Two discounts in a row do not add up: a 30% cut followed by a 10% cut leaves the price at 63/100, which is 37% off, not 40%. The second cut is taken from a price that has already dropped, so you multiply the factors 7/10 and 9/10 instead of adding the percentages. Here is the method, when percentages do add, and how to compare deals like 3-for-2. All prices are made-up examples.
The essentials
- A discount of p% is written as the factor 1 − p/100: 25% off is 3/4 and 40% off is 3/5.
- Two chained cuts multiply their factors; the percentages are not added.
- The order of the cuts does not change the final price, because multiplying fractions is commutative.
- A 50% rise followed by a 50% cut leaves the price at 3/4 of the start.
- Half off the second item saves 25% per item, and 3-for-2 saves 33.3…%.
Worked example: two 20% cuts on $60
A $60 coat drops 20%, and a week later drops another 20%. What does it cost in the end, and what is the total discount?
- Write the factor of each cut. 20% off leaves 1 − 1/5 = 4/5 of the price.
- Multiply the factors. 4/5 × 4/5 = 16/25.
- Apply the product to the price. 60 × 16/25 = 960/25 = $38.40.
- Find the total discount. 1 − 16/25 = 9/25, which is 36%.
Result: $38.40, which is 36% less than the starting price.
Check: cut by cut, $60 becomes $48 and then $38.40. The second cut takes off $9.60 (20% of $48), not $12 (20% of $60).
Why 20% plus 20% does not reach 40%
The first cut is figured on $60 and the second on $48. Since 48 is less than 60, the second 20% is worth less than the first, and the total saving ends up below the sum of the two percentages. The second cut has the same percentage, but the base has changed. That is why you work with the price that remains, which is what the factor measures.
From percentage to factor
Each percentage becomes a factor that tells you what part of the price you pay. A discount is subtracted from 1 and an increase is added to 1. The table lists the most common ones; if you need to turn another percentage into a fraction, the percent to fraction page gives the exact fraction.
| Change | Calculation | Factor |
|---|---|---|
| −10% | 1 − 1/10 | 9/10 |
| −25% | 1 − 1/4 | 3/4 |
| −30% | 1 − 3/10 | 7/10 |
| +10% | 1 + 1/10 | 11/10 |
| +50% | 1 + 1/2 | 3/2 |
To chain changes, just multiply the fractions in the right-hand column and convert the result to a percentage at the end. That way you avoid rounding in the middle steps.
The order does not change the price
A 30% cut followed by a 10% cut gives 7/10 × 9/10 = 63/100. In the opposite order you get the same product. The final price is 63% of the start, and the total discount is 37%. The commutative property of multiplying fractions guarantees that any reordering gives the same number.
Wikipedia puts it this way: percent changes can be applied in any order and have the same effect, but they do not add up in the usual way. That last part is the one that tends to surprise people.
Going up and down by the same percentage
If a $100 price rises 50% and then drops 50%, the factors are 3/2 and 1/2, and their product is 3/4. The final price is $75. The cut is applied to $150, the price after the increase, so it does not bring you back to where you started.
With small percentages the same thing happens, though it is less noticeable. In the Wikipedia example, a $200 price rises 10% and then falls 10%: 11/10 × 9/10 = 99/100, so it ends at $198. To get back exactly to the starting point, the two factors must be inverses, like a 25% rise followed by a 20% cut.
Half off the second item and 3-for-2
Quantity deals are compared the same way as discounts: work out what fraction of the regular price you pay. With half off the second item, you pay 1 + 1/2 = 3/2 of one item’s price for two items, which is 3/4 per item. The saving is 1/4, or 25%. With 3-for-2 you pay for 2 out of every 3, that is 2/3, and the saving is 1/3. That third is a repeating decimal, 0.3333…, so the discount is 33.3…% and any finite figure is only an approximation.
The table compares six offers on a $60 item. The last two rows are measured per item, and the comparison is fair only if you use the same number of items in all of them.
| Offer | Fraction you pay | Final price | Discount |
|---|---|---|---|
| One 40% cut | 3/5 | $36.00 | 40% |
| Two 20% cuts | 16/25 | $38.40 | 36% |
| 30% then 10% | 63/100 | $37.80 | 37% |
| One 30% cut | 7/10 | $42.00 | 30% |
| Half off the second item | 3/4 per item | $45.00 per item | 25% |
| 3-for-2 | 2/3 per item | $40.00 per item | 33.3...% |
The single 40% cut beats the two 20% cuts, because it acts in full on the complete price. Per item, 3-for-2 saves more than half off the second item, but it makes you buy three. To compare packages of different sizes, look at the unit price.
When percentages do add
Percentages add when they are all figured on the same price. A tag that says “20% off plus an extra 10% off the original price” describes two discounts on $60: $12 and $6, a total of $18 and 30% off. Before multiplying, read which base each percentage acts on.
If you need to take a tax out of a price that is already final, the calculation is a division, explained in how to remove sales tax from a price. Two other percentages that get confused because of their names appear in margin and markup. To check the calculations with the exact value of each step, use the percentage calculator; to turn a result like 16/25 into a percentage, use the fraction to percent conversion. If you only want the part of an amount that matches a fraction, the fraction of a quantity page does it. The fraction applications collect more cases like this one.
Practice with three prices
- A $90 book drops 10% and then another 10%. What does it cost, and what is the total discount?
- A $100 price rises 25% and then falls 20%. What does it cost at the end?
- A store offers 3-for-2 on T-shirts that cost $45 each. How much does each T-shirt cost if you buy three?
Show the answers
- Each cut multiplies by 0.9, and 0.9 × 0.9 = 0.81, so 90 × 81/100 = $72.90. The total discount is 19%.
- The factors are 5/4 and 4/5, which cancel, so the price goes back to $100.
- You pay 2/3 of the total: 45 × 2/3 = $30 per T-shirt.
Sources
- Wikipedia, Percentage: a percentage as a fraction of 100 equal parts, sequential percent changes that do not add up in the usual way, the $200 example with +10% and −10%, and the order not mattering (consulted October 17, 2026).