The two formulas behind every sale tag, pay rise and price rise — plus the traps to avoid
10 min read · Oct 3, 2026 · By PagalHo Tools Team
Almost every number that affects your money arrives as a percentage. Shop sales shout “30% off,” employers offer “a 5% raise,” newsreaders warn that “inflation hit 6%,” and your savings account quietly pays “4% a year.” Percentages are the common language of gains and losses — and most people speak it approximately at best.
That approximateness costs real money. Misreading a stacked discount, underestimating how a small annual raise compounds, or confusing a percentage-point move with a percent move can each quietly shift hundreds of pounds a year. The good news is that the entire subject rests on just two formulas, and once you own them, every one of these situations becomes transparent.
This guide builds those two formulas from scratch, applies them to the four places percentages matter most — discounts, salaries, inflation and investments — and then walks through the classic traps: stacked discounts, compounding raises, and the percentage-point confusion that trips up even professional commentators.
By the end you will be able to read any “percent off” tag, any pay offer and any inflation headline and know exactly what it means for your money — and you will know when to reach for a calculator instead of trusting your gut.
Percentage change answers one question: how big is the move relative to where it started? The formula is: percentage change = (new value − old value) ÷ old value × 100. If the result is positive, you have an increase; if negative, a decrease. That single formula covers price rises, pay cuts, weight loss and investment returns alike.
The second formula runs the calculation in reverse: given an old value and a percentage change, find the new value. New value = old value × (1 + rate), where the rate is the percentage as a decimal — so a 20% increase means multiplying by 1.20, and a 15% decrease means multiplying by 0.85. Memorise this one and you can price any discount in seconds.
A worked example ties them together. A jacket priced at £120 is reduced to £90. The change is (90 − 120) ÷ 120 × 100 = −25%, a 25% decrease. Going the other way: £120 × (1 − 0.25) = £120 × 0.75 = £90. Both directions, same arithmetic, no mystery.
One subtlety worth internalising early: percentage change is always relative to the starting value. A £30 rise on a £120 jacket is 25%, but the same £30 rise on a £60 shirt is 50%. Identical pounds, wildly different percentages — which is exactly why retailers and politicians prefer whichever framing flatters them.
The multiplier shortcut
Convert any percentage change into a single multiplier: +20% → ×1.20, −15% → ×0.85, +100% → ×2. Chain multiple changes by multiplying the multipliers. It is the fastest way to handle stacked discounts and compound growth without a calculator.
Discounts are the friendliest face of percentage decrease, and the formula above handles them directly: sale price = original price × (1 − discount rate). A £80 coat at 25% off costs £80 × 0.75 = £60. A £45 book at 10% off costs £45 × 0.90 = £40.50. Once the multiplier habit clicks, you can do most sale tags in your head while standing in the aisle.
The classic trap is the stacked discount. A shop offers “20% off, then an extra 10% off at the till.” Many shoppers add the percentages and expect 30% off — but the second discount applies to the already-reduced price. The true multiplier is 0.80 × 0.90 = 0.72, i.e. 28% off, not 30%. On a £200 item that two-point gap is £4 the shop keeps.
Retailers know this, which is why “up to 70% off” signs deserve scepticism: the “up to” almost always applies to one sad item at the back. The honest way to compare two competing sales is to convert each to its final multiplier and compare those — 0.70 versus 0.75 tells you instantly which deal is better, with no percentage fog.
One more discount shape: “buy one, get one half price.” On two identical items this is a 25% discount overall (you pay 1.5× the single price for 2× the goods), not 50%. It is still often a good deal — just not the deal the sign implies at first glance.
Pay rises are percentage increases wearing a suit. A £30,000 salary with a 5% raise becomes £30,000 × 1.05 = £31,500. Simple enough — but the interesting behaviour appears over multiple years, because each raise applies to the already-raised salary. This is compounding, and it quietly works in your favour.
Two consecutive 5% raises do not total 10%: they total 1.05 × 1.05 = 1.1025, i.e. 10.25%. The extra quarter point is small here, but stretch it over a career — or over an investment — and compounding dominates. A 7% annual return doubles money in about ten years, not fourteen, because each year’s growth earns its own growth the next year.
The same maths cuts the other way when comparing offers. A one-off £2,000 bonus versus a 4% permanent raise on £40,000 (£1,600 a year, compounding forever): the raise wins within two years and keeps winning. When evaluating job offers, always convert one-off payments and permanent percentage rises into the same multi-year frame before comparing.
And watch the oldest trick in the negotiation book: a “10% increase” that follows an earlier undisclosed cut. If your £50,000 salary was cut 10% to £45,000 and then “restored” with a 10% rise, you land at £49,500 — still £500 short. Percentages are not symmetric: a fall and an equal rise never quite cancel out.
The rule of 72
To estimate how long compounding takes to double your money, divide 72 by the annual percentage rate: at 6% it takes about 12 years, at 9% about 8. Bankers have used this shortcut for centuries — it falls straight out of the compound-growth formula.
Inflation is a percentage increase applied to the general price level — when inflation runs at 6%, something that cost £100 last year costs about £106 this year. The number that matters to you is the real change: if your pay rose 4% while prices rose 6%, your purchasing power fell by roughly 2%. Nominal versus real is the single most useful distinction in personal finance.
Inflation also has a sneaky sibling: shrinkflation, where the price stays the same but the quantity decreases. A chocolate bar shrinking from 100g to 90g at the same price is a 10% decrease in what you get — equivalent to an 11.1% price increase per gram (100 ÷ 90 = 1.111). Manufacturers prefer it because a smaller bar attracts less outrage than a higher price tag.
Both phenomena reward the same defensive habit: track unit prices, not sticker prices. Price per 100g, price per litre, price per wash — these strip away packaging tricks and expose the true percentage move. Most supermarkets now print unit prices on shelf labels precisely so shoppers can make this comparison.
Over long stretches, even modest inflation compounds brutally. At 3% a year, prices double in about 24 years — which is why anyone planning decades ahead, for retirement or a mortgage, must think in real (inflation-adjusted) terms rather than nominal ones.
Here is the confusion that derails newsreaders, politicians and dinner-party debates alike. If an interest rate rises from 5% to 7%, how big was the increase? Many people say “2%” — but that is wrong. The rate rose by two percentage points, which is a 40% relative increase (2 ÷ 5 × 100). The distinction matters enormously: a “2% rise” sounds trivial, while a “40% rise” sounds alarming, and both describe the same event.
The rule is simple: when you are talking about changes to something already expressed as a percentage — interest rates, inflation rates, election vote shares, unemployment rates — describe the absolute move in percentage points and the relative move in percent, and never mix them up. “Inflation fell from 8% to 6%: down two percentage points” is precise; “inflation fell 2%” is wrong (it actually fell 25% in relative terms).
This trap is a favourite of misleading headlines. “Unemployment up 50%!” sounds catastrophic — until you learn it moved from 4% to 6%, i.e. two percentage points. Always ask for the underlying numbers before reacting to a percentage-of-a-percentage claim.
In your own communication, default to percentage points for rate changes and keep percent for everything else. Your listeners will understand you better, and you will never accidentally manufacture a scare story.
You do not need a calculator for everyday percentages — the 10% method handles nearly everything. Find 10% by moving the decimal point one place left (£240 → £24), then build any percentage from tens and fives: 20% is two tens (£48), 15% is ten plus half of ten (£36), 5% is half of ten (£12). Tips, discounts and taxes all surrender to this method.
For increases, add instead of multiplying: a 15% tip on £60 is £6 + £3 = £9. For decreases, subtract: 30% off £90 is £27 off (three tens), so £63. With a little practice these become faster than reaching for your phone.
When precision matters — stacked discounts, compound raises, anything involving a contract — drop the shortcuts and use the exact multipliers or a proper calculator. Mental math is for estimates and sanity checks; the formula is for money.
The examples above are worth working through once by hand, because the arithmetic teaches the intuition. After that, let software handle the fiddly cases: PagalHo’s free percentage calculator computes percentage of a number, percentage increase and decrease, and “what percent is X of Y” instantly — the exact operations this guide covers.
Use it to check a sale tag before you buy, to compare two competing discounts by their true multipliers, or to translate a pay offer into multi-year terms. Each result shows the working, so you can see which formula applied and why.
Percentages stop being intimidating the moment you see them as multipliers. ×1.20 for growth, ×0.85 for a 15% cut, and percentage points for rate changes — with those three ideas, no headline, price tag or pay offer can mislead you again.
Subtract the old value from the new value, divide by the old value, and multiply by 100. Example: a salary rising from £40,000 to £43,000 is (43,000 − 40,000) ÷ 40,000 × 100 = 7.5% increase.
The same formula — the answer just comes out negative. A £120 jacket reduced to £90 is (90 − 120) ÷ 120 × 100 = −25%, i.e. a 25% decrease. Equivalently, the sale multiplier is 90 ÷ 120 = 0.75.
Because the second discount applies to the already-reduced price. The combined multiplier is 0.80 × 0.90 = 0.72, which is 28% off. Stacked percentage changes always multiply; they only add when applied to the same original base.
Percent describes a relative change; percentage points describe the absolute difference between two percentages. A rise from 5% to 7% is +2 percentage points, which equals a 40% relative increase. Mixing them up is one of the most common statistical errors in public debate.
Divide the sale price by the remaining multiplier: original = sale ÷ (1 − 0.25) = sale ÷ 0.75. So a £60 sale price after 25% off means the original was £60 ÷ 0.75 = £80.
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