Percentage Increase and Decrease | 百分比增减及应用

📚 Percentage Increase and Decrease | 百分比增减及应用

Percentages are everywhere – from shop discounts and bank interest to exam scores and population growth. Understanding how to calculate percentage increases and decreases is an essential mathematical skill. This article will guide you through the key concepts, formulas, and real-life applications, helping you master the topic with confidence. You will also learn how to avoid common mistakes and tackle exam-style questions effectively.

百分比在我们的生活中无处不在——从商店折扣和银行利息到考试成绩和人口增长。掌握如何计算百分比增减是一项基本的数学技能。本文将通过关键概念、公式和实际应用，帮助你自信地掌握这一主题。你还将学会如何避免常见错误，并高效应对考试题型。

1. What is Percentage Increase? | 什么是百分比增加？

A percentage increase occurs when an amount goes up by a certain proportion compared to its original value. For example, if a bag of rice weighs 5 kg and is increased by 20%, the new weight is more than the original. The increase is calculated as a fraction of 100, meaning you are adding a part of the original quantity to itself. The formula for percentage increase is: New Value = Original Value + (Percentage Increase ÷ 100) × Original Value.

当一个数值相对于原值按一定比例上升时，就发生了百分比增加。例如，一袋大米重 5 公斤，增加 20% 后，新的重量就大于原值。增加量按百分之几来计算，意味着你在原值的基础上增加了一部分。百分比增加的计算公式为：新值 = 原值 + (百分比增加 ÷ 100) × 原值。

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

To calculate a new amount after a percentage increase, multiply the original by (1 + percentage/100). For example, increasing £200 by 15%: New amount = 200 × (1 + 15/100) = 200 × 1.15 = £230. Another method is to find 15% of 200 first (£30), then add it to the original (£200 + £30 = £230). Both approaches give the same result. Always write the percentage as a decimal or fraction to simplify the multiplication.

要计算百分比增加后的新值，将原值乘以（1 + 百分比/100）。例如，将 200 英镑增加 15%：新值 = 200 × (1 + 15/100) = 200 × 1.15 = 230 英镑。另一种方法是先求出 200 的 15%（30 英镑），再加到原值上（200 + 30 = 230 英镑）。两种方法结果相同。计算时将百分比写成小数或分数，可以简化乘法。

New Amount = Original Amount × (1 + Percentage Increase ÷ 100)

新值 = 原值 × (1 + 百分比增加 ÷ 100)

3. What is Percentage Decrease? | 什么是百分比减少？

A percentage decrease means an amount reduces by a certain proportion. For instance, a laptop originally priced at £800 may be reduced by 25%. The reduction is a fraction of the original value. Just like increases, the percentage is based on the starting amount. The new value is what remains after subtracting the reduction from the original.

百分比减少意味着一个数值按一定比例降低。例如，一台原价 800 英镑的笔记本电脑降价 25%。减少量是原值的一部分。就像增加一样，百分比也是基于初始值计算的。新值是从原值中减去减少量后剩下的结果。

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

To find an amount after a percentage decrease, you can either calculate the reduction first and then subtract, or multiply directly using a multiplier. The multiplier for a decrease is (1 – percentage/100). For a 30% decrease on £450, the multiplier is 0.70, so new amount = 450 × 0.70 = £315. This method is much faster for complex problems.

要计算百分比减少后的值，可以先算出减少量再减去，也可以直接使用乘数进行计算。减少的乘数是（1 – 百分比/100）。对于 450 英镑减少 30%，乘数为 0.70，因此新值 = 450 × 0.70 = 315 英镑。对于复杂问题，这种方法要快得多。

New Amount = Original Amount × (1 – Percentage Decrease ÷ 100)

新值 = 原值 × (1 – 百分比减少 ÷ 100)

5. Finding the Original Amount (Reverse Percentages) | 求原值（逆向百分比）

Sometimes you know the final amount after a percentage change and need to find the original value. This is called reverse percentages. For instance, a shop sells a coat for £120 after a 20% reduction. You must divide by the multiplier used. If decreased by 20%, multiplier is 0.80. Original price = £120 ÷ 0.80 = £150. Be careful: you cannot simply add 20% to £120 because 20% of £120 is not the same as 20% of the original.

有时你已经知道经过百分比变化后的终值，需要求出原值。这称为逆向百分比。例如，某商店打完 20% 的折扣后，一件外套售价为 120 英镑。你需要除以所用乘数。减少 20%，乘数为 0.80，原价 = 120 ÷ 0.80 = 150 英镑。注意：不能简单地将 120 英镑直接加上 20%，因为 120 的 20% 与原值的 20% 不同。

Original Amount = Final Amount ÷ Multiplier

原值 = 终值 ÷ 乘数

6. Repeated Percentage Changes | 重复百分比变化

Repeated percentage changes happen when an amount goes up or down several times by different percentages. For example, a house value increases by 10% in the first year, and then by 5% in the second year. You cannot simply add the percentages. Instead, apply each multiplier in sequence: New value = Original × 1.10 × 1.05. This gives a compound effect. The same rule applies for mixed increases and decreases.

当一个数值多次以不同的百分比上升或下降时，就形成了重复百分比变化。例如，一栋房子的价值第一年上涨 10%，第二年上涨 5%。你不能简单地将百分比相加。应该依次应用每个乘数：新值 = 原值 × 1.10 × 1.05。这样就产生了复合效应。对于增减混合的情况，同样的规则也适用。

7. Percentage Change in Real Life: Discounts | 实际生活中的百分比变化：折扣

Discounts are a common application of percentage decreases. Shops often advertise ‘30% off’ or ‘Half price’. To find the sale price after a discount, multiply the original price by (1 – discount %/100). If there is a further discount on the already reduced price, you must apply another multiplier. Always read the problem carefully: some discounts are taken off the original price directly.

折扣是百分比减少的一种常见应用。商店常常打出“减价 30%”或“半价”的广告。要计算折扣后的售价，将原价乘以（1 – 折扣%/100）。如果在已降价的基础上再打折，就必须再次应用乘数。审题时要仔细：有些折扣是直接从原价上扣除的。

Original Price	Discount	Multiplier	Sale Price
£60	25%	0.75	£45
£85	40%	0.60	£51

折扣计算示例 | Discount Calculation Examples

8. Percentage Change in Real Life: Mark-ups and Profits | 实际生活中的百分比变化：加价与利润

Businesses use percentage increases to determine selling prices. A shop buys a product at cost price and then adds a mark-up (profit margin). If a phone case costs £8 to buy and the shop marks it up by 35%, the selling price = 8 × 1.35 = £10.80. The increase is the profit. Understanding mark-ups helps compare value across different retailers.

商家利用百分比增加来确定售价。商店以成本价购入商品，然后加上一定的加价（利润率）。如果一个手机壳进价为 8 英镑，商店加价 35%，则售价 = 8 × 1.35 = 10.80 英镑。增加的金额就是利润。理解加价有助于在不同零售商之间比较商品价值。

9. Percentage Change and Value Added Tax (VAT) | 百分比变化与增值税 (VAT)

In many countries, Value Added Tax (VAT) is a percentage of the price of goods and services. For example, the UK standard VAT rate is 20%. If a bike is priced at £350 before VAT, the final price is 350 × 1.20 = £420. To find the pre-VAT price from a total including VAT, divide by 1.20. This is another real-life use of reverse percentages.

在许多国家，增值税 (VAT) 是商品和服务价格的一个百分比。例如，英国标准增值税率为 20%。如果一辆自行车的不含税价格为 350 英镑，最终价格就是 350 × 1.20 = 420 英镑。要从含税总价求出不含税价格，则除以 1.20。这是逆向百分比在生活中的另一个应用。

10. Working with Multipliers | 使用乘数

Multipliers are the decimal equivalent of (1 ± percentage/100) and they make calculations quick and accurate. For an 8% increase, the multiplier is 1.08. For a 15% decrease, it’s 0.85. When you have several percentage changes, simply multiply the original by all multipliers in sequence. This method reduces errors and is especially useful for compound growth or depreciation.

乘数是（1 ± 百分比/100）对应的小数，它们使计算变得快速而准确。对于 8% 的增加，乘数为 1.08；对于 15% 的减少，乘数为 0.85。当存在多次百分比变化时，只需将原值按顺序乘以所有的乘数。这种方法能减少错误，对于复合增长或折旧问题尤其有用。

11. Common Mistakes to Avoid | 常见错误避免

One common error is confusing the original amount with the final amount when using reverse percentages. Another mistake is adding or subtracting percentages directly without using multipliers for repeated changes. Also, students sometimes calculate a percentage of the new amount instead of the original. Always identify whether you are finding a percentage of the starting value or the ending value, and draw a diagram if it helps.

一个常见错误是在进行逆向百分比计算时混淆原值与终值。另一个错误是在处理重复变化时直接加减百分比而不使用乘数。此外，学生有时会把百分比算成是新值的百分比，而不是原值的。一定要明确你是在求起始值的百分比还是终止值的百分比，如果必要可以画图辅助理解。

12. Practice Problems and Solutions | 练习题与解答

Let’s test your understanding. Problem 1: A coat originally costs £120. In a sale, the price is reduced by 35%. What is the sale price? Solution: Multiplier = 1 − 0.35 = 0.65, New price = 120 × 0.65 = £78. Problem 2: After a 20% increase, a computer costs $960. What was the original cost? Solution: Multiplier = 1.20, Original = 960 ÷ 1.20 = $800. Try more questions to build fluency, and remember to check if the answer makes sense in context.

我们来检验一下你的掌握情况。问题 1：一件外套原价 120 英镑，商场降价 35% 销售。售价是多少？解答：乘数 = 1 − 0.35 = 0.65，新价格 = 120 × 0.65 = 78 英镑。问题 2：一台电脑涨价 20% 后，售价为 960 美元。原价是多少？解答：乘数 = 1.20，原价 = 960 ÷ 1.20 = 800 美元。多做练习题可以提高熟练度，还要记得检查答案在情境中是否合理。

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Published by TutorHao | Mathematics Revision Series | aleveler.com

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