Why do two successive percentage changes not always cancel out?
Quick answer
A percentage change is calculated from a particular base amount. After an increase, the amount has changed, so the same percentage decrease is usually taken from a larger base than the original. For example, increasing 100 by 10% gives 110; decreasing that new amount by 10% removes 11, leaving 99. Track the current amount at every step rather than assuming the percentages cancel.
The question, the details, the next step.
Why doesn’t a 10% increase followed by a 10% decrease bring a number back where it started?
Because each percentage is applied to the amount at that step. Start with 100 units. A 10% increase adds 10, giving 110. The next 10% decrease is 10% of 110, which is 11, so the final amount is 99—not 100. Percentage change is proportional to the current value, so changing the base changes the size of the next change. [S1] The increase and decrease have the same rate, but they do not have the same amount to work on.
Can you show the reference amount at every stage so I can see exactly where it changes?
Here’s the sequence written out:
- Starting amount: 100 units.
- Increase: the base is 100. The change is 10% of 100 = 10, so the new amount is 110.
- Decrease: the base is now 110. The change is 10% of 110 = 11, so the new amount is 99.
The second calculation uses 110 as its reference amount, not the original 100. The decrease is therefore one unit larger than the increase. These are illustrative units; the same calculation works for any starting amount.
Does this happen only when the starting amount is 100, or is there a quick way to check other examples?
It happens for other starting amounts too. For a starting amount of 200, a 10% increase adds 20, giving 220. A 10% decrease from 220 removes 22, leaving 198. To check any sequence, write the current amount, calculate the next percentage from that amount, and then update it before applying another change.
For equal increases and decreases of rate r, expressed as a decimal, the two steps multiply the starting amount by (1 + r) and then by (1 − r). For example, at 10%, that is 1.10 × 0.90 = 0.99, so the result is 99% of the start. [S1] This relationship describes equal percentage changes in opposite directions; it does not mean every pair of different changes leaves the same kind of result.
What’s a simple layout I can use to solve a two-step problem without mixing up the bases?
Use one line for each step, and make the base explicit. For the 100-unit example:
- Start: 100 units.
- Increase by 10%: base 100; change +10; amount afterward 110.
- Decrease by 10%: base 110; change −11; amount afterward 99.
You can also write new amount = current amount × (1 + rate) for an increase, or current amount × (1 − rate) for a decrease. Here, 10% is written as 0.10. This helps keep the reference amount visible at each step. It doesn’t establish what a real-world price or quantity will be unless the stated rates and starting amount match that situation.
A practical checklist
Write down the starting amount before calculating.
For each change, identify the current amount that serves as its base.
Convert the percentage to a decimal or calculate that percentage of the base.
Update the amount before applying the next percentage.
Compare the final amount with the original to check whether the changes truly cancel.
A few more useful answers
What if the increase and decrease percentages are different?
Calculate each change from the amount at that stage. The final result depends on both rates and on their order, so do not assume they cancel.
How can I tell whether two changes will return to the starting amount?
For an increase by rate a followed by a decrease by rate b, the amount returns to its start when (1 + a)(1 − b) = 1, with rates written as decimals. Otherwise, calculate both steps and compare the result with the start.
Does the order of an increase and a decrease matter?
For one increase and one decrease, multiplying by their two factors gives the same final amount in either order. The intermediate amounts and the base used at each step still differ.
Sources
References used for this guide.
- [S1] OpenStax College Algebra: Exponential Functions and Multiplicative Change openstax.org · Checked