Discounts and price rises

20% off? Pay 80%.

A discount is the percent question you will meet most often in life. A game costs 500 Kč, 20% off. How much do you pay?

The long path: calculate the discount (20% of 500 = 100 Kč) and subtract: 500 − 100 = 400 Kč.

The short path: when there is 20% off, you pay 80% of the price. Straight away: 0.8 × 500 = 400 Kč. One multiplication instead of two steps.

A price rise works as a mirror: a rise of 15% means the new price is 115% of the old one. Times 1.15 and 완료.

And one thing that surprises adults too: a discount and a rise by the same percent do not cancel. Raise 100 Kč by 50% → 150 Kč. Then take 50% off → 75 Kč. You are not back at zero, you are in minus. Why? Because each percent is calculated from a different base. Today you will see through it.

Adam and Sofie stand in front of a window with a sign “EVERYTHING −30%”. Adam counts a hoodie at 900 Kč the old way: 30% of 900 is 270, so 900 − 270 = 630. Sofie has already said: “630.” Adam stares: “How?” — “30% off means I pay 70%. Seven times nine is 63, so 630.” At the till hangs a second sign: “Club members another −10%.” Sofie counts: 10% of 630 is 63, she pays 567. “Watch,” she tells Adam, “it is not −40% of the original price. The second discount is calculated from the already reduced price.”

The new price in one multiplication

A discount of p%: new price = old price × (100 − p)/100.

20% off → times 0.80. 35% off → times 0.65. 5% off → times 0.95.

A rise of p%: new price = old price × (100 + p)/100.

A rise of 15% → times 1.15. A rise of 100% → times 2 (double).

The method for a discount:

Step 1: how many percent do I pay? 100 − discount. Step 2: convert to a decimal. Step 3: multiply the old price. Step 4: check with an estimate: after 20% off it should come out a bit less than the original price, after 50% off a half.

Two discounts one after another multiply, they do not add. 30% off and then 10% means times 0.7 and then times 0.9, together times 0.63 — so 37% off, not 40%. The second discount is calculated from the already reduced price, so it is “smaller” than it looks on the sign.

The reverse gear: you know the price after a discount and the percent? Then you look for the base as in lesson 7. The price after 20% off is those 80% — the ladder: divide by 80, times 100.

Shoes at 1 250 Kč, 24% off. How much do you pay?

Step 1: I pay 100 − 24 = 76% of the price.

Step 2: 76% = 0.76.

Step 3: 0.76 × 1 250 = 950.

Result: 950 Kč. Estimate for a check: the discount is almost a quarter, a quarter of 1 250 is about 312, so a price around 940 — that checks out. The long path: 24% of 1 250 = 300, then 1 250 − 300 = 950. Both paths the same, the short one has one step fewer.

A ticket rose by 12% to a new price. The old one was 350 Kč.

Step 1: the new price is 100 + 12 = 112% of the old one.

Step 2: 112% = 1.12.

Step 3: 1.12 × 350 = 392.

Result: 392 Kč. Check in parts: 10% of 350 is 35, 2% is 7, together a rise of 42 Kč and 350 + 42 = 392. That checks out. A rise means multiplying by a number bigger than 1 — if after a “rise” you get less, you have a discount in your writing.

A phone after 15% off costs 5 100 Kč. How much did it cost originally?

Here I go backwards — I look for the base.

Step 1: 5 100 Kč is the price after the discount, so 85% of the original price.

Step 2: the ladder. One percent: 5 100 : 85 = 60.

Step 3: a hundred percent: 60 × 100 = 6 000.

Result: original price 6 000 Kč. Check forwards: 0.85 × 6 000 = 5 100. That checks out. A common mistake: add 15% to 5 100 (= 5 865). It does not work, because 15% was calculated from 6 000, not from 5 100.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. 시작 with ten. When it goes well, add more.

Calculate the price after a discount — try the shortcut “how much I pay” straight away. Write only the resulting amount in Kč, without the unit.

Practise the questions

On paper: a shopping check

Calculate the short way (one multiplication) and for each question write an estimate first.

1. A game at 600 Kč with 35% off. How much do you pay? Write by how many percent you multiply.

2. A ticket rose by 8% from 250 Kč. New price? Check in parts via 10% and 2%.

3. A T-shirt has 20% off and at the till another 10%. It cost 450 Kč. Calculate the final price and show it is not the same as 30% off.

4. Headphones after 40% off cost 720 Kč. How much did they cost before the discount? Go by the ladder via 1%.

Now you 💪

1. I can do a discount in one multiplication: 20% off = times 0.8.
2. I can do a rise: by 15% = times 1.15.
3. I know that discounts one after another multiply, and that a discount + a rise by the same percent do not cancel.

What to take from this lesson

• A discount of p% = you pay (100 − p)%. One multiplication.
• A rise of p% = times (1 + p/100).
• Two discounts one after another multiply, they do not add.
• Price after a discount + the discount percent → original price by the ladder via 1%.
• A discount and a rise by the same percent do not return the original price.

© 2026 Ing. Martin Polak / AlgoRhino · 콘텐츠 이용 약관