📚 Solving Percentages and Reverse Percentages | 百分比与反百分比的解题技巧
Percentages appear in almost every IGCSE Mathematics exam. From simple discounts to compound interest, understanding how to calculate a percentage of a quantity and how to reverse the process when the original value is unknown is essential. This article explains each method step by step, with paired Chinese translations to help you master both the English and Chinese mathematical terms.
在 IGCSE 数学考试中，百分比几乎无处不在。无论是简单的折扣还是复利，掌握如何计算一个量的百分比，以及当原值未知时如何反向求解，都是必备技能。本文逐步讲解每种方法，并配以中英双语，帮助你同时掌握英文和中文的数学术语。
1. What Is a Percentage? | 什么是百分比？
A percentage is a number expressed as a fraction of 100. The word ‘percent’ means ‘out of 100’. For example, 25% means 25 out of 100, or 25 ÷ 100 = 0.25.
百分比是表示“每一百中有多少”的数。英文单词“percent”意思是“每一百”。例如，25% 表示 100 中的 25，即 25 ÷ 100 = 0.25。
Percentages are used to compare quantities of different sizes because they standardise the comparison to a common base of 100.
百分比用于比较不同大小的数量，因为它将比较标准化为以 100 为共同基准。
2. Converting Percentages, Decimals, and Fractions | 百分比、小数和分数的互化
To convert a percentage to a decimal, divide by 100. For example, 37% = 37 ÷ 100 = 0.37. To convert a decimal to a percentage, multiply by 100. For example, 0.08 = 8%.
将百分比转换为小数，需除以 100。例如，37% = 37 ÷ 100 = 0.37。将小数转换为百分比，需乘以 100。例如，0.08 = 8%。
To convert a percentage to a fraction, write it over 100 and simplify. For example, 125% = 125/100 = 5/4.
将百分比转换为分数，可将其写为分母是 100 的分数，然后化简。例如，125% = 125/100 = 5/4。
Common conversions you should memorise:
你应该牢记的常见互化：
| Percentage | Decimal | Fraction |
| 50% | 0.5 | 1/2 |
| 25% | 0.25 | 1/4 |
| 10% | 0.1 | 1/10 |
| 33⅓% | 0.333… | 1/3 |
| 20% | 0.2 | 1/5 |
3. Finding a Percentage of a Quantity | 求一个量的百分比
To find p% of a quantity Q, multiply Q by p and divide by 100. In symbols:
要求一个量 Q 的 p%，只需将 Q 乘以 p 再除以 100。用符号表示：
p% of Q = Q × p ÷ 100
Example: Find 15% of 240.
例：求 240 的 15%。
15% of 240 = 240 × 15 ÷ 100 = 36
You can also convert the percentage to a decimal first: 0.15 × 240 = 36. Both methods give the same answer.
你也可以先将百分比化为小数：0.15 × 240 = 36。两种方法得到相同答案。
4. Percentage Increase and Decrease | 百分比的增加与减少
A percentage increase means adding the calculated percentage to the original amount. The fastest way is to multiply by a decimal multiplier equal to (1 + p/100).
百分比增加意味着在原始量上加上计算出的百分比。最快的做法是乘以一个等于 (1 + p/100) 的小数乘数。
If an amount increases by 12%, the new value is the original multiplied by 1.12. If it decreases by 12%, multiply by 0.88.
如果一个量增加 12%，新值等于原始值乘以 1.12。如果减少 12%，则乘以 0.88。
Example: A shop price of 80 pounds is increased by 15%. Find the new price.
例：一件商品价格 80 英镑，上涨 15%。求新价格。
New price = 80 × (1 + 0.15) = 80 × 1.15 = 92 GBP
For a decrease, suppose the same 80 pounds item is discounted by 20%.
如果同一件 80 英镑的商品降价 20%。
New price = 80 × (1 − 0.20) = 80 × 0.80 = 64 GBP
Notice that a 20% decrease is not simply the opposite of a 20% increase. A 20% increase followed by a 20% decrease does not bring you back to the original value.
注意，减少 20% 并不等同于增加 20% 的逆运算。先增加 20% 再减少 20% 并不会回到原始值。
5. Reverse Percentages | 反百分比
A reverse percentage problem gives you the final amount after a percentage increase or decrease, and asks you to find the original amount before the change.
反百分比问题给出的是经过百分比增加或减少后的最终数值，要求你找出变化前的原始值。
The key idea is to identify the multiplier that was used and then divide by that multiplier instead of multiplying.
关键思路是找出所使用的乘数，然后用除法而不是乘法来还原原值。
6. Reverse Percentage: After an Increase | 反百分比：增加之后求原值
If an amount becomes S after a p% increase, then:
如果一个量经过 p% 增加后变成 S，则：
Original = S ÷ (1 + p/100)
Example: After a 25% increase, a salary is 5000 pounds. What was the original salary?
例：经过 25% 加薪后，工资为 5000 英镑。原工资是多少？
Original = 5000 ÷ 1.25 = 4000 GBP
Check: 4000 increased by 25% is 4000 × 1.25 = 5000.
验算：4000 增加 25% 为 4000 × 1.25 = 5000。
7. Reverse Percentage: After a Decrease | 反百分比：减少之后求原值
If an amount becomes S after a p% decrease, then:
如果一个量经过 p% 减少后变成 S，则：
Original = S ÷ (1 − p/100)
Example: After a 30% reduction, a laptop costs 420 pounds. Find the original price.
例：笔记本电脑降价 30% 后售价为 420 英镑。求原价。
Original = 420 ÷ 0.70 = 600 GBP
Always check: 600 decreased by 30% is 600 × 0.70 = 420.
务必验算：600 减少 30% 为 600 × 0.70 = 420。
8. Dealing with Repeated Changes | 多次连续变化的处理
When a quantity changes multiple times, apply each multiplier in order. Do not add the percentages together unless the problem explicitly states simple percentage points.
当一个量发生多次变化时，应依次应用每个乘数。除非题目明确说明是简单的百分点相加，否则不要将百分比直接相加。
Example: A value of 200 is increased by 10% and then decreased by 10%. Find the final value.
例：数值 200 先增加 10%，再减少 10%。求最终值。
Final = 200 × 1.10 × 0.90 = 198
Notice the final value is 198, not 200. This shows that two 10% changes do not cancel out.
注意最终值是 198，不是 200。这说明两个 10% 的变化并不会相互抵消。
For reverse problems with repeated changes, divide by each multiplier in turn.
对于多次变化的反向问题，依次除以每个乘数即可。
9. Using a Percentage to Compare Two Quantities | 用百分比比较两个量
Sometimes an exam asks: ‘What percentage of A is B?’ To answer, divide B by A and multiply by 100.
有时考试会问：“B 是 A 的百分之几？”答案为 B 除以 A，再乘以 100。
Percentage = B ÷ A × 100%
Example: In a school, 45 students out of 180 take mathematics. What percentage take mathematics?
例：某校 180 名学生中有 45 人选修数学。选数学的占百分之几？
45 ÷ 180 × 100 = 25%
You can also write this as a fraction: 45/180 = 1/4 = 25%.
你也可以把它写成分数：45/180 = 1/4 = 25%。
10. Percentage Change | 百分比变化率
To express a change as a percentage, use the formula:
要将变化量表示为百分比，使用公式：
Percentage change = (Change ÷ Original value) × 100%
If the result is positive, it is a percentage increase; if negative, it is a percentage decrease.
如果结果为正，则为百分比增加；如果结果为负，则为百分比减少。
Example: A dress price rises from 40 to 50 pounds. Find the percentage increase.
例：一件裙子的价格从 40 英镑涨到 50 英镑。求增长率。
Change = 50 − 40 = 10; Percentage change = 10 ÷ 40 × 100 = 25%
Always divide by the original value, not the new value.
始终要除以原始值，而不是新值。
11. Common Errors and Pitfalls | 常见错误与陷阱
One common mistake is using the final value as the denominator when calculating percentage change. Remember to use the original value.
一个常见错误是计算百分比变化时把最终值当作分母。记住要用原始值作为分母。
Another error is subtracting or adding percentages directly in reverse problems. For example, if a price increased by 20% to 120, the original price is not 120 − 24 = 96. You must divide by 1.2: 120 ÷ 1.2 = 100.
另一个错误是在反百分比问题中直接加减百分比。例如，如果价格增加 20% 后为 120，原价不是 120 − 24 = 96，而必须除以 1.2：120 ÷ 1.2 = 100。
Also, be careful with multipliers. A decrease of 15% uses the multiplier 0.85, not 0.15. The multiplier for a decrease is always (1 − decimal equivalent).
同时要注意乘数。减少 15% 的乘数是 0.85，而不是 0.15。减少时的乘数总是 (1 − 小数形式的百分比)。
Finally, always check whether your answer makes sense. If a final value is lower than the original, the multiplier must be less than 1. Reverse problems should produce a plausible original value by multiplying your answer by the original multiplier as a check.
最后，始终检查答案是否合理。如果最终值小于原始值，则乘数必须小于 1。反百分比问题的答案应可通过“用你的答案乘以原乘数”来验算。
12. Worked Examination-Style Examples | 考试风格例题精讲
Example 1: A jacket is sold for 56 pounds after a 30% discount. Find the original price.
例 1：一件夹克打折 30% 后售价 56 英镑。求原价。
Original = 56 ÷ 0.70 = 80 GBP
Example 2: The population of a town increases by 8% to 6480. Find the population before the increase.
例 2：一个小镇的人口增加 8% 后为 6480 人。求增加前的人口。
Original = 6480 ÷ 1.08 = 6000
Example 3: Maria invests money in a bank. After an annual interest rate of 4%, her balance is 6240 pounds. Find the amount she invested.
例 3：Maria 将钱存入银行。在年利率 4% 后，她的余额为 6240 英镑。求她的本金。
Original = 6240 ÷ 1.04 = 6000 GBP
Always show your working clearly, especially in calculator questions. Set out the multiplier explicitly before dividing.
答题时务必清晰地写出计算步骤，尤其是在使用计算器的题目中。先写出乘数，再进行除法。
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