundred. Five percent is zero point zero five, NOT zero point five. Four: Percentage multipliers — increase means one plus the decimal, decrease means one minus the decimal. Use them — they're faster and more accurate. Five: Always show your working, check the required answer format, and use estimation to sense-check your answers. Six: Higher tier — practice the algebraic method for converting recurring decimals to fractions. You've got this. Fractions, decimals, and percentages are everywhere in the real world — sales, interest rates, statistics — so every time you master this topic, you're building skills that go way beyond the exam room. Good luck with your revision, and I'll see you in the next episode. Keep going — you're doing brilliantly. --- END OF SCRIPT
Key Terms & Definitions
- Numerator
- The top number in a fraction, representing how many parts we have.
- Denominator
- The bottom number in a fraction, representing the total number of equal parts the whole is divided into.
- Multiplier
- A single decimal number used to calculate a percentage increase or decrease in one step.
- Recurring Decimal
- A decimal number where a digit or sequence of digits repeats infinitely.
- Simplify
- To divide the numerator and denominator of a fraction by their highest common factor to find an equivalent fraction in its lowest terms.
- Equivalent Fractions
- Fractions that represent the same value or proportion, though they may look different (e.g., 1/2 and 2/4).
Worked Examples
Worked Example
Question: Calculate $\frac{3}{8}$ of $420$. (2 marks)
Solution: Step 1: Divide the quantity by the denominator to find one eighth. $420 \div 8 = 52.5$ Step 2: Multiply by the numerator to find three eighths. $52.5 \times 3 = 157.5$ Final answer: $157.5$
Worked Example
Question: A shop has a sale with $35\%$ off all prices. A coat normally costs $£80$. Calculate the sale price of the coat. (3 marks)
Solution: Step 1: Identify the percentage decrease multiplier. Multiplier $= 1 - 0.35 = 0.65$ Step 2: Multiply the original price by the multiplier. Sale Price $= 80 \times 0.65$ Final answer: $£52$
Worked Example
Question: Write the recurring decimal $0.2\dot{7}$ as a fraction in its simplest form. (3 marks) [Higher Tier]
Solution: Step 1: Set up the algebraic equation. Let $x = 0.27777...$ Step 2: Multiply by 10 to move the non-recurring digit past the decimal. $10x = 2.7777...$ Step 3: Multiply by 100 to move one full recurring cycle past the decimal. $100x = 27.7777...$ Step 4: Subtract the $10x$ equation from the $100x$ equation. $100x - 10x = 27.7777... - 2.7777...$ $90x = 25$ Step 5: Solve for x and simplify. $x = \frac{25}{90}$ Divide numerator and denominator by 5: Final answer: $\frac{5}{18}$
Practice Questions
Question: Convert $0.45$ to a fraction in its simplest form. (2 marks)
Answer:
Question: Calculate $15\%$ of $£340$. (2 marks)
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Question: A train ticket costs $£65$. The price increases by $12\%$. Work out the new price of the ticket. (3 marks)
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Question: Order the following from smallest to largest: $\frac{3}{5}$, $62\%$, $0.58$, $\frac{13}{20}$. (3 marks)
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Question: Prove algebraically that the recurring decimal $0.8\dot{3}$ can be written as $\frac{5}{6}$. (3 marks) [Higher Tier]
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