📚 Fractions, Decimals and Percentages: IGCSE Core Number Skills | 分数、小数与百分比：IGCSE核心数字技能
This revision guide brings together the essential IGCSE number skills for fractions, decimals and percentages. These three representations are connected, and mastering the conversions between them will help you solve exam questions accurately and efficiently, with or without a calculator.
本复习指南汇集了 IGCSE 数字部分关于分数、小数和百分比的核心技能。这三种表示方法相互关联，掌握它们之间的转换将帮助你准确、高效地解答考试题目，无论是否使用计算器。
Each section below pairs an English explanation with a Chinese explanation, so you can revise the same idea in either language. Work through the examples with a pen and paper rather than only reading them.
下文每一节都将英文讲解与中文讲解配对呈现，方便你用任意一种语言复习同一个知识点。请用纸笔跟着例题演算，而不要只是阅读。
1. Why Fractions, Decimals and Percentages Matter | 为什么分数、小数与百分比重要
In IGCSE Mathematics, fractions, decimals and percentages are three equivalent ways of describing parts of a whole. Being able to switch quickly between them is an essential number skill, and it is assumed knowledge in almost every later topic.
在 IGCSE 数学中，分数、小数和百分比是描述整体部分的三种等价方式。快速在它们之间转换是一项核心数字技能，而且几乎在后续所有主题中都被视为必备基础。
You will meet these forms in probability, ratio, compound interest, mensuration and data handling. A probability answer of 3/8 may need to be written as 0.375 or 37.5%; a ratio of 3 : 5 may need to be expressed as the fraction 3/8; an interest rate of 4% must be converted to 1.04 before it can be used as a multiplier.
你会在概率、比、复利、测量和数据处理等主题中遇到这些形式。概率答案 3/8 可能需要写成 0.375 或 37.5%；比 3 : 5 可能需要表示为分数 3/8；利率 4% 必须先转成 1.04 才能作为乘数使用。
Strong number skills give you a reliable foundation for problem solving across the whole syllabus. They also protect you from the most common source of lost marks in Paper 1 and Paper 2: careless conversion errors rather than genuine misunderstandings of the method.
扎实的数字技能为你在整个课程体系中解决问题提供了可靠基础。它们还能帮你避开 Paper 1 和 Paper 2 中最常见的失分来源：粗心的转换错误，而不是对方法本身的理解错误。
2. Equivalent Forms and the Key Conversions | 等价形式与关键转换
Some conversions must be known instantly, without working. The table below shows common equivalents that appear regularly in IGCSE papers, and these are worth memorising because they save time and reduce errors.
有些转换必须能够不假思索地立即写出。下表列出了 IGCSE 试卷中经常出现的常见等价关系，值得记忆，因为它们既省时又能减少错误。
| Fraction 分数 | Decimal 小数 | Percentage 百分比 |
| 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% |
| 1/10 | 0.1 | 10% |
| 1/20 | 0.05 | 5% |
| 1/3 | 0.333... | 33⅓% |
| 2/3 | 0.666... | 66⅔% |
The three basic rules cover everything else. To convert a fraction to a decimal, divide the numerator by the denominator. To convert a decimal to a percentage, multiply by 100. To convert a percentage to a decimal, divide by 100.
三条基本规则可以覆盖其余所有情况：分数化小数，用分子除以分母；小数化百分比，乘以 100；百分比化小数，除以 100。
Fraction → ÷ → Decimal → ×100 → Percentage
Percentage → ÷100 → Decimal → × denominator → Fraction
Notice that a decimal can always be written as a fraction over 10, 100, 1000 and so on, which gives you a reliable route back from any terminating decimal to an exact fraction.
注意：小数总能写成以 10、100、1000 等为分母的分数，这为你从任何有限小数回到精确分数提供了可靠路径。
3. Fractions: Proper, Improper and Mixed Numbers | 分数：真分数、假分数与带分数
A fraction has a numerator on top and a denominator below. In a proper fraction the numerator is smaller than the denominator, for example 3/7. In an improper fraction the numerator is larger, for example 9/7. A mixed number combines a whole number with a proper fraction, for example 1 2/7.
分数由分子在上、分母在下组成。真分数的分子小于分母，例如 3/7；假分数的分子大于分母，例如 9/7；带分数由一个整数与一个真分数组成，例如 1 2/7。
To reverse, divide the numerator by the denominator: 7 ÷ 2 = 3 remainder 1, giving 3 1/2.
把 3 1/2 化为假分数：用整数乘分母再加分子，3 × 2 + 1 = 7，所以是 7/2。反过来，用分子除以分母：7 ÷ 2 = 3 余 1，得到 3 1/2。
For calculations, improper fractions are usually safer than mixed numbers, because multiplication and division become straightforward. Convert to a mixed number only if the question asks for the answer in that form, or if the context makes it clearer.
计算时，假分数通常比带分数更安全，因为乘除法会变得直接。只有在题目要求以带分数作答，或情境中带分数更清晰时，才转回带分数。
- Proper fraction 真分数: numerator < denominator, value between 0 and 1
- Improper fraction 假分数: numerator ≥ denominator, value ≥ 1
- Mixed number 带分数: whole number + proper fraction
4. Simplifying and Equivalent Fractions | 约分与等值分数
u create an equivalent fraction by multiplying or dividing the numerator and denominator by the same non-zero number, which does not change the value.
等值分数表示相同的值：1/2 = 2/4 = 5/10 = 50/100。用同一个非零数同时乘或除分子和分母，就得到等值分数，其数值不变。
l fraction can be reduced further: 3 and 4 share no common factor, so 3/4 is in its simplest form.
约分时，用最大公因数（HCF）同时除分子和分母。对于 36/48，36 与 48 的 HCF 是 12，所以 36/48 = 3/4。务必检查结果是否还能继续约分：3 与 4 没有公因数，因此 3/4 已是最简形式。
A crucial warning for exam technique: you may only cancel common factors when multiplying. In a sum such as 1/2 + 3/4 you must not 'cancel' the 2 and the 4 across the addition sign. Simplifying applies to a fraction as a single quantity, not to separate terms joined by + or −.
一个关键的考试技巧提醒：只有在相乘时才可以约去公因式。在 1/2 + 3/4 这样的加法中，绝不能跨过加号把 2 和 4 "约掉"。约分针对的是作为一个整体的分数，而不是被 + 或 − 连接的独立各项。
5. The Four Operations with Fractions | 分数的四则运算
1/4, the lowest common denominator is 12: 2/3 = 8/12 and 1/4 = 3/12, so the sum is 11/12.
分数加减法需要通分。对于 2/3 + 1/4，最小公分母是 12：2/3 = 8/12，1/4 = 3/12，所以和为 11/12。
fraction is the same as multiplying by its reciprocal: 3/5 ÷ 2/7 = 3/5 × 7/2 = 21/10 = 2 1/10.
乘法更简单：分子乘分子，分母乘分母。对于 2/3 × 9/10，先约分（2 与 10，3 与 9）可得 3/5。除以一个分数等于乘以它的倒数：3/5 ÷ 2/7 = 3/5 × 7/2 = 21/10 = 2 1/10。
- Add / subtract 加减: find a common denominator, then work with the numerators only
- Multiply 乘: numerator × numerator, denominator × denominator; cancel first if possible
- Divide 除: keep the first fraction, change ÷ to ×, flip the second fraction
Two checks worth building into your routine: after addition the answer should be larger than either fraction, and after division the answer should be larger than the first fraction if the divisor is less than 1.
有两个值得养成的检查习惯：加法结果应大于其中任何一个分数；当除数小于 1 时，除法结果应大于被除数。
6. Converting Fractions to Decimals | 分数化为小数
ethod always works, but long division without a calculator requires care with place value.
把分数化为小数，就用分子除以分母。对于 3/8，计算 3 ÷ 8 = 0.375。这种方法总是有效，但在不能用计算器时做长除法，需要小心数位。
of 100 or 1000, then read off the decimal: 7/20 = 35/100 = 0.35, and 9/25 = 36/100 = 0.36.
如果分母是 10、100 或 1000 的因数，则有更快的路径：同时乘分子分母，化成分母为 100 或 1000 的分数，再直接读出小数：7/20 = 35/100 = 0.35，9/25 = 36/100 = 0.36。
prime factors are all 2s and 5s. So 1/8 = 0.125 terminates (8 = 2 × 2 × 2), while 1/3 = 0.333... and 5/6 = 0.8333... do not.
你还可以预判一个分数是否化为有限小数。把分母写成质因数形式：只有当质因数全是 2 和 5 时，小数才有限。所以 1/8 = 0.125 有限（8 = 2 × 2 × 2），而 1/3 = 0.333... 与 5/6 = 0.8333... 不有限。
7. Converting Decimals to Fractions | 小数化为分数
The number of decimal places tells you the denominator: one place means tenths, two places mean hundredths, three places mean thousandths, and so on. Then simplify.
小数位数决定分母：一位表示十分之几，两位表示百分之几，三位表示千分之几，依此类推。然后再约分。
0.36 = 36/100 = 9/25. Similarly 0.125 = 125/1000 = 1/8, and 2.75 = 275/100 = 11/4 = 2 3/4.
例如 0.36 有两位小数，所以 0.36 = 36/100 = 9/25。类似地 0.125 = 125/1000 = 1/8，2.75 = 275/100 = 11/4 = 2 3/4。
es need one extra step. For 0.4 written as a recurring decimal, or for values such as 0.15%, convert to a plain decimal first. Remember that 0.15% means 0.15 ÷ 100 = 0.0015, which is 15/10000 = 3/2000.
混合情形需要额外一步。例如把循环小数形式的 0.4，或把 0.15% 这类值，先化成普通小数。记住 0.15% 的意思是 0.15 ÷ 100 = 0.0015，即 15/10000 = 3/2000。
Percentages to fractions are the same idea with a fixed denominator of 100: 45% = 45/100 = 9/20, and 12.5% ply the numerator and denominator by 10 first if the percentage has a decimal point in it.
百分比化分数是同一思路，只需固定分母为 100：45% = 45/100 = 9/20，12.5% = 12.5/100 = 125/1000 = 1/8。若百分比本身带小数点，先把分子分母同乘 10。
8. Percentages and Their Link to Fractions and Decimals | 百分比与分数、小数的联系
The word 'per cent' means 'out of 100'. So a percentage is simply a fraction with denominator 100 written in shorthand: 37% = 37/100 = 0.37. This single idea explains every conversion in this guide.
"百分比"的意思是"每一百"。所以百分比只是把分母为 100 的分数写成简记形式：37% = 37/100 = 0.37。这一个观念就能解释本指南中的所有转换。
centage of an amount, convert the percentage to a decimal or fraction and multiply. For 15% of 240: 0.15 × 240 = 36, or alternatively 240 ÷ 100 × 15 = 36.
求一个量的百分之几，先把百分比化成小数或分数再相乘。求 240 的 15%：0.15 × 240 = 36，也可以 240 ÷ 100 × 15 = 36。
fraction and multiply by 100. If 18 out of 24 students pass, then 18/24 = 3/4 = 0.75 = 75%. Always identify the 'whole' correctly: it is the quantity that follows the word 'of' or 'out of'.
把一个量表示为另一个量的百分比，先写成分数再乘 100。24 名学生中有 18 人及格，则 18/24 = 3/4 = 0.75 = 75%。一定要正确识别"整体"，即跟着 "of" 或 "out of" 的那个量。
9. Percentage Change, Increase, Decrease and Reverse Percentages | 百分比变化、增减与逆运算
A percentage increase or decrease is handled with a single multiplier. An increase of 20% uses 1 + 0.20 = 1.20; a decrease of 35% ying once is faster and more accurate than finding the change and adding it on separately.
百分比的增加或减少可用一个乘数完成。增加 20% 使用 1 + 0.20 = 1.20；减少 35% 使用 1 − 0.35 = 0.65。一次乘法比先算变化量再加回去更快也更准确。
For example, a price of £80 increased by 15% becomes 80 × 1.15 = £92. A mass of 250 g decreased by 12% becomes 250 × 0.88 = 220 g.
例如，£80 的价格上涨 15% 后为 80 × 1.15 = £92；250 g 的质量减少 12% 后为 250 × 0.88 = 220 g。
Reverse percentages are the most commonly mis-answered type. If a price becomes £84 after a 20% increase, the original price is found by dividing, not subtracting: 84 ÷ 1.20 = £70. Subtracting 20% of 84 would wrongly give £67.20, because the 20% was calculated on the original £70, not on £84.
逆百分比是最容易答错的一类题。如果某价格在上涨 20% 后变为 £84，原价应通过除法而非减法得到：84 ÷ 1.20 = £70。若用 84 减去 84 的 20%，会错误地得到 £67.20，因为那 20% 是按原价 £70 计算的，而不是按 £84。
Compound interest applies a multiplier repeatedly. £500 invested at 4% per year for 3 years gives 500 × 1.04³ = 500 × 1.124864 = £562.43 to the nearest penny. Simple interest instead adds 4% of £500 three times, giving 500 + 3 × 20 = £560.
复利是把乘数重复使用。£500 以年利率 4% 存 3 年，终值为 500 × 1.04³ = 500 × 1.124864 = £562.43（精确到便士）。单利则是把 £500 的 4% 加三次，得 500 + 3 × 20 = £560。
10. Recurring Decimals and Comparing Values | 循环小数与数值比较
A recurring decimal repeats a block of digits forever, for example 0.333..., 0.1666... or 0.142857142857... Standard notation places a dot over the first and last digit of the repeating block, but in written work the ellipsis form 0.8333... is easier to read.
循环小数会无限重复一段数字，例如 0.333...、0.1666... 或 0.142857142857...。
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