How to Back Out a Percentage From a Total
Discover how to precisely determine the original amount before any percentage was applied. Master the technique to reverse engineer totals with added or subtracted percentages.
Backing out a percentage is a process used to determine an original value when a percentage has already been applied. This skill is valuable in various financial and mathematical situations, providing a clear understanding of an amount before any additions or subtractions were made.
Understanding the Core Calculation
Calculating an original amount from a total that includes a percentage adjustment involves recognizing that the total represents a modified version of the original 100%. When a percentage is added or subtracted, the total becomes either more or less than that initial 100%. To reverse the operation, divide the total by the factor representing the original 100% plus or minus the percentage change.
The general formula is: Original Amount = Total Amount / (1 ± Percentage as a Decimal). Division is used because the total amount already incorporates the percentage change, meaning the original amount was multiplied by a factor (1 plus or minus the percentage) to arrive at the total. Multiplying by an inverse percentage would not yield the correct original value. For instance, if an item increased by 10%, it is now 110% of its original value; to find the original, divide by 1.10, not multiply by 0.90.
When a Percentage Was Added
When a percentage, such as sales tax or a service charge, has been added to an original amount, the total represents the original value plus that additional percentage. To find the amount before the addition, you must account for this increase in the divisor. This method is applicable in scenarios like calculating the pre-tax price of an item or determining the base cost before a markup.
To perform this calculation, convert the percentage added into its decimal form by dividing it by 100. Then, add 1 to this decimal to create the adjustment factor. Finally, divide the total amount by this adjustment factor to reveal the original amount. For example, if a customer paid $53.25 for an item, and the sales tax rate was 6.5%:
1. Convert the sales tax rate to a decimal: 6.5% / 100 = 0.065.
2. Add 1 to the decimal: 1 + 0.065 = 1.065.
3. Divide the total price by this factor: $53.25 / 1.065 = $50.00.
Thus, the original price of the item before sales tax was $50.00.
When a Percentage Was Subtracted
Conversely, when a percentage, such as a discount, has been subtracted from an original amount, the total reflects the original value minus that reduction. To determine the original amount before the subtraction, the calculation must compensate for this decrease in the divisor. This approach is useful for uncovering the original price of a product after a discount or the full value of an asset following depreciation.
To execute this calculation, convert the percentage subtracted into its decimal form by dividing it by 100. Subtract this decimal from 1 to form the adjustment factor. Then, divide the total amount by this adjustment factor to ascertain the original amount. For instance, if a shirt was purchased for $36.00 after a 20% discount:
1. Convert the discount percentage to a decimal: 20% / 100 = 0.20.
2. Subtract this decimal from 1: 1 – 0.20 = 0.80.
3. Divide the discounted price by this factor: $36.00 / 0.80 = $45.00.
Therefore, the original price of the shirt before the discount was applied was $45.00.
Practical Applications
The ability to back out percentages extends beyond simple retail transactions, proving useful in diverse financial computations. For example, understanding a pre-fee service cost involves reversing an added percentage. If a professional service bill totals $525, and a 5% administrative fee was included, applying the “percentage added” method reveals the original service cost was $500.00.
Similarly, determining a pre-bonus salary from a total compensation package utilizes the “percentage added” calculation. If an employee’s total compensation for a period was $65,000, and this included a 30% performance bonus calculated on their base salary, the base salary could be identified. The bonus makes the total 130% of the base, so dividing $65,000 by 1.30 would show a base salary of $50,000.