Percentage Increase and Decrease | 百分比的增加与减少

📚 Percentage Increase and Decrease | 百分比的增加与减少

Percentage increase and decrease are fundamental concepts in IGCSE Mathematics. They allow us to quantify how much a quantity has grown or shrunk relative to its original size. This skill appears in nearly every exam paper, from simple calculations to compound interest problems.

百分比的增加与减少是IGCSE数学中的基础概念。它们使我们能够量化一个数量相对于其原始大小增长或缩减了多少。这项技能几乎出现在每一份试卷中，从简单的计算到复利问题都会涉及。

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1. What Is Percentage Change? | 什么是百分比变化？

Percentage change expresses a change in quantity (increase or decrease) as a proportion of the original amount, multiplied by 100. It always compares the change to the original value, not the new value.

百分比变化将一个数量的变化（增加或减少）表示为原始数量的比例，再乘以100。它总是将变化与原始值进行比较，而不是与新值比较。

The general formula is:

Percentage change = (change ÷ original value) × 100%

Or equivalently:

Percentage change = ((new value − original value) ÷ original value) × 100%

If the result is positive, it is a percentage increase. If negative, it is a percentage decrease.

如果结果为正，则为百分比增加；如果为负，则为百分比减少。

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2. The Multiplier Method | 乘法因子法

For IGCSE, the fastest and most reliable method is using a decimal multiplier. A percentage increase of x% corresponds to multiplying the original amount by (1 + x/100). A percentage decrease of x% corresponds to multiplying by (1 − x/100).

在IGCSE考试中，最快且最可靠的方法是使用小数乘法因子。增加x%相当于将原始数量乘以(1 + x/100)。减少x%相当于将原始数量乘以(1 − x/100)。

• A 15% increase → multiplier = 1 + 0.15 = 1.15

  增加15% → 乘法因子 = 1 + 0.15 = 1.15

• A 25% decrease → multiplier = 1 − 0.25 = 0.75

  减少25% → 乘法因子 = 1 − 0.25 = 0.75

• A 3% increase → multiplier = 1.03

  增加3% → 乘法因子 = 1.03

To find the new value, simply multiply the original amount by the multiplier:

要找到新值，只需将原始数量乘以乘法因子：

New value = Original value × Multiplier

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3. Calculating Percentage Increase | 计算百分比增加

To find the percentage increase between two values, use the formula with the increase (new value − original value) on top.

要求两个值之间的百分比增加，使用公式，将增加量（新值 − 原始值）放在分子上。

Example: A house price rises from £200,000 to £230,000. Find the percentage increase.

示例：一栋房子的价格从200,000英镑涨至230,000英镑。求百分比增加。

Step 1: Find the increase: 230,000 − 200,000 = 30,000

第一步：求增加量：230,000 − 200,000 = 30,000

Step 2: Divide by the original: 30,000 ÷ 200,000 = 0.15

第二步：除以原始值：30,000 ÷ 200,000 = 0.15

Step 3: Multiply by 100: 0.15 × 100 = 15%

第三步：乘以100：0.15 × 100 = 15%

verse. The multiplier is 230,000 ÷ 200,000 = 1.15, so the increase is 1.15 − 1 = 0.15 = 15%.

或者，反向使用乘法因子法。乘法因子为 230,000 ÷ 200,000 = 1.15，所以增加量为 1.15 − 1 = 0.15 = 15%。

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4. Calculating Percentage Decrease | 计算百分比减少

Percentage decrease follows the same logic, but the change is negative. Always divide by the original value.

百分比减少遵循相同的逻辑，但变化量为负。始终除以原始值。

Example: A laptop originally costing £800 is sold for £680. Find the percentage decrease.

示例：一台原本售价800英镑的笔记本电脑以680英镑售出。求百分比减少。

Step 1: Find the decrease: 800 − 680 = 120

第一步：求减少量：800 − 680 = 120

Step 2: Divide by original: 120 ÷ 800 = 0.15

第二步：除以原始值：120 ÷ 800 = 0.15

Step 3: Multiply by 100: 15%

第三步：乘以100：15%

680 ÷ 800 = 0.85, which means the multiplier is 0.85. The decrease is 1 − 0.85 = 0.15 = 15%.

使用乘法因子：680 ÷ 800 = 0.85，即乘法因子为0.85。减少量为 1 − 0.85 = 0.15 = 15%。

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5. Finding the New Value Directly | 直接求新值

When a quantity is increased or decreased by a given percentage, the multiplier method is the most efficient approach.

当一个数量按给定百分比增加或减少时，乘法因子法是最有效的方法。

Example: A salary of £24,000 is increased by 7%. Find the new salary.

示例：一份24,000英镑的工资增加了7%。求新工资。

Multiplier = 1 + 7/100 = 1.07

乘法因子 = 1 + 7/100 = 1.07

New salary = 24,000 × 1.07 = £25,680

新工资 = 24,000 × 1.07 = 25,680英镑

Example: A phone costing £450

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