How do you calculate percentages? Using maths to see through discount and price-rise traps

A shop window says in huge letters: “50% off everything, plus an extra 20% off at the till!” Is that 70% off in total? Or a product’s price first goes up by 20%, and a few months later it is discounted by 20%. Does the price go back to where it was? The answer to both questions is “no”, and the reason lies in how percentages work. In this article we go over percentages from the ground up and use maths to get out of the traps people fall into most often in everyday life.

What is a percentage?

A percentage expresses a number as parts out of 100. “25%” is read “twenty-five percent” and means $\frac{25}{100}$. In other words, a percentage is a fraction with a denominator of 100:

Turning percentages into decimals makes calculations much easier: 15% = 0.15; 8% = 0.08; 120% = 1.2.

Three basic percentage problems

Almost every percentage question fits one of three patterns.

1. Finding a percentage of a number: what is 15% of 360 TL (Turkish lira)?

2. Finding what percentage one number is of another: a student gets 24 right in a 30-question practice test. What percentage did they score?

3. Finding the whole from a percentage: if 40% of a class is 12 people, how many people are in the class?

Recognising these three patterns is enough to solve most of the problems you will meet.

Percentage increase and decrease: the multiplier method

The most powerful tool for discount and price-rise calculations is the idea of a multiplier:

• A 20% price rise means multiplying the price by $1 + 0.20 = 1.20$.
• A 20% discount means multiplying the price by $1 - 0.20 = 0.80$.
• A 35% discount means multiplying the price by $0.65$.

Example: an 850 TL coat has 30% off. The new price:

The general formula for percentage change is:

The denominator is always the old (starting) value. This is where most mistakes are made.

Common misconceptions: discount and price-rise traps

Most mistakes in percentage calculations come from forgetting that a percentage is always a percentage of something. The five traps below are the ones you meet most often, both in exams and when shopping.

Trap 1: a rise and a discount don’t cancel out

Take a 100 TL product: first its price rises by 20%, then it is discounted by 20%.

• After the rise: $100 \times 1.20 = 120$ TL
• After the discount: $120 \times 0.80 = 96$ TL

The product hasn’t gone back to its old price; it has become 4 TL cheaper! Why? Because the rise was calculated on 100 TL, but the discount on 120 TL. With multipliers it is very clear:

Changing the order makes no difference: if the discount comes first and the rise second, you still get $0.96$. So a 20% rise and a 20% discount together mean a 4% decrease.

Trap 2: successive discounts don’t add up

Back to the “50% + an extra 20% at the till” offer in the window. A pair of shoes costing 400 TL:

• First discount: $400 \times 0.50 = 200$ TL
• Extra discount: $200 \times 0.80 = 160$ TL

The 160 TL you pay is 40% of the original price. So the total discount is not 70% but 60%. With multipliers:

Because the extra discount is applied to a price that has already been reduced, the two percentages don’t add up.

Trap 3: “buy 3, pay for 2” and “second one half price”

• Buy 3, pay for 2: you pay for 2 items to get 3. The fraction you pay is $\frac{2}{3} \approx 0.667$, so the discount is about 33%.
• Second one 50% off: for two items you pay the price of $1 + 0.5 = 1.5$ items. The fraction you pay is $\frac{1.5}{2} = 0.75$, so the discount is 25%. What’s more, the discount is usually applied to the cheaper item; in that case the real discount rate is even lower.

Offers like these only work in your favour if you really need all of those items.

Trap 4: percent and percentage points are not the same thing

At a school, the pass rate in a subject rose from 40% to 50%. How much did it go up?

• In percentage points: it rose by $50 - 40 = 10$ points.
• In percent: it rose by $\frac{50 - 40}{40} \times 100 = 25\%$.

Both are correct, but they say different things. Paying attention to this distinction when the news discusses interest rates, inflation or election results prevents misunderstandings.

Trap 5: finding the price before VAT from the price including VAT

Say a product with a VAT (value added tax) rate of 20% costs 600 TL including VAT. To find the price before VAT, it is wrong to take 20% off 600:

• Wrong: $600 \times 0.80 = 480$ TL
• Right: $600 \div 1.20 = 500$ TL

Check: adding 20% VAT to 500 TL gives $500 \times 1.20 = 600$ TL. Because VAT was added on top of the price before VAT, going back means dividing instead of multiplying.

An LGS-style example question

In exams, percentage questions are often hidden inside a story. Let’s solve this one together (the LGS is Turkey’s national entrance exam for high schools):

A stationery shop first gives a 20% discount on a bag priced at 250 TL. Then, in the week schools open, it raises the discounted price by 10%. By what percentage has the bag’s final price changed compared with its original price?

Solution: let’s write down the multipliers: 0.80 for the 20% discount; 1.10 for the 10% rise.

Percentage change:

So the bag has become 12% cheaper than its original price. Thinking “20% off, 10% up, 10% off in total” is a common mistake. The multiplier method is the safest way to solve questions like this in a single line without errors.

Try it at home: leaflet detective

You will need: a supermarket leaflet or the offers on a shopping website, paper, a pen, a calculator.

1. Choose five products in the leaflet that show an “old price” and a “new price”.
2. For each one, work out the real discount rate with the formula.
3. Compare it with the discount rate printed in the leaflet.
4. Find an offer such as “buy 3, pay for 2” or “second one discounted” and work out the price per item.
5. For different pack sizes of the same product, compare the price per kilogram or per litre.

This small exercise both reinforces percentage calculations and shows that being a smart shopper starts with maths.

Summary

• A percentage is a fraction with a denominator of 100; 15% = 0.15.
• Calculate rises and discounts with multipliers: a 20% rise = ×1.20; a 20% discount = ×0.80.
• Successive percentages are not added but multiplied: 50% and 20% discounts make 60% in total.
• In percentage change, the denominator is always the starting value.
• Percent and percentage points are different ideas.
• To go from a price including VAT back to the price before VAT, divide by 1.20.

For more on ratios, proportion and percentages, take a look at our Maths topics and our maths articles.