3.2.2 Percent Increase and Decrease¶

Calculating percentage changes by finding the difference relative to the original value, including applications to price changes and quantity adjustments.

定义¶

Percent increase and decrease are methods for calculating the relative change in a quantity compared to its original value. Percent increase measures how much a quantity has grown as a percentage of the original amount, while percent decrease measures how much a quantity has shrunk as a percentage of the original amount. These calculations are fundamental in real-world applications such as pricing, population changes, and data analysis. The key principle is that the percentage change is always calculated relative to the original (initial) value, not the new value. For a percent increase, the new value is greater than the original value, resulting in a positive percentage. For a percent decrease, the new value is less than the original value, resulting in a positive percentage that represents the magnitude of the decrease.

核心公式¶

\(\text{Percent Increase} = \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100\%\)
\(\text{Percent Decrease} = \frac{\text{Original Value} - \text{New Value}}{\text{Original Value}} \times 100\%\)
\(\text{New Value} = \text{Original Value} \times (1 + r)\), where \(r\) is the decimal form of percent increase
\(\text{New Value} = \text{Original Value} \times (1 - r)\), where \(r\) is the decimal form of percent decrease
\(\text{Percent Change} = \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100\%\) (general formula, positive for increase, negative for decrease)

易错点¶

⚠️ ["Using the new value as the denominator instead of the original value: Students often calculate \(\frac{\text{New Value} - \text{Original Value}}{\text{New Value}} \times 100\%\), which gives an incorrect result. The denominator must always be the original value.", "Confusing the direction of change: When a quantity decreases and then increases back, students may incorrectly assume the percent decrease and percent increase are equal. For example, a 50% decrease followed by a 50% increase does not return to the original value.", "Forgetting to convert between decimal and percentage forms: Students may forget to multiply by 100% when converting from decimal form to percentage, or may incorrectly apply the percentage multiplier in calculations involving multiple changes.", "Misinterpreting compound percent changes: When multiple percent changes occur sequentially, students often add the percentages instead of multiplying the multipliers. For example, a 10% increase followed by a 20% increase is not a 30% total increase, but rather \(1.10 \times 1.20 - 1 = 32\%\)."]