A 20% rise followed by a 20% fall does not bring you back to the start. Enter a sequence of percentage changes to see the net change and the value after each step.
Enter each percentage change in order, negative for decreases.
Enter a starting value if you want the final amount.
Read the net change.
Two changes: Net % = a + b + ab ÷ 100
Many changes: Final = Start × Π(1 + rᵢ ÷ 100)
When a value changes by several percentages one after another, such as a 20% discount followed by an extra 10% off, or a salary rise one year and a cut the next, the changes compound. They do not simply add. This calculator applies each change in order to a starting value and shows the final value and the single net percentage change.
Shoppers use it to see through 'flat 50% + extra 20%' offers, investors to combine yearly returns, and aptitude students to answer successive percentage questions in seconds. For exactly two changes, it also shows the shortcut a + b + ab/100.
Enter the changes in order, using a minus sign for decreases, and optionally a starting value.
1. Write each percentage change as a multiplier: an increase of r% becomes (1 + r ÷ 100), a decrease becomes (1 − r ÷ 100).
2. Multiply the starting value by each multiplier in turn.
3. Final value = start × (1 + r₁ ÷ 100) × (1 + r₂ ÷ 100) × ...
4. Net change % = (final ÷ start − 1) × 100.
5. For two changes a and b, use the shortcut net % = a + b + a × b ÷ 100.
6. For more than two, apply the shortcut to the first two, then combine the result with the next change.
Two changes multiply: (1 + a/100)(1 + b/100) = 1 + a/100 + b/100 + ab/10,000. Subtracting 1 and multiplying by 100 gives a + b + ab/100. The last term is the change on the change. For two discounts of 20% and 10%, a = −20 and b = −10, so the net is −30 + 2 = −28%, a 28% discount, not 30%. For a 10% rise followed by a 10% fall, the net is −1%.
Because multiplication is commutative, applying the same changes in any order gives the same final value. A 20% cut then a 30% rise gives the same result as the rise first. However, the size of each step in rupees does depend on the order. A rise followed by an equal percentage fall always ends lower than the start, because (1 + r)(1 − r) = 1 − r², which is less than 1.
Repeated equal changes are compound growth: a value growing at r% for n years ends at start × (1 + r/100)ⁿ. Successive percentage change is the general case with different rates each period. Averaging the rates arithmetically overstates the true average growth when rates vary; the correct average rate is found from the nth root of the total multiplier, which is the idea behind CAGR.
An online store lists a winter jacket at ₹3,499 with 20% off, and applies a further 10% off at checkout with a bank card offer.
Apply each change in turn: 3,499 × (1 − 20/100) = 2,799.2 2,799.2 × (1 − 10/100) = 2,519.28
Net change: (2,519.28 ÷ 3,499 − 1) × 100 = -28%
Two-change shortcut: a + b + ab/100 = -20 + -10 + 2 = -28%
Answer: Net change -28%; Final value 2,519.28
Adding the percentages, so that 20% and 10% off become 30% off.
Thinking a rise and an equal fall cancel out.
Applying the second change to the original value instead of the new one.
Forgetting the minus sign for decreases.
Averaging yearly returns arithmetically to describe overall growth.
Checking the real discount in stacked sale offers.
Combining year-on-year salary changes or price changes.
Aptitude questions on population, depreciation and successive discounts.
Working out net portfolio change over several years with different returns.
Does the order of changes matter?
No, multiplication is commutative, so the net result is the same in any order.
How do I use it for population growth?
Enter each year's growth rate as a separate change.