📚 Fractions, Decimals and Percentages | 分数、小数与百分数
Fractions, decimals and percentages are three equivalent ways of describing parts of a whole. In IGCSE Mathematics, you need to convert fluently between these forms, compare them, and apply percentage change to real-life problems such as discounts, tax, interest and profit.
分数、小数和百分数是描述整体中部分的三种等价方式。在 IGCSE 数学中，你需要熟练地在这三种形式之间转换、比较大小，并将百分数变化应用于折扣、税收、利息和利润等实际问题。
1. Understanding Equivalent Forms | 理解等价形式
A fraction such as 3/4, a decimal 0.75, and a percentage 75% all express the same value. The denominator of a fraction tells how many equal parts the whole is divided into; the numerator tells how many parts are taken.
分数如 3/4、小数 0.75 和百分数 75% 都表示同一个值。分数的分母表示整体被分成多少等份，分子表示取了多少份。
Percent means 'out of 100', so 75% means 75 per 100. Being able to switch between these forms is the foundation for ordering values and solving percentage problems.
百分数意为“每一百份”，所以 75% 表示 100 份中的 75 份。能够在这三种形式之间切换是比较大小和解决百分数问题的基础。
3 ÷ 4 = 0.75 = 75%
2. Converting Fractions to Decimals | 分数化小数
To convert a fraction to a decimal, divide the numerator by the denominator. For example, 1/8 becomes 0.125 because 1 divided by 8 is 0.125.
将分数化为小数，用分子除以分母。例如，1/8 化为 0.125，因为 1 除以 8 等于 0.125。
Some fractions produce terminating decimals that stop after a finite number of digits. Others produce recurring decimals with a repeating pattern, such as 0.666... for 2/3. In the exam, write recurring decimals with a clear repeating notation.
有些分数会得到有限小数，小数点后有限位就停止。另一些分数则产生循环小数，例如 2/3 化为 0.666...。在考试中，请使用清晰的循环节表示法。
1 ÷ 8 = 0.125
2 ÷ 3 = 0.666...
3. Converting Decimals to Percentages | 小数化百分数
Multiply the decimal by 100 and add the % sign. This is equivalent to moving the decimal point two places to the right.
将小数乘以 100 并加上 % 号，相当于把小数点向右移动两位。
Be careful with small decimals: 0.07 is 7%, not 70%. Also, 0.7 is 70% because the decimal point moves two places and a zero is added.
对于较小的小数要格外小心：0.07 是 7%，不是 70%。另外，0.7 是 70%，因为小数点右移两位后需要补一个零。
0.45 × 100 = 45%
0.07 × 100 = 7%
4. Converting Percentages to Fractions and Decimals | 百分数化分数与小数
To convert a percentage to a fraction, write it over 100 and simplify if possible. For example, 45% equals 45/100, which simplifies to 9/20.
将百分数化为分数，先写成分母为 100 的分数，然后尽可能化简。例如，45% 等于 45/100，化简后为 9/20。
To convert a percentage to a decimal, divide by 100, which moves the decimal point two places left. Mixed number percentages should be written as improper fractions first.
将百分数化为小数，除以 100，即将小数点左移两位。带分数的百分数需要先化为假分数。
| Percentage | Fraction | Decimal |
| 25% | 1/4 | 0.25 |
| 10% | 1/10 | 0.1 |
| 12.5% | 1/8 | 0.125 |
5. Ordering Fractions, Decimals and Percentages | 排序
To compare values, convert all of them into the same form, usually decimals or percentages. Write each decimal to the same number of decimal places so that place value is clear.
比较大小时，将所有数值转换为同一种形式，通常是小数或百分数。将每个小数写成相同的小数位数，以便清晰地比较位值。
For example, compare 0.4, 3/8, 41% and 0.39. As decimals they become 0.400, 0.375, 0.410 and 0.390. Therefore the ascending order is 3/8, 0.39, 0.4, 41%.
例如，比较 0.4、3/8、41% 和 0.39。化为小数后分别是 0.400、0.375、0.410 和 0.390。因此从小到大排列为 3/8、0.39、0.4、41%。
0.4, 3/8, 41%, 0.39 → 0.400, 0.375, 0.410, 0.390
6. Percentage Increase and Decrease | 百分数增减
To increase a quantity by a percentage, multiply by (1 + r/100). To decrease a quantity by a percentage, multiply by (1 - r/100). The number inside the brackets is called the multiplier.
将某量增加一个百分数，乘以 (1 + r/100)；减少一个百分数，乘以 (1 - r/100)。括号内的数称为乘数。
Do not simply add or subtract the percentage amount unless the question asks only for the actual change. The multiplier method keeps calculations clear and consistent.
除非题目只要求实际变化量，否则不要简单地把百分数加到原数上。乘数法可以使计算更加清晰和一致。
Increase 80 by 15%: 80 × 1.15 = 92
Decrease 80 by 15%: 80 × 0.85 = 68
7. Reverse Percentages | 反向百分数
If a value after a percentage change is given, the original amount is found by dividing by the multiplier. If the final price includes 20% tax, divide by 1.20. If the value is after a 15% decrease, divide by 0.85.
如果已知百分数变化后的值，原值可通过除以乘数求得。若最终价格包含 20% 税，除以 1.20；若数值是减少 15% 后的结果，则除以 0.85。
Final = Original × multiplier ⇒ Original = Final ÷ multiplier
Example: a price of $92 after a 15% increase gives the original price as 92 ÷ 1.15 = $80.
例题：价格上涨 15% 后为 92 美元，则原价为 92 ÷ 1.15 = 80 美元。
8. Simple Interest | 单利
e I = P × r × t, where P is principal, r is the rate as a decimal, and t is time in years.
单利仅按原始本金计算。使用 I = P × r × t，其中 P 为本金，r 为年利率（小数），t 为年数。
The total amount after the investment period is the principal plus interest, P + I. Simple interest is common in IGCSE questions involving bank savings or loans.
投资期结束后的总额为本金加利息，即 P + I。单利常见于 IGCSE 中涉及银行存款或贷款的问题。
I = P × r × t
$500 at 4% for 3 years: I = 500 × 0.04 × 3 = $60
9. Compound Interest | 复利
he principal plus previously earned interest. The amount after n years is A = P(1 + r/100)n.
复利在本金加上已获得利息的基础上继续生息。n 年后的金额为 A = P(1 + r/100)n。
Use n as the number of compounding periods and adjust r if interest is compounded monthly or quarterly. If interest is compounded monthly, divide r by 12 and multiply n by 12.
注意 n 是复利计息次数，如果按月或按季度复利，需要相应调整 r。如果按月复利，将 r 除以 12，将 n 乘以 12。
A = P(1 + r/100)n
$500 at 4% for 3 years: A = 500 × 1.04³ ≈ $562.43
10. Common Exam Pitfalls | 常见考试误区
Common mistakes include confusing 0.5% with 5%, forgetting to simplify fractions, adding percentage changes instead of multiplying, and using the final value instead of the original in reverse percentage questions.
常见错误包括混淆 0.5% 和 5%、忘记化简分数、用加法处理连续的百分数变化而不是乘法、以及在反向百分数问题中误用最终值而不是原值。
Always check whether a percentage is based on which amount and whether the question asks for the change or the final amount. Write a clear method line before calculating.
始终检查百分数是基于哪一数量，以及题目要求的是变化量还是最终值。计算前写出清晰的方法步骤。
- Confusing decimal place: 0.5% is not 5%.
- Using wrong multiplier: decrease by 15% means multiply by 0.85, not by 0.15.
- Misinterpreting compound period: adjust both rate and number of periods.
11. Calculator Skills | 计算器技巧
Use the fraction key to keep answers exact. For compound interest, use the power key. Store the multiplier in memory or use the ANS key for repeated percentage increases.
使用分数键保持答案精确。计算复利时使用幂键。重复百分数增加时，可将乘数存入记忆或使用 ANS 键。
Round only at the final answer unless a question specifies otherwise. Intermediate rounding can cause significant errors in compound interest and reverse percentage problems.
除非题目另有规定，最后结果才进行四舍五入。中间步骤四舍五入可能在复利和反向百分数问题中造成明显误差。
1.05³ = 1.157625
12. Exam-style Examples | 考试风格例题
Example 1: A coat costs $80 plus 15% VAT. Find the total price. Solution: the multiplier is 1.15, so total = 80 × 1.15 = $92.
例题 1：一件外套价格为 80 美元，另加 15% 增值税。求总价。解：乘数为 1.15，所以总价 = 80 × 1.15 = 92 美元。
Example 2: After a 25% reduction, a bag costs $45. Find the original price. Solution: the multiplier for a 25% reduction is 0.75, so original = 45 ÷ 0.75 = $60.
例题 2：降价 25% 后，一个包售价 45 美元。求原价。解：降价 25% 的乘数为 0.75，所以原价 = 45 ÷ 0.75 = 60 美元。
Example 3: A salary grows by 5% each year for two years. Starting at $24000, find the new salary. Multiply by 1.05 twice: 24000 × 1.05² = 24000 × 1.1025 = $26460.
例题 3：年薪每年增长 5%，持续两年。起薪为 24000 美元，求新薪金。乘以 1.05 两次：24000 × 1.05² = 24000 × 1.1025 = 26460 美元。
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