Reverse Percentages

• The fundamental principle of reverse percentages is that multiplication and division are inverse operations. If we multiply an original value by a factor to get a final value, we must divide the final value by that same factor to return to the original.

• The Percentage Identity states that the final value $V_f$ is related to the original value $V_o$ by the relation $V_f = V_o \times (1 \pm r)$, where $r$ is the percentage change expressed as a decimal. To isolate $V_o$, we rearrange the formula to $V_o = \frac{V_f}{1 \pm r}$.

• This logic holds because the percentage is always a 'percentage of' the original. Because we do not know the original, we cannot simply calculate the percentage of the final value and add or subtract it.

The Multiplier Method

• Step 1: Identify the total percentage that the final value represents. For an increase, add the percentage to $100\%$; for a decrease, subtract it from $100\%$.

• Step 2: Convert this total percentage into a decimal multiplier by dividing by $100$. For example, $115\%$ becomes $1.15$, and $85\%$ becomes $0.85$.

• Step 3: Divide the final value by this multiplier to find the original $100\%$ value. This single-step calculation is the most efficient method for calculator-based work.

The Unitary Method

• Step 1: Equate the final value to its corresponding percentage. If a price is reduced by $20\%$, then the sale price is equal to $80\%$.

• Step 2: Find the value of $1\%$ by dividing the final amount by the current percentage. This establishes a base unit for the calculation.

• Step 3: Multiply the $1\%$ value by $100$ to reach the original $100\%$ total. This method is often easier for mental arithmetic or non-calculator problems.

• It is vital to distinguish between finding a percentage of a number and finding the original number. The following table summarizes the differences:

• Note that the multiplier remains the same for both directions; only the operation (multiplication vs. division) changes to reflect the direction of the calculation.

• Identify the 'Before' and 'After': Always read the question carefully to determine if the value provided is the original amount or the amount after a change. Keywords like 'including tax', 'after a discount', or 'original price' are essential clues.

• Perform a Sanity Check: If the value increased, the original must be smaller than the final value. If the value decreased, the original must be larger. If your answer doesn't follow this logic, you likely multiplied instead of divided.

• Avoid the 'Add Back' Trap: Never calculate the percentage of the final value and try to add or subtract it to get back to the start. For example, if a value increases by $10\%$, subtracting $10\%$ of the new value will not result in the original number because the second $10\%$ is calculated on a larger base.

• The Symmetry Fallacy: Many students believe that a $20\%$ increase followed by a $20\%$ decrease returns you to the original value. This is false because the decrease is applied to a larger number, resulting in a final value that is $96\%$ of the original ($1.20 \times 0.80 = 0.96$).

• Incorrect Multiplier Selection: Students often use the percentage change itself as the multiplier (e.g., dividing by $0.20$ for a $20\%$ increase). The divisor must always represent the total percentage ($1.20$), not just the change.

• Misidentifying the 100%: A common error is treating the final value as $100\%$. In reverse percentage problems, the final value is almost never $100\%$; it is always $100\% \pm \text{change}\%$.