Algebra Revision Notes
Subject: Mathematics | Level: GCSE | Exam Board: OCR
Master the essential connections between fractions, decimals, and percentages to unlock high marks across the GCSE Maths paper. This guide covers everything from basic conversions to complex percentage multipliers and recurring decimals.
Revision Notes & Key Concepts
Revision Podcast Transcript
Hello and welcome to your GCSE Maths revision podcast. I'm your tutor today, and we're diving into one of the most frequently tested topics on your exam paper: Fractions, Decimals, and Percentages. Whether you're sitting Foundation or Higher tier, this topic comes up again and again — so let's make sure you absolutely nail it. Grab a pen and paper, because this is one of those topics where writing things down as we go will really help. Let's get started. --- SECTION ONE: CORE CONCEPTS So, first things first — what are fractions, decimals, and percentages actually doing? They're all just different ways of expressing the same thing: a part of a whole. Think of a pizza. If you eat three slices out of four, you've eaten three-quarters of the pizza. That's the fraction three over four. As a decimal, that's zero point seven five. As a percentage, that's seventy-five percent. Same amount of pizza — three different ways to write it. That's the key insight that ties this whole topic together. Let's start with conversions, because that's where most marks are won and lost. Converting a fraction to a decimal is straightforward: you 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. Simple. The trick is to remember: top divided by bottom. Always. Going from a decimal to a percentage? Even easier. Just multiply by one hundred. So zero point seven five becomes seventy-five percent. You're literally just moving the decimal point two places to the right. Now, going from a percentage to a decimal — you do the reverse. Divide by one hundred. So forty-five percent becomes zero point four five. This is where candidates lose marks: they write forty-five percent as zero point four five — which is correct — but some students write it as zero point four five zero, which is fine, or worse, they write four point five, which is wrong. The decimal point moves two places to the LEFT when dividing by one hundred. Say it with me: divide by one hundred, move the point two places left. Converting a percentage to a fraction? Write the percentage over one hundred, then simplify. So forty percent becomes forty over one hundred, which simplifies to two-fifths. You find the highest common factor of forty and one hundred — which is twenty — and divide both by it. Forty divided by twenty is two. One hundred divided by twenty is fifty. So two over fifty... wait, let me redo that. Forty divided by twenty is two, one hundred divided by twenty is five. So it's two-fifths. There we go. Now let's talk about ordering mixed types — fractions, decimals, and percentages all jumbled together. The golden rule: convert everything to the same form before comparing. Decimals are usually easiest to compare. So if you've got one-half, zero point four, and thirty-eight percent, convert them all to decimals: one-half is zero point five, zero point four stays as zero point four, and thirty-eight percent is zero point three eight. Now you can order them: zero point three eight, zero point four, zero point five. Smallest to largest: thirty-eight percent, zero point four, one-half. Easy when you've got a common form to compare. Let's move on to calculating fractions of quantities. This is a bread-and-butter skill. To find a fraction of a quantity, you divide by the denominator and multiply by the numerator. So to find three-quarters of eighty: divide eighty by four to get twenty, then multiply by three to get sixty. Three-quarters of eighty is sixty. You can remember this as: divide first, multiply second. Or think of it as the denominator tells you how many equal parts to split into, and the numerator tells you how many of those parts you want. Now for percentage calculations — and this is where the really efficient method comes in: percentage multipliers. A percentage multiplier is a single number you multiply the original value by to apply a percentage change in one step. For a percentage increase, the multiplier is one plus the decimal version of the percentage. For a ten percent increase, the multiplier is one point one. For a twenty-five percent increase, it's one point two five. For a percentage decrease, the multiplier is one minus the decimal. A fifteen percent decrease gives a multiplier of zero point eight five. Why is this better than the traditional method? Because it's faster and less prone to error. If a jacket costs eighty pounds and is reduced by fifteen percent, you could calculate fifteen percent of eighty — which is twelve — and subtract to get sixty-eight. Or you could just do eighty multiplied by zero point eight five and get sixty-eight in one step. Same answer, fewer steps, less chance of a mistake. Examiners love to see multipliers used correctly, and for multi-step percentage problems — like compound interest — multipliers are essentially the only practical method. Speaking of compound interest: the formula is A equals P multiplied by the quantity one plus r over one hundred, all raised to the power n. Where A is the final amount, P is the principal or starting amount, r is the percentage rate, and n is the number of time periods. So if you invest five hundred pounds at three percent per year for four years, you calculate five hundred multiplied by one point zero three to the power four. That gives you approximately five hundred and sixty-two pounds and seventy-five pence. For Higher tier candidates, there's one more concept to master: converting recurring decimals to fractions. A recurring decimal is one where a digit or group of digits repeats forever. For example, zero point three recurring — written with a dot above the three — equals one-third. The algebraic method works like this: let x equal the recurring decimal. Multiply both sides by a power of ten to shift the recurring part. Then subtract the original equation to eliminate the recurring part, and solve for x. For example: let x equal zero point three recurring. Then ten x equals three point three recurring. Subtract: nine x equals three. So x equals three-ninths, which simplifies to one-third. This method earns full marks every time if you show all steps clearly. --- SECTION TWO: EXAM TIPS AND COMMON MISTAKES Right, let's talk about how to pick up every available mark in the exam. Common mistake number one: converting percentages to decimals incorrectly. Five percent is NOT zero point five. It is zero point zero five. This trips up so many candidates. Remember: five percent means five out of one hundred. Five divided by one hundred is zero point zero five. If you write zero point five, you've actually written fifty percent. That's a ten-times error — and it will cost you marks. Common mistake number two: using the wrong multiplier for percentage change. If a question says "increase by ten percent", the multiplier is one point one — not zero point one and not zero point nine. One point one. Because you're keeping the original one hundred percent and adding ten percent on top. Similarly, a ten percent decrease gives zero point nine — you're keeping ninety percent of the original. Common mistake number three: not simplifying fractions. If your answer is six-eighths, you must simplify to three-quarters. Examiners expect fractions in their simplest form unless told otherwise. Always check: can I divide the numerator and denominator by the same number? Common mistake number four: place value errors with decimals. When multiplying decimals, count the total number of decimal places in the question and make sure your answer has the same number. So zero point three multiplied by zero point four: three times four is twelve, and there are two decimal places in total, so the answer is zero point one two — not one point two, not zero point zero one two. Common mistake number five: confusing percentage change with percentage of. If a price goes from forty pounds to fifty pounds, the percentage change is ten divided by forty, multiplied by one hundred — which is twenty-five percent. Not ten divided by fifty. You always divide by the ORIGINAL value for percentage change. Exam tip: always show your working. For a four-mark calculation question, the marks are usually awarded for the method, the correct substitution, the correct calculation, and the final answer with appropriate rounding. Even if you get the final answer wrong, you can still earn two or three marks for correct working. Never just write an answer with no working for a multi-step question. Exam tip: read the question carefully for whether it wants an exact answer or a rounded one. If it says "give your answer as a fraction", do not give a decimal. If it says "give your answer correct to two decimal places", make sure you round properly — and write the answer to exactly two decimal places, even if the second decimal is a zero. Exam tip: use estimation to check your answer. If you calculate that forty percent of two hundred is eight hundred, that should immediately feel wrong — forty percent is less than half, so it must be less than one hundred. Estimation catches errors before you write them down. --- SECTION THREE: QUICK-FIRE RECALL QUIZ Right, let's test yourself. I'll ask a question, pause for a moment, then give the answer. Ready? Question one: What is three-fifths as a decimal? ... Three divided by five equals zero point six. Question two: What is zero point three five as a percentage? ... Multiply by one hundred: thirty-five percent. Question three: What multiplier would you use for a twenty percent increase? ... One point two. Question four: What multiplier would you use for a thirty percent decrease? ... Zero point seven. Question five: What is two-thirds as a decimal? Give it to three decimal places. ... Zero point six six seven. Note that two-thirds is actually a recurring decimal: zero point six recurring. Question six: Arrange in order from smallest to largest: one-half, forty-five percent, zero point four eight. ... Convert all to decimals: zero point five, zero point four five, zero point four eight. Order: forty-five percent, zero point four eight, one-half. Question seven: Higher tier — what fraction is equal to zero point one recurring? ... Let x equal zero point one recurring. Ten x equals one point one recurring. Subtract: nine x equals one. x equals one-ninth. How did you do? If you got all seven, you're in great shape. If you stumbled on any, go back and revisit that concept in your notes. --- SECTION FOUR: SUMMARY AND SIGN-OFF Let's wrap up with the key things to take away from today's session. Number one: fractions, decimals, and percentages are all equivalent forms. Converting between them is a core skill — practice until it's automatic. Number two: to convert a fraction to a decimal, divide the top by the bottom. To convert a decimal to a percentage, multiply by one hundred. To go the other way, reverse the operation. Number three: percentage multipliers are your best friend. For an increase of n percent, multiply by one plus n over one hundred. For a decrease, multiply by one minus n over one hundred. Number four: always simplify fractions to their lowest terms in your final answer. Number five: show all your working in multi-step questions. Method marks are there to be earned even if you make a calculation slip. Number six: for Higher tier, practise the algebraic method for converting recurring decimals to fractions — it's a reliable way to pick up marks that many candidates miss. That's it for today's session on Fractions, Decimals, and Percentages. You've covered conversions, ordering, fraction calculations, percentage multipliers, compound interest, and — for Higher tier — recurring decimals. Keep practising, use past papers, and remember: every mark counts. Good luck with your revision, and I'll see you in the next episode!
Key Terms & Definitions
- Numerator
- The top number in a fraction, representing how many parts of the whole are being considered.
- Denominator
- The bottom number in a fraction, representing the total number of equal parts the whole is divided into.
- Percentage Multiplier
- A decimal value used to calculate a percentage increase or decrease in a single multiplication step.
- Compound Interest
- Interest calculated on the initial principal and also on the accumulated interest of previous periods.
- Recurring Decimal
- A decimal number in which a digit or sequence of digits repeats infinitely.
- Equivalent Fractions
- Fractions that represent the same value or proportion, even though they look different (e.g., 1/2 and 2/4).
Worked Examples
Worked Example
Question: Calculate $$\frac{3}{8}$$ of £320. (2 marks)
Solution: Step 1: Divide the quantity by the denominator to find one part. $$320 \div 8 = 40$$ Step 2: Multiply by the numerator to find three parts. $$40 \times 3 = 120$$ Final answer: £120
Worked Example
Question: A car's value depreciates by 12% each year. The car was bought for £15,000. Calculate its value after 3 years. Give your answer to the nearest penny. (3 marks)
Solution: Step 1: Identify the percentage multiplier for a 12% decrease. $$100\% - 12\% = 88\% = 0.88$$ Step 2: Apply the compound depreciation formula. $$\text{Final Value} = 15000 \times 0.88^3$$ Step 3: Calculate the result. $$15000 \times 0.681472 = 10222.08$$ Final answer: £10,222.08
Worked Example
Question: Prove algebraically that the recurring decimal $$0.\dot{4}\dot{5}$$ can be written as the fraction $$\frac{5}{11}$$. (3 marks) [Higher Tier]
Solution: Step 1: Let $$x$$ equal the recurring decimal. $$x = 0.454545...$$ Step 2: Multiply by 100 because two digits recur. $$100x = 45.454545...$$ Step 3: Subtract the original equation ($$x$$) from the new equation ($$100x$$). $$100x - x = 45.454545... - 0.454545...$$ $$99x = 45$$ Step 4: Solve for $$x$$ and simplify the fraction. $$x = \frac{45}{99}$$ Divide numerator and denominator by 9: $$x = \frac{5}{11}$$ Final answer: As required.
Practice Questions
Question: Write $$0.65$$ as a fraction in its simplest form. (2 marks)
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Question: A shop has a sale with 15% off all prices. The normal price of a television is £420. Calculate the sale price. (3 marks)
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Question: Arrange the following in order of size, starting with the smallest: $$\frac{3}{5}$$, $$62\%$$, $$0.58$$, $$\frac{7}{10}$$. (2 marks)
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Question: Sarah invests £4000 in a savings account paying 2.5% compound interest per annum. Calculate the total amount in the account after 5 years. (3 marks)
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Question: Prove algebraically that $$0.2\dot{7}$$ can be written as $$\frac{5}{18}$$. (3 marks) [Higher Tier]
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