PRACTICAL GUIDES · Finance and work
How to calculate reverse percentages and percentage change
Percentage changes depend on the starting value. Work through increases and decreases, reverse a change and see why equal percentages do not always cancel out.
By Puffinmint · Updated
1. Decide which value represents 100%
Let A be the original value and B the new value, measured in the same unit. Divide the difference B − A by A and multiply by 100. Use a positive starting value for ordinary increase and decrease comparisons.
If A is zero, this percentage change is undefined: report the absolute difference. Negative starting values or a change of sign need an explicitly stated convention.
2. Calculate an increase and its return journey
From 80 to 100, the difference is 20. Divide 20 by 80 to get one quarter, or 25%. The bar represents one extra part relative to four equal parts in the original value.
Returning from 100 to 80 also changes the amount by 20, but the base is now 100: this is a 20% decrease. Always check the denominator before comparing percentages.
From 80 to 100
- Starting values
- 80 → 100
- Result
- 25% increase
Worked steps
Find the difference.
Divide by the original value.
Convert the ratio to a percentage.
3. Reverse a percentage to find the original
If 80 remains after a 20% decrease, it represents 80% of the original. Divide 80 by four fifths to recover 100. Adding 20% to 80 would use the wrong base.
After a p% increase, divide the final amount by 1 + p/100; after a decrease, divide by 1 − p/100. A 100% decrease leaves zero, from which the original cannot be recovered without more information.
Find the original before a decrease
- Starting values
- 80 after a 20% decrease
- Result
- Original value: 100
Worked steps
Find the fraction of the original that remains.
Divide the final amount by that remaining fraction.
4. Separate successive changes from percentage points
Start at 100, increase by 20% to reach 120, then decrease 120 by 20%. You subtract 24 and finish at 96: a net decrease of 4%. Multiply the change factors rather than adding percentages based on different amounts.
A rate moving from 20% to 25% rises by 5 percentage points, or by 25% relative to its old rate. Round at the end and say which measure you mean.
Sources and method
Frequently asked questions
Why do a 20% rise and a 20% fall not cancel out?
The second percentage uses the changed amount. Starting at 100 gives 120 and then 96, leaving a loss of 4 out of 100.
Which existing calculator should I use?
Percentage calculator compares values and shows the steps. Discount calculator can also recover the original price before a discount. Choose the operation that matches your question.