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Percentage Formula Cheat Sheet

Print a nine-formula percentage reference with worked examples, the right denominator, reverse calculations and percentage-change caveats.

Version: 1.0

Percentage formulas at a glance

Nine common percent calculations with worked examples, the correct base and the important zero-denominator exceptions. Keep this alongside your work or print it as a reference.

1. Find a percentage of a value

Part = (rate ÷ 100) × whole

Example: 20% of 150 = 30

Check: The whole is the base.

2. Part as a percentage of whole

Rate (%) = (part ÷ whole) × 100

Example: 18 of 60 = 30%

Check: Whole must not be zero.

3. Recover the whole

Whole = part ÷ (rate ÷ 100)

Example: 45 is 15% of 300

Check: Rate must not be zero.

4. Increase by a rate

Final = original × (1 + rate ÷ 100)

Example: 80 increased by 25% = 100

Check: Rate is applied to the original value.

5. Decrease by a rate

Final = original × (1 − rate ÷ 100)

Example: 250 decreased by 12% = 220

Check: A 12% decrease is not reversed by a 12% increase.

6. Signed percentage change

Change (%) = ((new − old) ÷ old) × 100

Example: 80 → 100 = +25%; 120 → 90 = −25%

Check: Old value must not be zero; this is directional.

7. Undo a known percentage increase

Original = final ÷ (1 + rate ÷ 100)

Example: 120 after a 20% increase started at 100

Check: Use the adjustment rate, not the final value as the base.

8. Undo a known percentage decrease

Original = final ÷ (1 − rate ÷ 100)

Example: 72 after a 10% decrease started at 80

Check: This sheet uses rates from 0% to below 100%; at 100% the original cannot be recovered.

9. Unsigned percentage difference

Difference (%) = |A − B| ÷ ((A + B) ÷ 2) × 100

Example: 80 and 120 differ by 40%

Check: Uses the mean as its base; not the same as change.

Avoid these common mix-ups

Percentage change uses the old value as its denominator and keeps its sign. Percentage difference uses the mean of two nonnegative values and is unsigned. A percentage point change subtracts two percentages directly: moving from 20% to 25% is 5 percentage points but a 25% relative increase.

When the required denominator is zero, a relative percentage answer can be undefined. Reverse a decrease only when its rate is below 100%; do not divide by zero. Work with unrounded intermediate values and round only the final display.

How to use this percentage formula sheet

Find the question type, identify the original or whole value, then use that row. For change, the old value supplies the denominator; for difference between nonnegative numbers, use their mean. Percent and percentage points are distinct units.

  • For 20% of 150, convert 20% to 0.20 and multiply: 30.
  • For a rise from 20% to 25%, the change is five percentage points, but the relative percentage increase is 25%.
  • For a known-rate reverse change, divide by the increase or decrease multiplier rather than subtracting the rate from the final answer.
  • A zero denominator makes the corresponding ratio undefined; round the final value, not intermediate calculations.

Sources and references