Sales, tips, and taxes all do the same tiny trick: they nudge a number up or down by a share of itself. Learn the one move and you can read any sticker.
Start hereTo go up by a percentage, you add that share on top. To go down, you take it off. Either way, the shortcut is to multiply by one friendly number.
A percentage means "per hundred" — so 20% is 20 out of every 100, or the fraction 0.20. A multiplier is the single number you multiply by to make the change happen in one step. Add 20% and you multiply by 1.20; take off 20% and you multiply by 0.80. That's the entire secret, and the rest of this page is just you getting comfortable with it.
Picture a chocolate bar snapped into exactly 100 tiny squares. When someone says 20%, they mean 20 of those 100 squares — no more, no less. That's why every percentage can be written as a decimal by sliding the point two places: 20% becomes 0.20, 5% becomes 0.05, and a whole 100% becomes 1.00 (the entire bar).
So when a price changes by a percentage, you first work out how big that share is, then decide whether to glue it on top or peel it off. A $50 jacket at 20% has a share of $50 × 0.20 = $10. Add it: $60. Remove it: $40. Same $10, two directions.
Percentage = per hundred. The bigger the starting price, the bigger the share — 20% of $50 is $10, but 20% of $500 is $100. The percentage stays the same; the actual money grows with the price.
You could always find the share and then add or subtract it. That's two steps. The faster way — the one shops, calculators, and mathematicians actually use — is to fold it into one multiplication.
Start from 1, which stands for the whole original price (100%). Then:
So a 20% rise on $50 is simply $50 × 1.20 = $60, and a 20% cut is $50 × 0.80 = $40. One tap, no leftover pieces. Notice the neat mirror: a decrease multiplier plus its matching increase multiplier always add up to 2 (0.80 + 1.20 = 2). If you can build the multiplier, you can do this in your head.
| Change | Do this to 1 | Multiplier |
|---|---|---|
| Up 5% | 1 + 0.05 | ×1.05 |
| Up 15% (a tip) | 1 + 0.15 | ×1.15 |
| Up 100% (doubled) | 1 + 1.00 | ×2.00 |
| Down 10% | 1 − 0.10 | ×0.90 |
| Down 25% (quarter off) | 1 − 0.25 | ×0.75 |
| Down 50% (half price) | 1 − 0.50 | ×0.50 |
Type a starting price, then drag from a deep discount on the left to a markup on the right. The after tag and the working update the instant you move.
Left of centre removes a share (a sale); right of centre glues one on top (a markup or tip). Dead centre is 0% — the price never moves.
A sale takes a share away, so subtract from 1: 1 − 0.20 = 0.80. Then $50 × 0.80 = $40.00. You saved $10, and the "sale price" and the "amount saved" always add back up to the original ($40 + $10 = $50).
A tip adds a share on top, so add to 1: 1 + 0.15 = 1.15. Then $28 × 1.15 = $32.20. If you just want the tip by itself, that's $28 × 0.15 = $4.20 — and $28 + $4.20 lands on the same $32.20.
Sales tax is an increase, so 1 + 0.07 = 1.07, giving $40 × 1.07 = $42.80. The extra $2.80 is the tax. Prices on the shelf are often before tax, which is why the total at the till is a little higher than the sticker.
Here's where the multiplier method really shines. A $120 pair of shoes is 25% off, and then 7% tax is added to the reduced price. Do it in two multiplies:
Or chain both multipliers at once: $120 × 0.75 × 1.07 = $96.30. Order doesn't change the answer here, because multiplying is multiplying — but always take the discount off the original price first unless a sign says otherwise, since that's how real shops ring it up.
Take 20% off and then add 20% back and you do not return to the start. $100 × 0.80 = $80, then $80 × 1.20 = $96 — you land $4 short. The second 20% is a share of the smaller $80, not the original $100. Percentages always measure against whatever number they sit next to.
Set a price, choose a discount, choose a tax, and watch the two multipliers work in a row — the exact move behind a real checkout total.
The discount multiplier is under 1 (it shrinks), the tax multiplier is over 1 (it grows). Chained together they give the real total.
Sometimes you see both prices and want to know the change. The recipe is: find the difference, then compare it to the starting price.
A backpack drops from $80 to $60. The difference is $20. Compare that to the original: 20 ÷ 80 = 0.25 = 25% off. Always divide by the number you started from — the "before" price — because the percentage is measured against where you began.
Going up works the same way. If a $40 game rises to $46, the difference is $6, and 6 ÷ 40 = 0.15 = 15% increase. Difference over original, times 100 for a percent. That's the whole reverse trick.
Pick an answer for each — you'll get instant feedback and a running score. No calculator needed if you build the multiplier.
A percentage is a share out of 100 — write it as a decimal (20% → 0.20).
Up: 1 + the decimal (×1.20). Down: 1 − the decimal (×0.80).
New price = old price × multiplier. Chain them for sale-then-tax.