Percentage Increase and Decrease | 百分比的增加和减少

📚 Percentage Increase and Decrease | 百分比的增加和减少

Percentage change is a fundamental concept in mathematics that helps us understand how quantities grow or shrink relative to their original values. Whether calculating a price discount, a salary raise, or the growth of a population, knowing how to work out percentage increase and decrease is essential for everyday life and further studies in mathematics.

百分比变化是数学中的一个基本概念，帮助我们理解数量相对于原始值的增长或缩小。无论是计算价格折扣、工资涨幅还是人口增长，掌握如何计算百分比的增加和减少对于日常生活和进一步的数学学习都至关重要。

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

Percentage change measures the extent to which a value has increased or decreased, expressed as a fraction of 100. It is a powerful tool for comparison because it allows us to compare changes in numbers of different sizes on a common scale.

百分比变化衡量一个值增加或减少的程度，并以百分数表示。它是一种强大的比较工具，因为它能让我们在统一尺度上比较不同数量大小的变化。

The formula for percentage change is:

百分比变化的公式为：

Percentage Change = (Change ÷ Original Value) × 100%

百分比变化 = (变化量 ÷ 原值) × 100%

Here, ‘Change’ refers to the difference between the new value and the original value (New – Original). A positive result indicates an increase, while a negative result shows a decrease.

这里，“变化量”是指新值与原始值之差（新值 – 原值）。结果为正表示增加，结果为负表示减少。

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2. Understanding Percentage Increase | 理解百分比增加

Percentage increase occurs when a quantity becomes larger. Common examples include a rise in test scores, a growth in savings account interest, or a markup on goods in a shop.

百分比增加发生在数量变大时。常见的例子包括考试成绩提高、储蓄账户利息增长或商店商品加价。

To find the percentage increase, first work out the actual increase (New – Original), then divide the increase by the original amount, and finally multiply by 100 to convert it into a percentage.

要计算百分比增加，首先求出实际增加量（新值 – 原值），然后将增加量除以原值，最后乘以100将其转换为百分比。

For instance, if a video game originally costs £50 and its price rises to £70, the increase is £20. The percentage increase is (20 ÷ 50) × 100% = 40%.

例如，如果一款电子游戏原价为50英镑，价格上涨到70英镑，增加量为20英镑。百分比增加为(20 ÷ 50) × 100% = 40%。

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3. Step-by-Step Calculation of Percentage Increase | 百分比增加的分步计算

Let’s break down the calculation into clear steps to avoid mistakes.

让我们将计算分解为清晰的步骤以避免错误。

Step 1: Find the increase by subtracting the original value from the new value.

步骤1：用新值减去原值，求出增加量。

Step 2: Divide the increase by the original value.

步骤2：将增加量除以原值。

Step 3: Multiply the result by 100 to get the percentage.

步骤3：将结果乘以100得到百分比。

Step 4: Add the percentage sign (%) to your answer.

步骤4：在答案后加上百分号（%）。

Example: Original = 80, New = 100. Increase = 100 – 80 = 20. Divide: 20 ÷ 80 = 0.25. Multiply by 100: 0.25 × 100 = 25%. So the percentage increase is 25%.

例子：原值 = 80, 新值 = 100。增加量 = 100 – 80 = 20。除以：20 ÷ 80 = 0.25。乘以100：0.25 × 100 = 25%。因此百分比增加为25%。

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4. Worked Examples of Percentage Increase | 百分比增加的实例

Example 1: A school had 600 students last year and now has 720. The increase is 720 – 600 = 120. Percentage increase = (120 ÷ 600) × 100% = 20%.

例子1：一所学校去年有600名学生，现在有720名。增加量为720 – 600 = 120。百分比增加 = (120 ÷ 600) × 100% = 20%。

Example 2: A plant grows from 15 cm to 18 cm. The increase is 3 cm. Percentage increase = (3 ÷ 15) × 100% = 20%.

例子2：一株植物从15厘米长到18厘米。增加量为3厘米。百分比增加 = (3 ÷ 15) × 100% = 20%。

Example 3: A packet of flour weighed 1.5 kg; after adding more, it weighs 2.1 kg. Increase = 0.6 kg. Percentage increase = (0.6 ÷ 1.5) × 100% = 40%.

例子3：一袋面粉原本重1.5千克，添加后重2.1千克。增加量 = 0.6千克。百分比增加 = (0.6 ÷ 1.5) × 100% = 40%。

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5. Understanding Percentage Decrease | 理解百分比减少

Percentage decrease measures how much a quantity has reduced compared to its original value. Examples include a price discount, a drop in temperature, or a decrease in the number of members in a club.

百分比减少衡量一个数量相对于原始值的减少程度。例子包括价格折扣、温度下降或俱乐部会员人数的减少。

The formula is similar: Percentage Decrease = (Decrease ÷ Original Value) × 100%, where Decrease = Original – New.

公式类似：百分比减少 = (减少量 ÷ 原值) × 100%，其中减少量 = 原值 – 新值。

Step 1: Find the decrease by subtracting the new value from the original value.

步骤1：用原值减去新值，求出减少量。

Step 2: Divide the decrease by the original value.

步骤2：将减少量除以原值。

Step 3: Multiply by 100 to express it as a percentage.

步骤3：乘以100，表示为百分比。

Step 4: Write the answer with a % sign.

步骤4：写上百分号。

Example: A jacket originally costs £90, now on sale for £72. Decrease = 90 – 72 = 18. Percentage decrease = (18 ÷ 90) × 100% = 20%.

例子：一件夹克原价90英镑，现售价72英镑。减少量 = 90 – 72 = 18。百分比减少 = (18 ÷ 90) × 100% = 20%。

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6. Worked Examples of Percentage Decrease | 百分比减少的实例

Example 1: A car depreciates from £15,000 to £12,000 in one year. Decrease = £3,000. Percentage decrease = (3,000 ÷ 15,000) × 100% = 20%.

例子1：一辆汽车一年内从15,000英镑贬值到12,000英镑。减少量为3,000英镑。百分比减少 = (3,000 ÷ 15,000) × 100% = 20%。

Example 2: A rainwater tank contained 50 litres but after a dry spell only 35 litres remain. Decrease = 15 litres. Percentage decrease = (15 ÷ 50) × 100% = 30%.

例子2：一个雨水箱原本装有50升水，干旱后只剩下35升。减少量为15升。百分比减少 = (15 ÷ 50) × 100% = 30%。

Example 3: The number of apples on a tree fell from 80 to 60. Decrease = 20. Percentage decrease = (20 ÷ 80) × 100% = 25%.

例子3：一棵树上的苹果数量从80个减少到60个。减少量 = 20。百分比减少 = (20 ÷ 80) × 100% = 25%。

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7. Finding the Original Value After a Percentage Change | 求百分比变化后的原值

Sometimes we know the final value and the percentage increase or decrease, and we need to find the original value. This is a common exam question.

有时我们知道最终值和百分比增加或减少，需要求出原值。这是考试中常见的问题。

For a percentage increase, if the new value is N and the percentage increase is p%, then original = N ÷ (1 + p/100). For a percentage decrease, original = N ÷ (1 – p/100).

对于百分比增加，如果新值为N，百分比增加为p%，则原值 = N ÷ (1 + p/100)。对于百分比减少，原值 = N ÷ (1 – p/100)。

Example of increase: A pair of trainers costs £120 after a 20% increase. The original price = 120 ÷ 1.2 = £100.

增加例子：一双运动鞋涨价20%后价格为120英镑。原价 = 120 ÷ 1.2 = 100英镑。

Example of decrease: A smartphone is sold for £340 after a 15% discount. The original price = 340 ÷ 0.85 = £400.

减少例子：一部智能手机打折15%后售价340英镑。原价 = 340 ÷ 0.85 = 400英镑。

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8. Using Multipliers for Efficiency | 使用乘数以提高效率

A more efficient way to perform percentage change calculations is to use multipliers. For a percentage increase of p%, multiply the original by (1 + p/100). For a decrease, multiply by (1 – p/100).

进行百分比变化计算更高效的方法是使用乘数。对于增加p%，将原值乘以(1 + p/100)。对于减少p%，乘以(1 – p/100)。

For example,

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