📚 Percentage Increase and Decrease | 百分比增减
Understanding how quantities grow or shrink by a given percentage is a fundamental skill in mathematics. Whether we are calculating a price after a discount, working out a salary rise, or finding the original amount before a change, percentage increase and decrease provide a clear method to describe relative change. This topic builds on basic percentage concepts and introduces both forward and reverse calculations that are essential for real‑world problem‑solving.
理解数量如何按给定百分比增长或减少是数学中的一项基本技能。无论是计算折扣后的价格，计算加薪后的工资，还是求出变化前的原始数量，百分比增减都提供了描述相对变化的清晰方法。本专题建立在基础百分比概念之上，同时引入正向和逆向计算，这两者对于解决实际问题都至关重要。
1. What is a Percentage Change? | 什么是百分比变化？
s larger, we call it a percentage increase; if it is smaller, it is a percentage decrease.
百分比变化衡量的是增加或减少的量相对于原始数量的大小，并以百分数表示。计算公式为（变化量 ÷ 原始量）× 100。如果新值较大，我们称之为百分比增加；如果新值较小，则称之为百分比减少。
In many exam questions you will first need to find the change in value by subtracting the original amount from the new amount. Then divide that change by the original and multiply by 100 to convert it to a percentage.
在许多考试题目中，你首先需要用新量减去原始量来求出值的变化。然后将该变化量除以原始量，再乘以 100，将其转换为百分数。
| Term | Meaning |
|---|---|
| Percentage increase | New value = original × (1 + r/100) |
| Percentage decrease | New value = original × (1 − r/100) |
| Multiplier | 1 ± (rate/100) |
2. Using Multipliers | 使用乘数
Instead of calculating the change separately and adding or subtracting, we can use a single multiplier. For an increase of r%, the multiplier is (1 + r/100). For a decrease of r%, it is (1 − r/100). Multiplying the original amount by this decimal gives the new amount directly.
我们可以使用一个单一的乘数，而不必分开计算变化量再加或减。对于增加 r%，乘数为（1 + r/100）。对于减少 r%，乘数为（1 − r/100）。将原始量乘以这个小数，就能直接得到新量。
For example, to increase £80 by 15%, the multiplier is 1 + 15/100 = 1.15. New amount = 80 × 1.15 = £92. To decrease £80 by 15%, multiplier = 1 − 0.15 = 0.85, giving 80 × 0.85 = £68.
例如，将 80 英镑增加 15%，乘数为 1 + 15/100 = 1.15。新金额 = 80 × 1.15 = 92 英镑。将 80 英镑减少 15%，乘数为 1 − 0.15 = 0.85，结果为 80 × 0.85 = 68 英镑。
Multipliers are especially useful when several percentage changes happen one after another, because you can simply multiply the multipliers together to find the overall effect.
当连续发生多次百分比变化时，乘数尤其有用，因为你只需将各个乘数相乘，就能得到总体效果。
3. Finding the Original Amount Before a Percentage Change | 求百分比变化前的原始量
Reverse percentages allow us to work backwards when we know the new amount and the percentage change. If the new amount represents a certain percentage of the original, we can set up an equation or divide by the multiplier to find the original.
逆向百分比让我们在知道新量和百分比变化的情况下反推。如果新量代表原始量的某个百分比，我们可以建立方程，或者除以乘数来求出原始量。
For an increase: if the new price after a 20% rise is £240, then £240 = original × 1.20. So original = 240 ÷ 1.20 = £200. For a decrease: a sale price of £72 after a 40% discount means £72 = original × 0.60. Original = 72 ÷ 0.60 = £120.
对于增加情况：如果上涨 20% 后的新价格为 240 英镑，那么 240 英镑 = 原始量 × 1.20。因此原始量 = 240 ÷ 1.20 = 200 英镑。对于减少情况：折扣 40% 后的售价为 72 英镑，这意味着 72 英镑 = 原始量 × 0.60。原始量 = 72 ÷ 0.60 = 120 英镑。
A common mistake is to subtract the percentage from the new amount. Always divide by the multiplier corresponding to the remaining or total percentage.
一个常见的错误是从新量中减去这个百分比。始终除以对应剩余或整体百分比的乘数。
4. Expressing a Change as a Percentage | 将变化量表示为百分比
Sometimes we are given two values and asked what the percentage increase or decrease is between them. First calculate the absolute change, then divide by the original value, and finally multiply by 100.
有时我们会得到两个值，并被要求求出它们之间的百分比增加或减少。首先计算绝对变化量，然后除以原始值，最后乘以 100。
The formula can be written as:
Percentage change = (difference ÷ original) × 100
该公式可以写成：
百分比变化 =（差值 ÷ 原始值）× 100
For instance, a town's population grows from 50 000 to 57 500. The difference is 7 500. Original is 50 000. So percentage increase = (7500 ÷ 50000) × 100 = 15%.
例如，一个小镇的人口从 50 000 增长到 57 500。差值为 7 500。原始值为 50 000。因此百分比增加 =（7500 ÷ 50000）× 100 = 15%。
5. Repeated Percentage Changes | 连续的百分比变化
When a quantity is increased or decreased by the same percentage several times, we apply the multiplier repeatedly. For example, compound interest on savings is a repeated percentage increase.
当某个量多次以相同的百分比增加或减少时，我们会重复应用乘数。例如，储蓄的复利就是一种重复的百分比增加。
If £1000 is invested at 5% annual interest, after one year: 1000 × 1.05 = £1050. After two years: 1050 × 1.05, or 1000 × 1.05² = £1102.50. In general, after n years, the amount is original × (1.05)ⁿ.
如果 1000 英镑以 5% 的年利率投资，一年后：1000 × 1.05 = 1050 英镑。两年后：1050 × 1.05，或 1000 × 1.05² = 1102.50 英镑。一般而言，n 年后，金额为 原始量 × (1.05)ⁿ。
For a repeated decrease, such as depreciation of a car by 15% per year, the multiplier is 0.85. After t years, value = original × (0.85)ᵗ.
对于重复减少的情况，例如汽车每年贬值 15%，乘数为 0.85。t 年后，价值 = 原始量 × (0.85)ᵗ。
6. Working with More Than One Change | 处理多次不同的百分比变化
Sometimes a price goes through a rise and then a discount, or two successive percentage changes. To find the overall percentage change, multiply the multipliers. Do not add or subtract the percentages directly.
有时价格会经历一次上涨然后一次折扣，或者两次连续的百分比变化。要计算总体百分比变化，将乘数相乘。不要直接加或减百分比。
Example: a shop increases a bike's price by 20% then offers a 10% discount. Overall multiplier = 1.20 × 0.90 = 1.08, which is an 8% increase from the original. This is not simply 20% − 10% = 10%.
示例：一家商店将自行车价格提高 20%，然后提供 10% 的折扣。总体乘数 = 1.20 × 0.90 = 1.08，即比原始价格增加 8%。这并非简单的 20% − 10% = 10%。
Finding the original value after a series of changes requires reversing the order: divide the final amount by the last multiplier first, then by the previous multiplier, and so on.
经过一系列变化后求原始值需要逆序操作：先将最终金额除以最后一个乘数，再除以前一个乘数，依此类推。
7. Percentage Profit and Loss | 百分比利润与亏损
In business contexts, percentage profit or loss is always calculated on the cost price (the original amount the seller paid), not the selling price. The change is the difference between selling price and cost price.
在商业情境中，百分比利润或亏损始终以成本价（卖方支付的原价）为基础计算，而非售价。变化量是售价与成本价之间的差值。
If a shop buys a toy for £20 (cost price) and sells it for £28, the profit is £8. Percentage profit = (8 ÷ 20) × 100 = 40%. If the toy were sold for £15, the loss would be £5, giving a percentage loss of (5 ÷ 20) × 100 = 25%.
如果一家商店以 20 英镑（成本价）买进一个玩具，并以 28 英镑卖出，利润为 8 英镑。百分比利润 =（8 ÷ 20）× 100 = 40%。如果该玩具以 15 英镑卖出，亏损为 5 英镑，百分比亏损 =（5 ÷ 20）× 100 = 25%。
Be careful: the original is always the cost price, even when the question gives the selling price and asks for profit percentage. You may need to work backwards from selling price to cost price using reverse percentages.
请注意：原始值始终是成本价，即使题目给出了售价并要求计算利润百分比也是如此。你可能需要利用逆向百分比从售价反推成本价。
8. Percentage Error | 百分比误差
In science and measurement, percentage error tells us how far an estimated or measured value is from the true value. The formula is:
Percentage error = (|estimated − actual| ÷ actual) × 100
在科学和测量中，百分比误差告诉我们估计值或测量值与真实值相差多远。公式为：
百分比误差 =（|估计值 − 实际值| ÷ 实际值）× 100
If a student estimates a length as 55 cm but the true length is 50 cm, the error is |55 − 50| = 5 cm. Percentage error = (5 ÷ 50) × 100 = 10%. The actual value is always used as the denominator.
如果一个学生估计长度为 55 厘米，但真实长度为 50 厘米，误差为 |55 − 50| = 5 厘米。百分比误差 =（5 ÷ 50）× 100 = 10%。始终以实际值作为分母。
9. Real‑Life Contexts and Word Problems | 实际生活情境与文字题
Percentage changes appear everywhere: tax (VAT) added to a bill, discounts during sales, population growth, interest rates, and inflation. When solving word problems, identify the original amount, the rate, and whether the change is an increase or decrease.
百分比变化无处不在：账单上附加的增值税（VAT）、促销折扣、人口增长、利率和通货膨胀。在解决文字题时，要识别原始量、比率，以及该变化是增加还是减少。
Always double‑check whether you need to find the new amount, the original amount, or the percentage change itself. Underlining key numbers and percentages in the question helps to avoid confusion.
务必仔细核对题目要求的是求新量、原始量还是百分比变化本身。在题目中划出关键数字和百分比有助于避免混淆。
10. Common Mistakes and How to Avoid Them | 常见错误及如何避免
Using the wrong original value for the percentage. Remember: the denominator is always the older or starting value, not the new one.
使用了错误的原始值来计算百分比。记住：分母始终是较早或起始的值，而不是新值。Adding and subtracting percentages directly instead of using multipliers. 10% then 20% rise is not a 30% rise; it is 1.10 × 1.20 = 1.32, a 32% rise.
直接将百分比相加或相减，而不是使用乘数。先增加 10% 再增加 20%，并非增加 30%；而是 1.10 × 1.20 = 1.32，即增加 32%。Forgetting to reverse the operation when finding the original. For a 15% decrease, the new amount is 85% of original, so divide by 0.85, not multiply by 1.15.
在求原始量时忘记逆向运算。对于减少 15%，新量是原始量的 85%，因此应除以 0.85，而不是乘以 1.15。Confusing percentage point change and percentage change. A rise from 10% to 15% is a 5 percentage point increase, but the percentage increase is (5/10)×100 = 50%.
混淆百分点变化与百分比变化。从 10% 上升到 15% 增加了 5 个百分点，但百分比增加是 (5/10)×100 = 50%。
Careful reading and systematic working with clear steps will eliminate most of these errors.
仔细审题，并条理清晰地分步解题，就能消除大部分错误。
11. Practice Questions and Methods | 练习题与解题方法
Here are some typical questions to test understanding:
以下是一些典型的题目，用来检验理解程度：
- A laptop costs £480 before VAT at 20%. What is the price including VAT?
一台笔记本电脑不含增值税的价格为 480 英镑，增值税率为 20%。含税价格是多少？ - After a 25% reduction, a jacket is sold for £54. What was its original price?
一件夹克降价 25% 后售价为 54 英镑。它的原价是多少？ - A house value increased from £200 000 to £234 000. Find the percentage increase.
一栋房子的价值从 200 000 英镑增长到 234 000 英镑。求百分比增加。 - A car depreciates by 12% each year. If it costs £15 000 new, what is it worth after 3 years?
一辆汽车每年贬值 12%。如果新车价格为 15 000 英镑，3 年后它的价值是多少？
Approach each by identifying the known values, choosing the correct multiplier or formula, and writing down the calculation step by step. For reverse problems, always end with a sense check: if you found the original, apply the percentage to see if you get back to the given new value.
解答每道题时，要识别已知值，选择正确的乘数或公式，并逐步写下计算过程。对于逆向问题，最后要进行合理性检查：如果求出了原始量，应用该百分比看看能否得出题目所给的新值。
12. Summary and Key Takeaways | 总结与关键要点
Percentage increase and decrease are essentially about using multipliers to move between original and new values. The multiplier method provides a unified approach for both forward and reverse calculations. Always identify whether the question asks for the new amount, the original amount, or the percentage itself, and write down the relationship clearly before calculating.
百分比增减的核心在于使用乘数在原值和新值之间进行转换。乘数法为正向和逆向计算提供了统一的方法。始终要识别题目要求的是新量、原始量还是百分比本身，并在计算前清晰地写出关系式。
Mastering these techniques builds confidence not only for examinations but also for interpreting everyday financial information, from shopping discounts to news about economic changes.
掌握这些技巧不仅能增强考试信心，还能帮助解读日常财务信息，从购物折扣到有关经济变化的新闻。
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