Percentages, Ratios and Discounts: The Mental Maths of Everyday Life

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Discounts, tax, tips, interest, statistics — it is all the same handful of percentage tricks. Master a few conversions and you will never again be fooled by a "20% then 10% off" sale or a "50% more" label.

"Per Hundred" Is the Whole Idea

A percentage is just a fraction with a fixed denominator of 100. "35%" means "35 out of every 100," which is the number 35 ÷ 100 = 0.35. That single reframe unlocks everything:

• Percent → decimal: divide by 100 (move the decimal two places left). 7% = 0.07, 150% = 1.5.

• Decimal → percent: multiply by 100. 0.62 = 62%.

• Fraction → percent: divide then × 100. 3/8 = 0.375 = 37.5%.

To take P% of a number, multiply the number by P/100. Finding 15% of 240 is 240 × 0.15 = 36. The most useful mental shortcut is that 10% is the number divided by 10, and you build from there: 15% = 10% + half of 10%; 20% = double 10%. So 15% of 240 = 24 + 12 = 36, done in your head.

Discounts: Multiply, Don't Subtract

A sale of "30% off" means you pay 70%. The fast move is to multiply the price by (100 − 30)% = 0.70 rather than working out the discount and subtracting it. A €80 item at 30% off costs 80 × 0.70 = €56 in one step.

The trap is successive discounts. "20% off, then an extra 10% off" is not 30% off. The second cut only applies to the already-reduced price: 0.80 × 0.90 = 0.72, so you pay 72% — a total discount of 28%, not 30%. Store the rule: percentage changes multiply; they never simply add.

Percentage Change: It Matters What You Divide By

To find how much something changed, divide the change by the starting value, then × 100:

> change % = (new − old) / old × 100

A price from €50 to €65 rose by 15/50 = 30%. Reversed — €65 back to €50 — falls by 15/65 ≈ 23%, not 30%. The base is different, so the percentages differ. This is the single most abused idea in headlines. If a stock rises 50% then falls 50%, you have lost money: 1.5 × 0.5 = 0.75, so you end at 75% of the start, down 25%.

Ratios and Proportions: Scaling a Recipe or a Map

A ratio compares parts: a 2:3 mix of rice to water, or a map scale of 1:50,000. Ratios are fractions in disguise — 2:3 means 2 parts out of 5 total (40%) and 3 out of 5 (60%). To split a quantity in a ratio, add the parts and divide by that total. Split 250 in the ratio 2:3 → 250 ÷ 5 = 50 → 100 and 150.

A proportion says two ratios are equal, and it is the workhorse for scaling: if 3 apples cost €2, then 12 apples cost? 2/3 = x/12, so x = 12 × 2/3 = €8. Cross-multiply and solve — that one pattern handles recipes, map distances, unit conversions and speed problems alike.

Percent, Percentage Point, and the Confusion That Fooled Everyone

There is a difference between a percentage and a percentage point. If a tax rate goes from 20% to 25%, it rose by 5 percentage points but by 25% in relative terms (5 is a quarter of 20). News reports blur these constantly, and the gap between the two readings can be large. When a figure that is itself a percentage changes, always ask: points or percent?

Why This Pays Off

Every one of these is a two-line calculation, but strung together they cover shopping, tipping, loans, statistics, exam scores, battery percentages, sports stats and cooking. Learn the four moves — convert to decimal, multiply for discounts, divide-the-change-by-the-start, cross-multiply proportions — and mental arithmetic stops being a parlor trick and becomes an everyday defence against being mislead by a number.

You can check any of these in seconds with our percentage calculator, which handles "what is X% of Y," "X is what percent of Y," increases and decreases, and percentage change, and the loan calculator shows the same percentages working at scale when interest is spread over years of payments.

See also

• When Averages Lie: Mean, Median, and the Story They Hide
• What a Central Bank Actually Does — Interest Rates, Money and the Last Resort
• How Inflation Quietly Changes What Your Money Buys
• Compound Interest Explained: The Math Behind Both Your Savings and Your Debt