Why Don’t Equal Percentage Increases and Decreases Cancel?

Percentages · Changing the Reference Value

A percentage is a share of a particular starting value. Once that value changes, the next percentage is applied to a different base.

Why Does the Starting Value Matter?

Twenty percent means twenty parts out of one hundred, but you still need to know twenty percent of what. On a value of 100, it is 20. On a value of 120, it is 24. Using the same rate does not guarantee the same absolute change.

That is why adding 20% and then subtracting 20% is different from adding 20 and subtracting 20. Fixed amounts cancel. Percentage amounts depend on the base used at each step.

OpenStax models discounts as a rate multiplied by the relevant price. The key question in a sequence is which price is relevant now: the original price or the already-adjusted one.

What Does the Multiplier Formula Show?

An increase of rate r multiplies a value by 1 + r. A decrease of the same rate multiplies it by 1 − r. Multiplying these factors gives 1 − r², so the final value is below the starting value whenever 0 < r < 1.

The squared decimal rate is 0.04, equivalent to 4%. Do not square the written percentage number 20 and label the result 400%. Convert the rate to a decimal before using this formula.

Reversing the order gives the same final value in this simple model: 100 × 0.80 × 1.20 = 96. That is because multiplication commutes. Fixed fees, intermediate rounding, thresholds, or changes applied to different bases can break that equivalence.

What Percentage Actually Reverses an Increase or Decrease?

To reverse a 20% increase from 100 to 120, remove 20 from a base of 120. The required decrease is 20/120 = 1/6, or approximately 16.67%. It is smaller than 20% because the new base is larger.

To reverse a 20% decrease from 100 to 80, add 20 to a base of 80. The required increase is 20/80 = 25%. This is larger than 20% because the new base is smaller.

A 100% decrease leaves zero. No finite percentage increase of zero restores a positive starting value: zero multiplied by any finite factor remains zero. This is a boundary of the percentage model, not a missing calculator setting.

Why Aren’t Successive Discounts Added Together?

A 20% discount followed by a 10% discount on the reduced price uses 0.80 × 0.90 = 0.72. A price of 100 becomes 72, a total discount of 28%, not 30%. The second discount removes 8 because it is 10% of 80.

For discount rates a and b, the combined discount rate is a + b − ab. The product term removes the overlap that simple addition would count twice. Use the Discount Calculator for a price check, but confirm whether a promotion applies each discount successively or to the original price.

How Is a Percentage-Point Change Different?

If a pass rate rises from 40% to 50%, it increases by 10 percentage points. Relative to the original rate, however, its percentage increase is (50 − 40)/40 × 100 = 25%. These describe the same movement using different measures.

Do not put a percentage-point difference directly into a successive-change multiplier. First identify whether the problem concerns a measured rate, an amount, or a relative change. This distinction is especially useful when comparing test results and survey proportions.

How Can You Check a Multi-Step Percentage Calculation?

1. Write the positive starting amount and identify each step’s base.
2. Convert rates to decimals and use one multiplier for each successive change.
3. Keep unrounded values until the requested final rounding step.
4. Compare the final amount with the original: (final − original)/original × 100.

Practice with a 30% increase followed by a 30% decrease. Starting at 200 gives 260, then 182. The net decrease is 18/200 = 9%, matching 0.30² = 0.09. The Percentage Change Calculator can check the original-to-final comparison.