Maths

Percentages: Why Your Brain Lies to You (And How to Fix It)

Percentages should be simple. They're just "out of 100." And yet, they trip up smart people constantly. That "70% off!" sale sign? Probably not as good as you think. The "prices increased by 50%, then decreased by 50%" scenario? You're not back where you started. And don't get me started on how many people mess up tip calculations.

The Basics (That Aren't Actually Basic)

A percentage is a fraction of 100. That's it. 25% means 25 out of 100, or a quarter. 50% is half. 100% is the whole thing. To find a percentage of something, multiply by the decimal version:

  • 20% of 80 = 0.20 × 80 = 16
  • 15% of 200 = 0.15 × 200 = 30
  • 7.5% of 400 = 0.075 × 400 = 30

The Three Questions Everyone Asks

"What is X% of Y?"
Multiply: Y × (X ÷ 100). What's 15% of £80? £80 × 0.15 = £12.

"X is what percentage of Y?"
Divide and multiply by 100: (X ÷ Y) × 100. 45 is what percentage of 180? (45 ÷ 180) × 100 = 25%.

"X is Y% of what?"
Divide: X ÷ (Y ÷ 100). 30 is 20% of what? 30 ÷ 0.20 = 150. That's genuinely 90% of percentage problems right there.

Increases and Decreases

Increasing by a percentage: To increase £100 by 20%: £100 × 1.20 = £120. The "1.20" is 100% (the original) plus 20% (the increase). Works every time.

Decreasing by a percentage: Same logic, but subtract: £100 × 0.80 = £80 (that's a 20% decrease).

The Trap: Percentage Changes Don't Cancel Out

This catches everyone at some point. If something increases by 50% and then decreases by 50%, you're NOT back where you started.

Start with £100.
Increase by 50%: £100 × 1.50 = £150
Decrease by 50%: £150 × 0.50 = £75

You've lost £25. The percentages are the same, but they're applied to different base amounts. This is why "prices slashed by 30% after last month's 30% increase" isn't the bargain it sounds like.

Finding the Original Price

You see something on sale for £60 after a 25% discount. What was the original price? The sale price is 75% of the original (100% - 25% = 75%).
So: Original × 0.75 = £60
Original = £60 ÷ 0.75 = £80

Stacking Discounts: The Maths Nobody Does

"30% off, plus an extra 10% at checkout!" Most people think: 30% + 10% = 40% off. Nope. Here's what actually happens:
Start: £100
First discount (30%): £100 × 0.70 = £70
Second discount (10% of the new price): £70 × 0.90 = £63
Total discount: £37, which is 37% — not 40%.

Quick Mental Maths Tricks

Finding 10%: Move the decimal point one place left. 10% of £85 = £8.50. Finding 1%: Move it two places left. 1% of £85 = £0.85. Building other percentages from 10%:

  • 5% = half of 10%
  • 15% = 10% + 5%
  • 20% = 10% × 2
  • 25% = 10% × 2 + 5%

So for a 15% tip on a £47 meal:

  • 10% = £4.70
  • 5% = £2.35
  • 15% = £7.05

VAT: The One Everyone Needs

UK VAT is 20%. Here's how to handle it:

Adding VAT: Price × 1.20 = Price with VAT. £50 × 1.20 = £60. Removing VAT: Price with VAT ÷ 1.20 = Price without VAT. £60 ÷ 1.20 = £50. Finding just the VAT amount: Price with VAT ÷ 6 = VAT amount. £60 ÷ 6 = £10. (This works because VAT is 1/6 of the VAT-inclusive price. 20% of 100 is the same as 1/6 of 120.)

One common mistake: people try to remove VAT by calculating 20% and subtracting it. If something costs £60 including VAT, they calculate 20% of £60 (£12) and subtract it to get £48. Wrong. The correct answer is £50.

Percentage Points vs Percentages

This trips up even news reporters. If interest rates go from 2% to 3%, that's:

  • An increase of 1 percentage point
  • An increase of 50% (because 1 is 50% of 2)

Both are correct, but they mean different things. Politicians and advertisers love exploiting this confusion.

The Honest Truth

Percentages aren't hard. But they're easy to mess up when you're not paying attention, and marketers know this. "Up to 70% off!" usually means one item is 70% off and everything else is 10%. When in doubt, do the actual maths. Your brain's intuition about percentages is probably wrong.

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