Percentage Increase and Decrease | 百分比增加与减少

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

From price rises in shops to population growth and discounts on sale items, percentage increase and decrease are all around us. Mastering these concepts helps you make sense of everyday numbers and builds a strong foundation for more advanced mathematics. In this article, we will explore what percentage change means, learn the key formulas, practise using multipliers, and tackle reverse percentages and compound changes. By the end, you will feel confident working with percentage increase and decrease in any context.

从商店涨价到人口增长，再到打折商品的折扣，百分比增加和减少无处不在。掌握这些概念有助于你理解日常数字，并为更高阶的数学打下坚实基础。在本文中，我们将探讨百分比变化的含义，学习关键公式，练习使用乘数法，并攻克反向百分比和复合变化。学完本文后，你将能自信地处理各种情境下的百分比增加与减少问题。

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

A percentage change describes how much a quantity has increased or decreased compared to its original value, expressed as a fraction of 100. If the price of a game rises from £40 to £48, it has increased by £8. To express this as a percentage increase, we compare the change (£8) to the original (£40): (8 ÷ 40) × 100 = 20%. So, the price increased by 20%. Similarly, if the price drops from £40 to £30, the decrease is £10, giving a percentage decrease of (10 ÷ 40) × 100 = 25%.

百分比变化描述一个量相对于其原始值增加或减少了多少，并以100的分数形式表示。如果一款游戏的价格从40英镑涨到48英镑，它就增加了8英镑。要表示为百分比增加，我们将变化量（8）与原始值（40）进行比较：(8 ÷ 40) × 100 = 20%。因此，价格上涨了20%。同样，如果价格从40英镑跌到30英镑，减少量为10英镑，那么百分比减少为 (10 ÷ 40) × 100 = 25%。

It is essential to always use the original amount as the denominator when calculating percentage change. Many mistakes occur when students divide by the new amount instead. Remember: the original value is the baseline from which the change is measured.

计算百分比变化时，始终以原始数量作为分母至关重要。很多错误都源于学生误用新数量作为分母。请记住：原始值是衡量变化所依据的基线。

2. The Basic Formula for Increase and Decrease | 增减的基本公式

For any percentage increase or decrease, the core formula is:

对于任何百分比增加或减少，核心公式为：

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

百分比变化 = (数值变化量 ÷ 原始值) × 100

Here, the ‘Change in Value’ is the absolute increase or decrease (new value minus original value). If the result is positive, it indicates an increase; if negative, a decrease. When the new value is smaller than the original, the change will be negative, so you can simply ignore the negative sign and call it a percentage decrease.

这里的“数值变化量”指绝对的增加或减少（新值减去原始值）。结果为正表示增加，为负表示减少。当新值小于原始值时，变化量为负，你可以忽略负号，直接称为百分比减少。

For example, a plant grows from 25 cm to 35 cm. The change is 10 cm, original is 25 cm, so percentage increase = (10 ÷ 25) × 100 = 40%. If a jacket was £80 and is now £60, the change is –£20, original £80, so the percentage decrease = (20 ÷ 80) × 100 = 25%.

例如，一株植物从25厘米长到35厘米。变化量为10厘米，原始值为25厘米，因此百分比增加 = (10 ÷ 25) × 100 = 40%。如果一件夹克原价80英镑，现价60英镑，变化量为–20英镑，原始值为80英镑，那么百分比减少 = (20 ÷ 80) × 100 = 25%。

3. Using Multipliers for Quick Calculations | 使用乘数快速计算

Instead of first finding the change and then adding or subtracting, you can use a decimal multiplier. To increase an amount by a given percentage, multiply by (1 + r), where r is the percentage expressed as a decimal. To decrease, multiply by (1 – r). This method is fast and reduces errors.

与其先求出变化量再进行加减，不如使用一个小数乘数。要将一个量增加给定的百分比，乘以 (1 + r)，其中 r 是用小数表示的百分比。要进行减少，则乘以 (1 – r)。这种方法快速且能减少错误。

Increase by 15%: multiplier = 1 + 0.15 = 1.15
Decrease by 8%: multiplier = 1 – 0.08 = 0.92
Increase by 2.5%: multiplier = 1 + 0.025 = 1.025

增加15%：乘数 = 1 + 0.15 = 1.15
减少8%：乘数 = 1 – 0.08 = 0.92
增加2.5%：乘数 = 1 + 0.025 = 1.025

Example: A £250 tablet is discounted by 30%. The new price = 250 × 0.70 = £175. If a salary of £24,000 rises by 5%, the new salary = 24,000 × 1.05 = £25,200. The multiplier instantly gives the final value without computing the intermediate change.

示例：一台250英镑的平板电脑打七折（减价30%）。新价格 = 250 × 0.70 = 175英镑。如果24,000英镑的工资上涨5%，新工资 = 24,000 × 1.05 = 25,200英镑。乘数法可立即得出最终值，无需计算中间变化量。

4. Calculating the Original Value (Reverse Percentages) | 计算原值（反向百分比）

When you know the value after a percentage change and need to find the original amount, you must divide by the multiplier, not multiply. This is often called a reverse percentage problem. For instance, if a bag costs £84 after a 20% increase, the multiplier was 1.20, so the original price = 84 ÷ 1.20 = £70.

当你知道百分比变化后的值，需要求出原始量时，你必须除以乘数，而不是乘以。这通常被称为反向百分比问题。例如，如果一个包涨价20%后售价84英镑，乘数为1.20，那么原价 = 84 ÷ 1.20 = 70英镑。

A common mistake is to mistakenly decrease the given amount by the same percentage. Taking 20% off £84 gives £67.20, which is incorrect. Always divide by the appropriate multiplier. For a 15% reduction, if the sale price is £34, the multiplier was 0.85, so original = 34 ÷ 0.85 = £40.

一个常见错误是用相同的百分比去减少已知量。从84英镑中扣除20%得到67.20英镑，这是错误的。务必除以相应的乘数。若降价15%后售价为34英镑，乘数为0.85，因此原价 = 34 ÷ 0.85 = 40英镑。

5. Compound Percentage Changes | 复合百分比变化

When a quantity undergoes successive percentage changes, we apply several multipliers one after another. The overall effect is found by multiplying the individual multipliers. Adding the percentages is almost always incorrect. If an investment of £1,000 increases by 10% in the first year and then 5% in the second year, the overall multiplier is 1.10 × 1.05 = 1.155. The final value is £1,000 × 1.155 = £1,155, not £1,000 + 10% + 5% = £1,150.

当一个量经历连续的百分比变化时，我们需要依次应用多个乘数。总效果由各乘数相乘得出。将百分比相加几乎总是错误的。如果一笔1,000英镑的投资第一年增长10%，第二年增长5%，总乘数为 1.10 × 1.05 = 1.155。最终值为 1,000 × 1.155 = 1,155英镑，而不是 1,000 加 10% 再加 5% 得出的 1,150 英镑。

For decreases the logic is the same. A price is cut by 20% then by another 20%. The combined multiplier is 0.80 × 0.80 = 0.64, meaning a 36% overall decrease – not 40%. Be careful: the order does not affect the final amount because multiplication is commutative.

对于减少，逻辑相同。一件商品先降价20%，再降价20%。综合乘数为 0.80 × 0.80 = 0.64，意味着总降价幅度为36%，而非40%。注意：顺序不影响最终金额，因为乘法满足交换律。

6. Linking Fractions, Decimals and Percentages | 分数、小数和百分比的关联

Percentage increase and decrease are much easier if you can fluently convert between fractions, decimals and percentages. Knowing that 50% = ½ = 0.5, 25% = ¼ = 0.25, and 10% = ⅒ = 0.1 allows you to build multipliers mentally. Even uncommon percentages like 17.5% can be handled by writing it as 0.175.

如果你能熟练地在分数、小数和百分比之间进行转换，处理百分比增加和减少就会容易得多。知道 50% = ½ = 0.5, 25% = ¼ = 0.25, 10% = ⅒ = 0.1，就能心算构造乘数。即使是17.5%这样不常见的百分比，也可以写成0.175来处理。

Percentage	Decimal	Fraction	Multiplier for increase
50%	0.5	½	1.5
25%	0.25	¼	1.25
10%	0.1	⅒	1.1
1%	0.01	1/100	1.01

Practice writing any percentage as a decimal by dividing by 100, and recognise benchmark equivalents. This will speed up your work and improve accuracy.

练习将任何百分比除以100写成小数，并记住常见的等价关系。这将加快你的计算速度并提高准确性。

7. Common Mistakes and How to Avoid Them | 常见错误及避免方法

Mistake 1: Using the new value as the denominator. Always use the original value. Mistake 2: Confusing percentage points with percentage change. An increase from 10% to 20% is a 10 percentage point rise, but a 100% increase. Mistake 3: Adding consecutive percentage changes. Never add multipliers; multiply them. Mistake 4: Applying discount backwards incorrectly. If a price includes 20% VAT, dividing by 1.20 recovers the pre-tax price, not multiplying by 0.80.

错误一：将新值作为分母。始终使用原始值。错误二：混淆百分点与百分比变化。从10%上升到20%是上升了10个百分点，但增幅为100%。错误三：将连续的百分比变化相加。绝不能相加乘数，应相乘。错误四：错误地反向应用折扣。如果价格已包含20%增值税，除以1.20才能恢复税前价，而不是乘以0.80。

To avoid these, always write down the original value, identify the multiplier, and check whether the problem asks for the final amount or the original amount. Use estimation to see if your answer is sensible.

为避免这些错误，请务必写下原始值，确定乘数，并检查题目要求的是最终量还是原始量。通过估算检验答案是否合理。

8. Real-World Applications: Discounts, Tax and Profit | 实际应用：折扣、税收与利润

Percentage increase and decrease are everywhere in commerce. A shop offers a 15% discount on a £60 shirt: the sale price is £60 × 0.85 = £51. A restaurant bill of £80 includes a 12.5% service charge: total paid = £80 × 1.125 = £90. A trader buys goods for £500 and sells them for £650; the percentage profit is based on the cost price: (150 ÷ 500) × 100 = 30%.

百分比增加和减少在商业中无处不在。一家商店对售价60英镑的衬衫打八五折：销售价为 60 × 0.85 = 51英镑。一张80英镑的餐厅账单包含12.5%的服务费：支付总额 = 80 × 1.125 = 90英镑。一个商人以500英镑买入商品，以650英镑卖出；成本利润率以成本价为基准：(150 ÷ 500) × 100 = 30%。

When calculating profit, always clarify whether the percentage is based on the cost price or selling price. In many exams, unless stated otherwise, profit percentage is on the cost price.

计算利润时，务必明确百分比是基于成本价还是售价。在许多考试中，除非特别说明，利润率均以成本价为基准。

9. Worked Examples Step by Step | 逐步解题示例

Example 1: A laptop originally costs £540. In a sale, its price is reduced by 25%. What is the sale price?
Multiplier = 1 – 0.25 = 0.75. Sale price = 540 × 0.75 = £405.

示例1：一台笔记本电脑原价540英镑。促销期间降价25%。促销价是多少？
乘数 = 1 – 0.25 = 0.75。促销价 = 540 × 0.75 = 405英镑。

Example 2: After a 20% increase, a painting is valued at £720. Find the original value.
Multiplier = 1.20. Original = 720 ÷ 1.20 = £600.

示例2：一幅画升值20%后价值720英镑。求原始价值。
乘数 = 1.20。原始价值 = 720 ÷ 1.20 = 600英镑。

Example 3: A phone contract rises from £18 per month to £19.80 per month. Calculate the percentage increase.
Change = £1.80. Percentage increase = (1.80 ÷ 18) × 100 = 10%.

示例3：手机月费从18英镑涨到19.80英镑。计算百分比增加。
变化量 = 1.80英镑。百分比增加 = (1.80 ÷ 18) × 100 = 10%。

Example 4: A car loses value by 40% in year 1 and by 30% of its new value in year 2. If it cost £15,000 new, what is it worth after two years?
After year 1: £15,000 × 0.60 = £9,000. After year 2: £9,000 × 0.70 = £6,300. (The second decrease is 30% of the remaining value, not the original).

示例4：一辆汽车第一年贬值40%，第二年又在剩余价值上贬值30%。如果新车价格为15,000英镑，两年后价值多少？
第一年后：15,000 × 0.60 = 9,000英镑。第二年后：9,000 × 0.70 = 6,300英镑。（注意第二次贬值是剩余价值的30%，而非原始价值。）

10. Practice Questions and Summary | 练习与总结

Test your understanding with these questions. Answers are provided below.

用以下问题测试你的理解。答案见后。

Increase 240 by 35%.
A pair of trainers is reduced by 15% to £102. Find the original price.
What is the overall percentage change if a price is increased by 10% and then decreased by 20%?
A population of 8,000 increases by 5% each year for two years. Find the population after two years.
A jacket is marked up by 40% and later sold at a 25% discount on the marked price. If the original cost was £50, find the final selling price.

将240增加35%。
一双运动鞋降价15%后售价102英镑。求原价。
如果价格先涨10%再降20%，总百分比变化是多少？
一个8000的人口每年增长5%，连续两年。求两年后的人口。
一件夹克在成本价基础上加价40%标价，随后按标价打七五折出售。若成本为50英镑，求最终售价。

Answers: 1. 324 2. £120 3. 12% decrease (1.10 × 0.80 = 0.88) 4. 8,820 5. £52.50 (Marked price £70, then ×0.75).

答案：1. 324 2. 120英镑 3. 下降12% (1.10 × 0.80 = 0.88) 4. 8,820 5. 52.50英镑（标价70英镑，再×0.75）。

In summary, mastering percentage increase and decrease opens the door to handling finances, shopping deals and data interpretation with ease. Always identify the original value, choose the right multiplier, and check whether you are finding a final amount or working backwards. Practise both single and compound changes to build confidence.

总之，掌握百分比增加和减少能让你轻松应对财务、购物折扣和数据解读。一定要确定原始值，选择合适的乘数，并检查你是要求最终量还是逆向求解。通过练习单一和复合变化来树立信心。

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Percentage Increase and Decrease | 百分比增加与减少