Percentages, Decimals and Fractions — GCSE Maths Revision

Three Names for the Same Number

A fraction, a decimal and a percentage are three ways of writing the same value. $\frac{3}{4}$, $0.75$ and $75\%$ are not three related numbers — they are one number wearing three outfits. "Per cent" literally means "per hundred", so $75\%$ is just $\frac{75}{100}$ with the fraction bar hidden.

Why does GCSE care so much? Because each form is best at a different job. Fractions are exact and best for calculation by hand: $\frac{1}{3}$ is precise where $0.333$ is already rounded. Decimals are what your calculator speaks and what multipliers are built from. Percentages are how the real world reports change — interest rates, discounts, exam results. A large share of number questions on every paper amount to translating between the three and knowing which form makes the arithmetic easiest.

This guide covers the whole ladder: converting in all six directions, finding a percentage of an amount with and without a calculator, percentage change, reverse percentages, and the multiplier method that powers compound interest and depreciation. For structured practice on each rung, the percentages topic page and the fractions topic page break the skills down by tier and board.

Converting in All Six Directions

There are three forms, so there are six possible conversions, and each has a one-line rule.

Fraction → decimal: divide the top by the bottom. $\frac{5}{8} = 5 \div 8 = 0.625$. On a non-calculator paper, use short division or build from known facts.

Decimal → percentage: multiply by 100 (shift the digits two places). $0.625 = 62.5\%$.

Percentage → decimal: divide by 100. $35\% = 0.35$, and watch single digits: $4\% = 0.04$, not $0.4$.

Decimal → fraction: use place value, then simplify. $0.36 = \frac{36}{100} = \frac{9}{25}$.

Fraction → percentage: convert to a decimal first, then multiply by 100 — or scale the denominator to 100 when it divides neatly. $\frac{7}{40} = 0.175 = 17.5\%$.

Percentage → fraction: write it over 100 and simplify. $64\% = \frac{64}{100} = \frac{16}{25}$.

Learn the classics until they are instant: $\frac{1}{2} = 0.5 = 50\%$, $\frac{1}{4} = 0.25 = 25\%$, $\frac{1}{5} = 0.2 = 20\%$, $\frac{1}{8} = 0.125 = 12.5\%$, $\frac{1}{10} = 0.1 = 10\%$, and $\frac{1}{3} = 0.333...$ (a recurring decimal — Higher tier also asks you to convert recurring decimals back into fractions). Ordering questions rely on this fluency: to put $\frac{2}{5}$, $0.45$ and $38\%$ in size order, convert all three to decimals ($0.4$, $0.45$, $0.38$) and the order reads itself.

Percentage of an Amount Without a Calculator

Non-calculator papers love this skill, and the technique is to build any percentage out of easy blocks. The three blocks: $10\%$ (divide by 10), $1\%$ (divide by 100), and $50\%$ (divide by 2). Halving a block you already have gives $5\%$ from $10\%$, or $2.5\%$ from $5\%$.

So $35\%$ is $10\% + 10\% + 10\% + 5\%$, or faster, $3 \times 10\% + 5\%$. A classic exam favourite is $17.5\%$: build it as $10\% + 5\% + 2.5\%$, each block half the one before. For $17.5\%$ of £60: $10\%$ is £6, $5\%$ is £3, $2.5\%$ is £1.50, total £10.50.

Set your working out as a little ledger, one block per line, then add. It keeps the arithmetic honest and shows the examiner a method worth marks even if one line slips. This block-building is worth drilling until it is automatic — it also powers quick mental checks on calculator papers, where a sense of roughly what the answer should be catches mistyped digits.

Percentage of an Amount With a Calculator: Think in Multipliers

On a calculator paper, forget the blocks: convert the percentage to a decimal and multiply. $23\%$ of 640 is $0.23 \times 640 = 147.2$. One conversion, one multiplication, done.

That decimal is called a multiplier, and it is the single most powerful idea in this whole topic. An increase of $15\%$ means you end up with $100\% + 15\% = 115\%$ of what you started with, so the multiplier is $1.15$. A decrease of $8\%$ leaves $100\% - 8\% = 92\%$, so the multiplier is $0.92$. Increase £240 by $15\%$: $240 \times 1.15 = £276$. Reduce £85 by $8\%$: $85 \times 0.92 = £78.20$. One multiplication replaces the two-step "find the percentage, then add or subtract it" routine, with half the opportunities for error.

Multipliers are not just a shortcut — they are the required method later. Reverse percentages, compound interest and depreciation all depend on them, so make writing the multiplier the first line of every percentage-change answer now. Quick self-test: an increase of $4\%$ is $\times 1.04$; a decrease of $30\%$ is $\times 0.7$; a decrease of $4\%$ is $\times 0.96$, not $\times 0.6$.

Percentage Change: Comparing Two Values

Percentage change questions hand you a before and an after and ask how big the change was relative to where you started. The formula:

$\text{percentage change} = \dfrac{\text{change}}{\text{original}} \times 100$

The word doing all the work is original. You always divide by the starting value, never the final one, because the question is "what fraction of what I had did I gain or lose?". A price rising from £250 to £290 changes by £40, and $\frac{40}{250} \times 100 = 16\%$ — a $16\%$ increase. Dividing by 290 instead would give $13.8\%$, a wrong answer that looks plausible, which is exactly why examiners set it as a trap.

The same formula handles profit and loss ("a trader buys for £250 and sells for £290 — find the percentage profit": same numbers, same $16\%$) and error questions at Higher tier. It is also worth noticing what percentage change is not: a fall from £290 to £250 is a decrease of $\frac{40}{290} \times 100 \approx 13.8\%$, not $16\%$, because the original value has changed sides. Percentage change is direction-sensitive.

Reverse Percentages: Finding the Original Amount

Reverse percentage questions give you the value after a change and ask for the value before it. The tell-tale wording: "after a $20\%$ increase, the price is...", "in a sale, prices are reduced by $15\%$; the sale price is... — find the original price".

The method is the multiplier method run backwards. Going forward multiplies by the multiplier, so coming back divides by it. If a price rose $20\%$ to reach £72, then $72 = \text{original} \times 1.2$, so the original is $72 \div 1.2 = £60$.

The trap is tempting and almost universal: taking the percentage of the final value and subtracting. $20\%$ of £72 is £14.40, and $72 - 14.40 = £57.60$, which is wrong — the $20\%$ in the question was $20\%$ of the original £60 (which is £12), not of £72. Whenever you finish a reverse percentage, check forwards: £60 increased by $20\%$ is $60 \times 1.2 = 72$ ✓. If your check does not land back on the number in the question, you have divided when you should not have, or subtracted when you should have divided.

Compound Change: Interest, Depreciation and Repeated Percentages

Compound change is what happens when a percentage change is applied repeatedly, each time to the new total rather than the original. Savings interest compounds: year two's interest is earned on year one's interest as well as the principal. The multiplier method makes it one calculation:

$\text{final amount} = \text{start} \times (\text{multiplier})^n$

for $n$ repeats. £500 at $3\%$ compound interest for 4 years is $500 \times 1.03^4 = £562.75$ (to the nearest penny) — not the $500 + 4 \times 15 = £560$ that simple interest would give, because each year's $3\%$ acts on a slightly bigger balance.

Depreciation is the same machine with a multiplier below 1. A car bought for £12,000 losing $18\%$ of its value each year is worth $12000 \times 0.82^3 = £6{,}616.42$ after 3 years. And successive different changes just multiply their multipliers: a $20\%$ rise followed by a $20\%$ fall is $\times 1.2 \times 0.8 = \times 0.96$ — an overall $4\%$ decrease, not a return to the start, because the $20\%$ fall acted on a larger amount than the $20\%$ rise did. That surprise is a favourite Higher-tier question, and the multiplier method is the only clean way to see it.

How These Questions Are Marked, and the Slips That Cost Most

Percentage questions are generous with method marks if your working is visible. Stating the multiplier, showing the division in a reverse percentage, writing $\frac{\text{change}}{\text{original}} \times 100$ with your numbers in it — each of these is typically a marked step. A bare wrong answer scores nothing; the same wrong answer under two lines of correct method usually keeps most of the marks.

The slips to guard against are a short list. Dividing by the new value instead of the original in percentage change. Treating a reverse percentage as a forward one. Using a wrong multiplier for a decrease ($\times 0.6$ for a $4\%$ fall instead of $\times 0.96$). Adding percentages of different bases as if a $20\%$ rise then $20\%$ fall cancel out. And on money answers, forgetting to round to the nearest penny when the question asks.

Finally, always sense-check the size of your answer. A $23\%$ share of something must be a bit less than a quarter of it; an original price must be bigger than its sale price; compound interest must beat simple interest over the same years. Ten seconds of "is this plausible?" catches more dropped marks than any amount of rechecking button presses. For calculator-paper habits that make this automatic, see calculator paper tips — and Exam Ladder's practice engine can serve percentage questions across every GCSE Maths topic until the method is reflex, with mock exams graded on real exam-board grade boundaries.