Calculation
Mathematical Formula For Calculating Reverse Percentage
$$\text{Original Value} = \frac{\text{Percentage Value}}{\text{Percentage Rate}} \times 100$$
Example Broken Down With Steps
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\text{Given: Final Value = 120, Percentage Increase = 20\%}
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\text{Step 1: Add 20% to 100%: } 100 + 20 = 120%
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\text{Step 2: Convert 120% to decimal: } \frac{120}{100} = 1.2
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\text{Step 3: Divide final value by 1.2: } \frac{120}{1.2} = 100
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\text{Result: The original price was 100.}
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Further Explained
Example: The final price after a 20% increase is 120. Find the original price.
Steps:
- Add the percentage increase to 100%:
$$ 100 + 20 = 120\% $$ - Convert 120% to a decimal:
$$ \frac{120}{100} = 1.2 $$ - Divide the final price by 1.2:
$$ \frac{120}{1.2} = 100 $$ - Conclusion: The original price was 100.
Mathematical Formula For Reverse Percentage
To find the original value before a percentage was added or removed, use division — not multiplication. This is a common mistake that leads to incorrect results.
Original (before increase) = Final Value / (1 + Percentage / 100)
Original (before decrease) = Final Value / (1 − Percentage / 100)
The Common Mistake
Many people try to reverse a percentage by applying the opposite operation. For example, if a price includes 20% tax, they try to remove the tax by calculating 20% of the final price and subtracting it. This gives the wrong answer.
Example Broken Down With Steps
A shirt costs $120 after a 20% tax is added. What was the price before tax?
Wrong approach:
120 − (20% × 120) = 120 − 24 = 96 ← incorrect
Correct approach:
Step 1: The final price includes 100% + 20% = 120% of the original
Step 2: Divide by 1.20: 120 / 1.20 = 100
Result: The original price was $100
Further Explained
The wrong approach fails because 20% of $120 is not the same as 20% of the original price. The tax was calculated on $100 (giving $20 in tax), but when you try to reverse it by calculating 20% of $120, you get $24, which is too much.
Another Example
An item is on sale for $75 after 25% off. What was the original price?
Step 1: $75 represents 100% − 25% = 75% of the original
Step 2: Divide by 0.75: 75 / 0.75 = 100
Result: The original price was $100
The rule is simple: always divide the final value by the percentage factor. Use the calculator above to verify the result.