Discounts & Percentages

The same backpack: store A takes 50 off the price, store B takes 20% off. Which is cheaper? The tag alone cannot say — at a price of 180 the flat cut wins, at 400 the percentage wins. To compare deals you need one shared language: percentages.

Percent: out of every hundred

30% means exactly what it says: 30 out of every 100. It is just a fraction and a decimal wearing different clothes:

$30\% = \frac{30}{100} = 0.3$

A few are worth knowing by heart: $50\% = \frac{1}{2}$ , $25\% = \frac{1}{4}$ , $10\% = \frac{1}{10}$ , $75\% = \frac{3}{4}$ . There is more to fractions in Fractions. Percentages are everywhere: chance of rain, battery left, win rates in Probability for Beginners — all their home turf.

30% off means paying 70%

When a shop cuts 30% off, it removes 30% of the original price and you pay the rest:

$100\% - 30\% = 70\%$

So "30% off" and "paying 70%" are the same sentence said two ways.

Try it: what does the backpack cost?

The backpack costs 260, and today it is 30% off. The discount removes $260 \times 0.3 = 78$ , so you pay $260 - 78 = 182$ . One step works too: $260 \times 0.7 = 182$ . Two roads, same answer.

InteractiveDiscount Calculator

Original priceDiscount 30%30

200

-30%

140

You save

10% is the king of mental math

To take 10% of anything, slide the decimal point one place left: 10% of 180 is 18. From there everything follows: 20% is two of those, 5% is half of a 10%, 15% = 10% + 5%... most discounts can be done in your head.

20% off, then 10% off — how much is that?

Some stores stack two deals: members take 20% off, then another 10% off. Sounds close to 30% off, but it is not — discounts multiply, they do not add:

$0.8 \times 0.9 = 0.72$

You pay 72% of the original price, which is 28% off in total — 2% more expensive than a straight 30% off. Check it on 100: after 20% off it is 80, after another 10% off it is 72, while a flat 30% off would leave 70.

Discounts multiply, they do not add

The total saving from "20% off then 10% off" is $1 - 0.8 \times 0.9 = 28\%$ , not $30\%$ . When you see stacked discounts, multiply the pay rates first.

Which deal wins depends on the price

Now the opening question can be settled. A flat 50 off saves the same at every price; 20% off saves more as the price climbs:

Price 180: after the flat cut 130, after 20% off 144 — the flat cut wins;
Price 400: after the flat cut 350, after 20% off 320 — the percentage wins.

The tipping point is a price of 250, where both deals save exactly 50. So next time a sale tempts you, ask one question first: what percent of the original price am I paying?

Check yourself

Quick quiz

1. Shoes cost 300 and are 30% off. What do you pay?
2. 20% off, then another 10% off — what total discount is that?
3. 50% equals which fraction and decimal?