📚 Fractions, Decimals and Percentages: IGCSE Number Skills | IGCSE 数学：分数、小数与百分数核心技能
In IGCSE Mathematics, fractions, decimals and percentages are not separate topics; they are three ways of representing the same numerical idea. A single value such as one half can be written as 1/2, as 0.5, or as 50%, and every method in this topic is really about moving fluently between those three forms. This revision guide connects the key conversion methods, the arithmetic of fractions, percentage multipliers, reverse percentages, compound interest, and the common pitfalls that cost marks in both Core and Extended papers.
在 IGCSE 数学中，分数、小数和百分数并不是三个孤立主题，而是同一个数值概念的三种表达方式。比如同一个"一半"，可以写成 1/2、0.5 或 50%，本专题的所有方法本质上都是在熟练地在这三种形式之间转换。本复习指南串联核心换算方法、分数运算、百分数乘数、逆向百分数、复利计算，以及 Core 与 Extended 试卷中常见的失分陷阱。
1. Three Ways of Writing the Same Value | 同一数值的三种写法
mber of parts per hundred. Because 100 = 10², percentages sit naturally between the other two forms: a percentage is simply a fraction with denominator 100, which converts to a decimal by moving the decimal point two places.
分数用两个整数表示"部分与整体"的关系，小数用十进制位值写出数值，百分数则把数值表示为"每一百份中的多少份"。由于 100 = 10²，百分数天然处在另外两者之间：百分数就是分母为 100 的分数，把它化成小数只需把小数点向左移动两位。
The table below lists the equivalences you should recognise instantly in an exam, without reaching for a calculator.
下表列出了考试中应当"一眼识别"的常见等值关系，无需借助计算器。
| Fraction 分数 | Decimal 小数 | Percentage 百分数 |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 1/8 | 0.125 | 12.5% |
| 1/3 | 0.333... | 33.3% (1 d.p.) |
Notice that 1/8 = 0.125 = 12.5%, not 1.25% to a percentage is one of the most frequent single-mark losses in the whole number strand.
注意 1/8 = 0.125 = 12.5%，而不是 1.25%。在"小数化百分数"时点错小数点，是整个数与代数板块中最常见的单分失分点之一。
2. Equivalent Fractions and Simplest Form | 等值分数与最简分数
An equivalent fraction is obtained by multiplying or dividing both the numerator and denominator by the same non-zero integer. The value of the fraction stays unchanged because you are effectively multiplying by 1. To write a fraction in its simplest form, divide both parts by their highest common factor (HCF).
等值分数是将分子和分母同时乘或除以同一个非零整数得到的。分数的值不变，因为本质上乘的是 1。要把分数化为最简形式，需要将分子和分母同时除以它们的最大公因数 HCF。
A mixed number such as 2 3/5 can also be written as an improper fraction by multiplying the whole number by the denominator and adding the numerator, keeping the same denominator.
带分数如 2 3/5 也可以化为假分数，方法是用整数部分乘以分母再加上分子，分母保持不变。
2 3/5 = (2 × 5 + 3) / 5 = 13/5
The reverse direction matters too: to change 13/5 back into a mixed number, divide 13 by 5 to get 2 with remainder 3, giving 2 3/5. Extended papers often expect the answer as a mixed number when the question is phrased in ordinary language such as "hours" or "kilograms".
反方向同样重要：把 13/5 化回带分数，用 13 除以 5 得商 2 余 3，即 2 3/5。当题目用"小时""千克"等日常单位提问时，Extended 试卷通常期望答案写成带分数形式。
Example: Simplify 18/24.
例题：化简 18/24。
18 ÷ 6 / 24 ÷ 6 = 3/4
The HCF of 18 and 24 is 6, so the simplest form is 3/4. If you cannot spot the HCF immediately, cancel repeatedly by common factors (2, then 3) until no further common factor remains - the final answer is identical and full marks are still awarded.
18 和 24 的最大公因数是 6，因此最简形式是 3/4。如果一时看不出最大公因数，可以反复用公因数约分（先除 2，再除 3），直到没有公因数为止，最终结果完全相同，同样可以获得满分。
3. Converting Between Fractions, Decimals and Percentages | 分数、小数与百分数的互化
ply by 100, which moves the decimal point two places to the right. Percentage to decimal: divide by 100. Percentage to fraction: write it over 100 and simplify. Decimal to fraction: use place value, so 0.36 = 36/100 = 9/25.
分数化小数：分子除以分母。分数化百分数：分数乘以 100。小数化百分数：乘以 100，即小数点向右移动两位。百分数化小数：除以 100。百分数化分数：写成分母 100 的分数再约分。小数化分数：利用位值，例如 0.36 = 36/100 = 9/25。
Worked example: express 7/20 as a decimal and as a percentage, and express 0.85 as a fraction in its simplest form.
例题：把 7/20 化成小数和百分数，并把 0.85 化成最简分数。
7 ÷ 20 = 0.35 = 35% | 0.85 = 85/100 = 17/20
Extended candidates should also handle a percentage with a decimal part, such as 12.5%, which is 0.125 or 1/8. Treat it exactly like any other percentage: divide by 100 first, then simplify the resulting fraction.
Extended 考生还需处理带小数的百分数，例如 12.5%，它等于 0.125 或 1/8。处理方法与普通百分数完全相同：先除以 100，再对得到的分数约分。
4. Terminating and Recurring Decimals | 有限小数与循环小数
A fraction in simplest form produces a terminating decimal if and only if its denominator has no prime factors other than 2 and 5. If the denominator contains any other prime factor, such as 3, 7 or 11, the decimal will be recurring.
一个最简分数的分母如果只含有质因数 2 和 5，它就化成有限小数；如果分母还含有其他质因数（如 3、7、11），则一定化成循环小数。
- 1/2 = 0.5 and 1/5 = 0.2 are terminating decimals.
- 1/3 = 0.333... and 1/6 = 0.1666... are recurring decimals.
- 1/7 = 0.142857142857..., a repeating block of six digits.
- 1/2 = 0.5、1/5 = 0.2 是有限小数。
- 1/3 = 0.333...、1/6 = 0.1666... 是循环小数。
- 1/7 = 0.142857142857...，循环节为六位数字。
When a question asks you to write a recurring decimal, examiners accept the standard dot notation or three or more repeated digits followed by an ellipsis. Always give enough digits: writing 0.33 for 1/3 is usually penalised, whereas 0.333... is accepted.
当题目要求写出循环小数时，考官接受标准的循环点写法，或写出三位以上重复数字再加省略号。务必给出足够位数：把 1/3 写成 0.33 通常会被扣分，写成 0.333... 才可接受。
5. Comparing and Ordering Values | 大小比较与排序
To place a mixture of fractions, decimals and percentages in order, convert every value into the same form. A decimal is usually the safest common form because it can be compared digit by digit, but percentages work equally well when the numbers are close.
要把分数、小数和百分数混合排序，应先把所有数值统一成同一种形式。通常小数是最安全的统一形式，因为可以逐位比较；当数值彼此接近时，统一成百分数也同样方便。
Example: arrange 3/5, 0.58, 62% and 5/9 in ascending order.
例题：把 3/5、0.58、62% 和 5/9 按从小到大排列。
3/5 = 0.6 0.58 62% = 0.62 5/9 = 0.555... → 5/9 < 0.58 < 3/5 < 62%
Note that 3/5 and 0.58 look almost identical as decimals, so keep at least three decimal places while comparing to avoid a careless reversal. Rounding too early is a classic source of ordering errors.
注意 3/5 与 0.58 的小数形式非常接近，比较时至少要保留三位小数，以免粗心颠倒顺序。过早四舍五入是排序题最典型的错误来源。
6. Fractions of Amounts and the Four Operations | 分数的取值与四则运算
The word "of" ternative method, 3/4 × 240, gives the same result and is often quicker with a calculator.
"of"（的）在数学中表示乘。求 240 的 3/4，可算 240 ÷ 4 × 3 = 180；另一种方法 3/4 × 240 结果相同，用计算器时往往更快。
- Addition and subtraction: rewrite with a common denominator, then combine the numerators.
- Multiplication: multiply numerators together and denominators together; cancel common factors first to keep numbers small.
- Division: keep the first fraction, change ÷ to ×, and flip the second fraction.
- Mixed numbers: convert to improper fractions before multiplying or dividing.
- 加减法：先通分，再合并分子。
- 乘法：分子乘分子、分母乘分母；先约分可以让数字更小、更不易出错。
- 除法：保持第一个分数不变，把 ÷ 改为 ×，并把第二个分数取倒数。
- 带分数：在乘除之前先化为假分数。
Worked example: evaluate 2/3 + 3/4 and 5/6 ÷ 2/9.
例题：计算 2/3 + 3/4 与 5/6 ÷ 2/9。
2/3 + 3/4 = 8/12 + 9/12 = 17/12 = 1 5/12
5/6 ÷ 2/9 = 5/6 × 9/2 = 45/12 = 15/4 = 3 3/4
Always leave the final answer in its simplest form, and convert improper fractions to mixed numbers when the context makes that more natural. Showing the intermediate common denominator can also earn method marks if a slip occurs later.
最终答案务必化为最简形式；当情境更自然时，把假分数写成带分数。写出通分的中间步骤，即使后面算错，也可能获得方法分。
7. Percentages of Amounts: Multipliers | 百分数取值与乘数法
The most reliable technique is the multiplier method. A percentage increase of 20% corresponds to multiplying by 1.20; a percentage decrease of 15% corresponds to multiplying by 0.85, because 100% − 15% = 85%. This avoids the two-step error of finding the change and then adding or subtracting it.
最可靠的方法是乘数法。增加 20% 相当于乘以 1.20；减少 15% 相当于乘以 0.85，因为 100% − 15% = 85%。这样可以避免"先求变化量、再加减"两步走带来的错误。
Example: increase £240 by 15%, then decrease the result by 15%.
例题：把 240 英镑增加 15%，再把结果减少 15%。
240 × 1.15 = £276 → 276 × 0.85 = £234.60
The final value is not £240: a 15% increase followed by a 15% decrease never returns to the starting amount, because the second percentage applies to a larger base. Examiners deliberately set this up as a misconception question.
最终结果并不是 240 英镑：先增加 15% 再减少 15% 绝不会回到原值，因为第二次百分数是作用在更大的基数上。考官常特意用这一点设计"概念误区"题。
For simple percentage-of questions, write the percentage as a decimal and multiply: 45% of 320 = 0.45 × 320 = 144. For percentage change, use the formula below.
对于简单的"求百分数之值"问题，把百分数写成小数再相乘：320 的 45% = 0.45 × 320 = 144。求百分数变化则使用下面的公式。
Percentage change = (Change ÷ Original) × 100%
8. Reverse Percentages: Finding the Original Amount | 逆向百分数：求原始值
In a reverse percentage problem you know the final value and the percentage change, and you must recover the original amount. The key idea is that the final value equals the original multiplied by a multiplier, so the original equals the final value divided by that multiplier.
在逆向百分数问题中，已知最终值和百分数变化，要求还原原始值。核心思路是：最终值 = 原始值 × 乘数，因此原始值 = 最终值 ÷ 乘数。
Example: after a 20% discount, a jacket costs £64. Find the original price.
例题：打折 20% 后，一件外套售价 64 英镑，求原价。
Original = 64 ÷ 0.80 = £80
A frequent error is to add 20% of £64 to £64, giving £76.80. That is wrong because the 20% discount was calculated from the original price, not from the sale price, so the base is different. Ask yourself each time: "20% of what?"
最常见的错误是把 64 英镑的 20% 加到 64 上，得到 76.80 英镑。这是错的，因为 20% 的折扣是按原价计算的，而不是按售价计算，两者的基数不同。每次做题都要问自己："这个 20% 是相对于谁？"
Reverse percentages also appear in Value Added Tax, profit and loss, population change and exam-style "the price before the sale" questions. In every case, identify the multiplier first, then divide.
逆向百分数还出现在增值税、利润与亏损、人口变化以及"打折前价格"等题型中。无论哪种情境，都先确定乘数，再做除法。
9. Simple and Compound Interest, Growth and Decay | 单利、复利与增长衰减
ars is P × r/100 × n. Compound interest is calculated on the running balance, so each year's interest is slightly larger than the previous year's.
单利只按原始本金计算利息，因此 n 年后的利息总额为 P × r/100 × n。复利按当期余额计算利息，所以每一年的利息都比上一年多。
Simple: I = P × r/100 × n | Compound: A = P × (1 + r/100)ⁿ
Example: £2000 is invested at 4% per year for 3 years. Compare simple and compound interest.
例题：2000 英镑按年利率 4% 投资 3 年，比较单利与复利。
Simple: 2000 × 1.12 = £2240 | Compound: 2000 × 1.04³ = £2249.73
Depreciation and decay work the same way but with a multiplier below 1: a car losing 12% of its value each year is worth P × 0.88ⁿ after n years. A common error is to calculate 12% × n and subtract it once, which is simple rather than compound decay.
折旧与衰减的算法相同，只是乘数小于 1：一辆车每年贬值 12%，n 年后价值为 P × 0.88ⁿ。常见错误是把 12% × n 一次性减掉，那是单利式衰减而非复利式衰减。
Round money answers to two decimal places, and never round the intermediate power - keep full accuracy until the final line.
货币答案四舍五入到两位小数；中间的幂运算不要提前取近似值，应保持完整精度直到最后一行。
10. Common Errors and Exam Technique | 常见错误与应试技巧
Most lost marks in this topic come from a small set of repeated mistakes rather than from a lack of understanding. Reading the question twice - once for the numbers, once for the required form of the answer - prevents most of them.
本专题的失分，大多来自少数几个反复出现的错误，而不是理解上的欠缺。把题目读两遍——第一遍抓数字，第二遍确认答案要求的格式——就能避免其中大部分。
- Writing a percentage without the % sign, or writing 0.35% when the answer is 35%.
- Forgetting to simplify a fraction, or leaving a final answer as an improper fraction when a mixed number is expected.
- Adding percentages directly across compounded changes: 10% then 10% is 21%, not 20%.
- Rounding mid-calculation and losing an accuracy mark at the end.
- Giving a negative or unrealistic answer in a real-life context without checking it against the situation.
- 百分数答案漏写 % 符号，或把 35% 写成 0.35%。
- 忘记约分，或题目要求带分数时却留下假分数。
- 把连续变化的百分数直接相加：先 10% 再 10% 是 21%，不是 20%。
- 计算中途取近似值，导致最后丢失精度分。
- 得到负数或不现实的结果却不结合情境检查。
For "show that" down the multiplier or conversion factor you are using. Even a one-line statement such as "multiplier = 1.15" can secure a method mark, and it makes your reasoning easy for the examiner to follow.
遇到"证明"类和多步骤题目，把你使用的乘数或换算系数写出来。哪怕只是一行"乘数 = 1.15"，也可能拿到方法分，同时让考官容易看懂你的思路。
11. Core and Extended: What the Examiner Expects | Core 与 Extended 的要求差异
Core candidates need secure fluency in converting between the three forms, finding a fraction or percentage of an amount, simple percentage increase and decrease, and straightforward ordering questions. Answers are usually numerical, and a clear two-step method is sufficient.
Core 考生需要熟练掌握三种形式的互化、求某数的分数或百分数、简单的百分数增减，以及基础的大小排序。答案通常为数值，写出清晰的"两步法"即可。
Extended candidates additionally meet reverse percentages with algebraic framing, compound interest and depreciation over n years, recurring decimals expressed algebraically, and percentage problems embedded in ratio, proportion or algebraic contexts. The method must be shown, and the final answer must be in the form requested.
Extended 考生还要应对用代数形式表述的逆向百分数、n 年的复利与折旧、用代数方法表示循环小数，以及嵌入比、比例或代数情境中的百分数问题。这一类题必须写出方法，最终答案必须符合题目要求的格式。
In both tiers, a calculator paper still rewards written method. Write the multiplier or the conversion step before pressing equals, because the mark scheme often allocates one mark for the correct setup and one for the correct value.
无论哪个层级，允许使用计算器的试卷仍然奖励书面方法。按下等号之前先写出乘数或换算步骤，因为评分标准常常是"正确的列式"给一分、"正确的数值"再给一分。
12. Quick Revision Checklist | 快速复习清单
Use this checklist in the final week before the exam. Tick each item only when you can complete it accurately without notes and within a reasonable time.
在考前最后一周使用这份清单。只有当你能够不看笔记、在合理时间内准确完成时，才在该项前打勾。
- Convert fluently between fractions, decimals and percentages, including 1/8, 1/3, 2/3 and 12.5%.
- Simplify fractions using the HCF and convert between mixed numbers and improper fractions.
- Add, subtract, multiply and divide fractions and mixed numbers, leaving answers in simplest form.
- Identify whether a fraction gives a terminating or recurring decimal from its denominator.
- Order a mixed list of values by converting to a common form.
- Use multipliers for percentage increase, decrease and repeated change.
- Solve reverse percentage problems by dividing by the multiplier.
- Apply simple interest, compound interest, growth and depreciation formulae correctly.
- Check units, rounding and the required answer format before finishing.
- 熟练互化分数、小数与百分数，包括 1/8、1/3、2/3 和 12.5%。
- 用最大公因数约分，并在带分数与假分数之间自由转换。
- 完成分数的加减乘除与带分数运算，答案化为最简形式。
- 根据分母判断分数化成有限小数还是循环小数。
- 把混合数值统一形式后排序。
- 用乘数法处理百分数增加、减少与连续变化。
- 用"除以乘数"解决逆向百分数问题。
- 正确套用单利、复利、增长与折旧公式。
- 交卷前检查单位、取近似值与答案格式要求。
Fractions, decimals and percentages reward accuracy and routine more than insight. Build the conversion habits, always write the multiplier, and keep every answer in the form the question demands. That discipline alone converts a fragile topic into reliable marks.
分数、小数与百分数这一专题，更奖励准确与熟练，而非灵光一现。养成换算习惯，坚持写出乘数，并让每个答案符合题目要求的格式。仅凭这份纪律，就能把这个易错专题变成稳定的得分点。
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