3.2.3 Reverse Percentage Problems¶
Finding the original value when given a percentage change or finding what percentage one value represents of another.
定义¶
Reverse percentage problems involve finding an unknown original value when given information about a percentage change or percentage relationship. These problems require working backwards from a known final value, percentage change, or percentage comparison to determine the original quantity. Common scenarios include: (1) finding the original price before a discount or after a markup, (2) determining what percentage one value represents of another, and (3) calculating the base amount when given a percentage of that base. The key is to set up an equation where the percentage relationship is expressed algebraically, then solve for the unknown original value using the formula: \(\text{Original Value} = \frac{\text{Final Value}}{1 \pm \text{Percentage Rate}}\) or \(\text{Original Value} = \frac{\text{Part}}{\text{Percentage Rate}}\).
核心公式¶
- \(\text{Original Value} = \frac{\text{Final Value}}{1 + r}\) (when percentage increase is given)
- \(\text{Original Value} = \frac{\text{Final Value}}{1 - r}\) (when percentage decrease is given)
- \(\text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100\%\)
- \(\text{Original Value} = \frac{\text{Part}}{\frac{p}{100}}\) where \(p\) is the percentage
- \(\text{Final Value} = \text{Original Value} \times (1 + r)\) or \((1 - r)\) (forward relationship)
易错点¶
- ⚠️ Confusing the direction of the percentage change: applying the percentage to the final value instead of the original value, or vice versa. For example, if a price increased by 20%, students might incorrectly divide by 1.20 instead of recognizing they need to work backwards.
- ⚠️ Incorrectly converting between percentage and decimal form, such as using 20 instead of 0.20, or forgetting to convert the final answer back to percentage form when required.
- ⚠️ Misidentifying which value is the 'whole' or 'base' in the percentage relationship. Students often use the wrong denominator when calculating what percentage one value represents of another.
- ⚠️ Adding or subtracting the percentage rate directly from 1 with the wrong sign, such as using \((1 + r)\) when they should use \((1 - r)\) for a discount problem, leading to answers that are off by a factor.