Percentage Increase and Decrease | 百分比增减

📚 Percentage Increase and Decrease | 百分比增减

Percentage increase and decrease are fundamental concepts in mathematics that appear frequently in everyday life, from calculating discounts during sales to understanding population growth. The ability to work with percentages is essential for IGCSE Edexcel Mathematics, as it forms the basis for many problem-solving questions. This article will guide you through the key methods and applications, ensuring you can tackle any percentage change problem with confidence.

百分比增减是数学中的基本概念，在日常生活中频繁出现，从计算销售折扣到理解人口增长。掌握百分比对于 IGCSE Edexcel 数学至关重要，因为它是许多解题问题的基础。本文将引导你掌握关键方法和应用，确保你能自信地解决任何百分比变化问题。

1. Understanding Percentage Change | 理解百分比变化

Percentage change measures how much a quantity has increased or decreased relative to its original value. It is expressed as a percentage. The formula is:

百分比变化衡量一个数量相对于其原始值增加或减少了多少，以百分比表示。公式为：

Percentage change = (Change ÷ Original value) × 100%

lue. If the new value is greater, the change is positive (increase); if smaller, it is negative (decrease).

其中变化量 = 新值 − 原始值。如果新值更大，变化为正（增加）；如果更小，则为负（减少）。

grows from 30 cm to 45 cm, the change is +15 cm. The percentage increase = (15 ÷ 30) × 100% = 50%. Always use the original value as the denominator.

例如，一株植物从 30 cm 长到 45 cm，变化量为 +15 cm。增长百分比 = (15 ÷ 30) × 100% = 50%。总是以原始值作为分母。

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2. Calculating Percentage Increase | 计算百分比增长

To calculate a percentage increase:

计算百分比增长：

% Increase = [(New − Original) ÷ Original] × 100%

Example: A laptop's price rises from £400 to £460. Find the percentage increase.

示例：一台笔记本电脑的价格从 £400 上涨到 £460。求增长百分比。

Increase = £460 − £400 = £60. % Increase = (60 ÷ 400) × 100% = 15%.

涨幅 = £60。增长百分比 = (60 ÷ 400) × 100% = 15%。

hod: New = Original × (1 + percentage ÷ 100), but here we calculate the percentage itself.

你也可以使用乘数法：新值 = 原始值 × (1 + 百分比 ÷ 100)，但这里我们是直接求百分比。

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

Similarly, the formula for percentage decrease is:

类似地，百分比减少的公式为：

% Decrease = [(Original − New) ÷ Original] × 100%

Example: A student's test score dropped from 80 marks to 68. Find the percentage decrease.

示例：一名学生的考试成绩从 80 分降到 68 分。求减少百分比。

Decrease = 80 − 68 = 12. % Decrease = (12 ÷ 80) × 100% = 15%.

减少量 = 12。减少百分比 = (12 ÷ 80) × 100% = 15%。

Always ensure you subtract the smaller from the larger to obtain a positive change.

始终确保用较大的数减去较小的数，以得到正的变化量。

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4. Finding the Original Value | 求原始值

Often you are given the new value after a percentage increase or decrease and asked to find the original. Use the reverse percentage method.

经常给定百分比增减后的新值，要求求出原始值。使用逆向百分比法。

For an increase of r% New = Original × (1 + r ÷ 100). To find Original, divide: Original = New ÷ (1 + r ÷ 100).

对于增加 r%：新值 = 原始值 × (1 + r ÷ 100)。要求原始值，进行除法：原始值 = 新值 ÷ (1 + r ÷ 100)。

For a decrease of r%: Original = New ÷ (1 − r ÷ 100).

对于减少 r%：原始值 = 新值 ÷ (1 − r ÷ 100)。

Example: After a 25% increase, a salary is £50000. Find the original salary.

示例：增加 25% 后，薪水为 £50000。求原始薪水。

Original = £50000 ÷ 1.25 = £40000.

原始值 = £50000 ÷ 1.25 = £40000。

Be careful: you cannot simply subtract 25% of £50000. That would give £37500, which is incorrect.

注意：不能简单地从 £50000 中减去 25%。那样会得到 £37500，这是错误的。

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5. Using Multipliers | 使用乘数

A multiplier is a quick way to apply a percentage change directly. For an increase of r%, the multiplier is (1 + r ÷ 100). For a decrease, it is (1 − r ÷ 100).

乘数是一种直接应用百分比变化的快捷方法。对于增长 r%，乘数为 (1 + r ÷ 100)；对于减少，乘数为 (1 − r ÷ 100)。

Percentage change	Multiplier
Increase by 12%	1.12
Decrease by 8%	0.92
Increase by 150%	2.5

he appropriate multiplier gives the new value directly. For example, to increase 200 by 30%: 200 × 1.30 = 260.

乘以对应的乘数可以直接得到新值。例如，将 200 增加 30%：200 × 1.30 = 260。

Multipliers are especially useful for successive changes.

乘数法在连续变化时尤其有用。

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6. Successive Percentage Changes | 连续百分比变化

When two or more percentage changes occur one after another, you must multiply the multipliers – do not add or subtract the percentages.

当两个或多个百分比变化接连发生时，必须将乘数相乘——而不是简单地将百分比相加或相减。

Example: A price is increased by 10% and then decreased by 10%. Does it return to the original?

示例：价格先增加 10%，再减少 10%。它会回到原值吗？

No. Start with 100. Increase by 10%: 100 × 1.10 = 110. Decrease by 10% final value is 99, not 100. The overall multiplier is 1.10 × 0.90 = 0.99, representing a 1% decrease overall.

不会。从 100 开始。增加 10%：100 × 1.10 = 110。减少 10%：110 × 0.90 = 99。最终值为 99，不是 100。总的乘数为 1.10 × 0.90 = 0.99，表示总体减少了 1%。

For three successive changes, multiply all three multipliers

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