Percentage Increases & Decreases

A Multiplier is a single decimal value that represents the total percentage of the original amount after a change has occurred. It is calculated by converting the percentage change to a decimal and adding it to (for increase) or subtracting it from (for decrease) the number 1.

For an Increase of $p\%$, the multiplier is $M = 1 + \frac{p}{100}$. For example, a 15% increase uses a multiplier of $1.15$, effectively calculating 115% of the original value in one step.

For a Decrease of $p\%$, the multiplier is $M = 1 - \frac{p}{100}$. For example, a 20% decrease uses a multiplier of $0.80$, as you are finding the remaining 80% of the original value.

Reverse Percentage problems involve finding the original value when only the final value and the percentage change are known. This is a common area of error where students mistakenly apply the percentage to the final value instead of the unknown original.

The logical approach is to recognize that $\text{Original} \times \text{Multiplier} = \text{New Value}$. Therefore, to find the original, you must divide the new value by the multiplier: $\text{Original} = \frac{\text{New Value}}{\text{Multiplier}}$

If a value has increased by 10% to reach 220, the original is not $220 - 10\%$. Instead, it is $220 \div 1.1 = 200$. This ensures the 10% is calculated based on the 200, not the 220.

Identify the Base: Always ask, "What was the value before the change?" This value must be your denominator in the formula $\frac{\text{Change}}{\text{Original}} \times 100$.

Sanity Check: If a value decreases by 50% and then increases by 50%, the result will always be less than the original ($0.5 \times 1.5 = 0.75$). If your answer returns to the original, you have likely added the percentages instead of multiplying them.

Rounding Errors: In multi-step problems, keep the multipliers in their exact decimal or fraction form until the final step. Rounding intermediate multipliers can lead to significant inaccuracies in the final result.