📚 Fractions, Decimals and Percentages | 分数、小数与百分比
Numbers appear in many forms, and three of the most common are fractions, decimals and percentages. In IGCSE Mathematics, mastering how to convert between these forms, compare them, and use them in calculations is essential. This article breaks down every key concept you need, with clear reasoning in both English and Chinese, so you can tackle exam questions with confidence.
数字有多种表示形式，其中最常见的三种是分数、小数和百分比。在IGCSE数学中，掌握这三者之间的转换、比较以及运算是必备技能。本文以中英对照的方式，详细解析每一个关键概念，帮助你自信应对考试题目。
1. Understanding the Basics | 基础概念
A fraction represents a part of a whole and is written as a/b, where a is the numerator and b is the denominator. A decimal uses a decimal point to show parts of a whole, based on powers of ten. A percentage is a fraction whose denominator is 100, shown by the % symbol. In IGCSE, you must be fluent with all three, especially in problem-solving contexts like money, ratio and measures.
分数表示整体的一部分，写作 a/b 的形式，其中 a 是分子，b 是分母。小数通过小数点表示部分，基于十的幂次。百分比是分母为 100 的分数，用符号 % 表示。在IGCSE考试中，你需要熟练运用这三种形式，尤其是在涉及金钱、比例和测量的应用题中。
| Form | Example | Meaning |
|---|---|---|
| Fraction | 3/4 | 3 parts out of 4 |
| Decimal | 0.75 | 75 hundredths |
| Percentage | 75% | 75 per hundred |
2. Converting Between Fractions and Decimals | 分数与小数互转
cimal terminates or recurs, you can write it as a fraction by placing the digits over the appropriate power of 10 for terminating decimals, or by using algebraic methods for recurring decimals. In IGCSE, you will often need to convert a fraction to a decimal to compare sizes or to use in money problems.
将分数化为小数，用分子除以分母即可。例如，3/8 = 3 ÷ 8 = 0.375。如果小数是有限小数或循环小数，可以将其写成分数：有限小数直接把小数部分放在 10 的适当次幂上；循环小数则需用代数方法。在IGCSE考试中，经常需要把分数转换为小数来进行大小比较或在货币问题中使用。
Fraction → Decimal: 5/8 = 5 ÷ 8 = 0.625
分数转化为小数：5/8 = 5 ÷ 8 = 0.625
3. Converting Between Fractions and Percentages | 分数与百分比互转
, first turn it into a decimal (numerator ÷ denominator), then multiply by 100 and add the % equivalent fraction with denominator 100. For instance, 7/20 = (7×5)/(20×5) = 35/100 = 35%. To convert a percentage to a fraction, write it over 100 and simplify. So 45% = 45/100 = 9/20. This skill is frequently tested in IGCSE non‑calculator papers.
将分数化为百分比，先化为小数（分子 ÷ 分母），再乘以 100 并加上百分号。如果分母是 100 的因数，也可以直接将分数化为分母为 100 的等值分数。例如，7/20 = (7×5)/(20×5) = 35/100 = 35%。将百分比化为分数，写成分母为 100 的分数再化简。所以 45% = 45/100 = 9/20。这个技能在IGCSE非计算器卷中经常考查。
3/25 = 0.12 → 0.12 × 100 = 12%
3/25 = 0.12 → 0.12 × 100 = 12%
4. Converting Between Decimals and Percentages | 小数与百分比互转
Converting a decimal to a percentage is straightforward: multiply by 100 and add %. Thus, 0.85 = 0.85 × 100 = 85%. To go from a percentage to a decimal, divide by 100. So 6% mal. In IGCSE, this quick conversion is essential for interpreting data, discounts and interest rates.
小数与百分比的转换非常直接：小数乘以 100 加百分号。因此 0.85 = 0.85 × 100 = 85%。百分比转小数则除以 100。所以 6% = 6 ÷ 100 = 0.06。记住，小数点向右移动两位得到百分比，向左移动两位得到小数。在IGCSE考试中，这种快速转换对于理解数据、折扣和利率至关重要。
| Decimal | Percentage |
|---|---|
| 0.2 | 20% |
| 0.05 | 5% |
| 1.25 | 125% |
5. Ordering Fractions, Decimals and Percentages | 比较大小
, decimals and percentages, the safest method is to convert all of them into the same form – usually decimals or percentages. Compare 2/5, 0.45 and 38%. As decimals: 2/5 = 0.4, 0.45 = 0.45, 38% = 0.38. In ascending order: 0.38, 0.4, 0.45 → 38% f question appears frequently in IGCSE, often asking you to place values on a number line.
比较分数、小数和百分比的大小时，最稳妥的方法是把它们全部转换为同一种形式——通常转换为小数或百分比。例如比较 2/5、0.45 和 38%。化为小数：2/5 = 0.4，0.45 = 0.45，38% = 0.38。从小到大排列：0.38, 0.4, 0.45 → 38%, 2/5, 0.45。这类题目在IGCSE中很常见，有时还要求在数轴上标出各值。
Order: 3/10, 0.28, 31% → as decimals: 0.3, 0.28, 0.31 → 0.28, 0.3, 0.31 → 0.28, 3/10, 31%
排序：3/10, 0.28, 31% → 化为小数：0.3, 0.28, 0.31 → 0.28, 0.3, 0.31 → 0.28, 3/10, 31%
6. Adding and Subtracting Fractions | 分数加减
or mixed numbers, convert to improper fractions first. In IGCSE, you may need to simplify your final answer or give it as a mixed number.
分数相加减，必须先化为同分母。如果分母不同，找出分母的最小公倍数（LCM），然后把每个分数通分。例如，1/6 + 3/4，6 和 4 的 LCM 是 12：1/6 = 2/12，3/4 = 9/12。然后相加：2/12 + 9/12 = 11/12。带分数需先化为假分数。在IGCSE考试中，可能要求将最后结果化简或写成带分数形式。
2⅓ + 1¼ = 7/3 + 5/4 = 28/12 + 15/12 = 43/12 = 3 7/12
2⅓ + 1¼ = 7/3 + 5/4 = 28/12 + 15/12 = 43/12 = 3又7/12
7. Multiplying and Dividing Fractions | 分数乘除
fraction (the divisor) and then multiplying. So, 4/9 ÷ 2/3 = 4/9 × 3/2 = 12/18 = 2/3. This 'keep‑change‑flip' rule is a core IGCSE tool.
分数乘法很简单：分子乘分子，分母乘分母。例如，2/5 × 3/7 = (2×3)/(5×7) = 6/35。能化简的一定要化简。带分数先化为假分数。分数除法则是把第二个分数（除数）倒过来，再相乘。所以，4/9 ÷ 2/3 = 4/9 × 3/2 = 12/18 = 2/3。这条“保留、变号、倒数”规则是IGCSE的核心解题工具。
1½ × 2/5 = 3/2 × 2/5 = 6/10 = 3/5
1½ × 2/5 = 3/2 × 2/5 = 6/10 = 3/5
8. Fractions of an Amount | 求一个数的几分之几
In IGCSE word problems, this technique appears in sharing profit, discounts, and recipe adjustments. If the fraction is of a quantity that is already fractional, the same rule applies: multiply the amount by the fraction, simplifying as you go.
求一个数的几分之几，用这个数除以分母，再乘以分子。例如，64 公斤的 3/8：64 ÷ 8 = 8，然后 8 × 3 = 24 公斤。在IGCSE应用题中，这种方法常用于分配利润、计算折扣和调整配方。如果原量本身也是分数，同样适用：用这个量乘以分数，边算边化简。
5/6 of 90 = (90 ÷ 6) × 5 = 15 × 5 = 75
90 的 5/6 = (90 ÷ 6) × 5 = 15 × 5 = 75
9. Percentage of an Amount | 求一个数的百分之几
can be done by converting the percentage to a decimal or fraction and multiplying. For 30% of 250, use 0.30 × 250 = 75, or 30/100 × 250 = 75. A non‑calculator method is to find 10% first (divide by 10) and then scale. 10% of 250 = 25, so 30% th methods because some papers do not allow calculators. Always show your working clearly.
求一个数的百分之几，可以把百分比化成小数或分数再相乘。求 250 的 30%，用 0.30 × 250 = 75，或者 30/100 × 250 = 75。没有计算器时，可先求 10%（除以 10），再按倍数计算。250 的 10% = 25，所以 30% = 3 × 25 = 75。IGCSE考试中部分试卷禁用计算器，因此两种方法都要练习，并清晰展示计算步骤。
| Percentage | Non‑calculator method |
|---|---|
| 5% | Find 10% then halve it |
| 15% | Find 10% and 5%, then add |
| 90% | Find 10% and subtract from 100% |
10. Percentage Increase and Decrease | 百分比增减
To increase an amount by a percentage, add the percentage to 100% to find the multiplier. For a 15% increase, the multiplier is 1.15 (since 100% + 15% = 115% = 1.15). Multiply the original amount by 1.15. For a 20% decrease, the multiplier is 0.80 (100% − 20% = 80% ust the increase/decrease. In IGCSE, this is vital for compound interest and depreciation.
增加一个百分比，用 100% 加上这个百分比得到乘数。例如增加 15%，乘数为 1.15（100% + 15% = 115% = 1.15）。把原数乘以 1.15 即可。减少 20%，乘数为 0.80（100% − 20% = 80% = 0.80）。一定要看清题目问的是最终量还是变化量。在IGCSE中，这个知识点在复利和折旧问题中非常关键。
Increase 200 by 8%: 200 × 1.08 = 216
200 增加 8%：200 × 1.08 = 216
11. Reverse Percentages | 反向百分比
the amount after a percentage change and need to find the original. If a price after a 25% r was 1.25. The original = 125 ÷ 1.25 = 100. For a decrease, if a coat costs 68 after a 15% reverse percentages in sale prices or tax problems, so identify the multiplier carefully.
反向百分比用于已知变化后的量，求原始量。如果价格上涨 25% 后为 125，乘数是 1.25，则原价 = 125 ÷ 1.25 = 100。降价的情形：一件外套降价 15% 后售价 68 元，乘数为 0.85，原价 = 68 ÷ 0.85 = 80。IGCSE考题常把反向百分比隐藏在售价或税务问题中，因此要小心辨认乘数。
After 30% off, a laptop is £420. Original = 420 ÷ 0.70 = £600
降价 30% 后笔记本电脑为 £420。原价 = 420 ÷ 0.70 = £600
12. Real-life Applications | 实际应用
Fractions, decimals and percentages appear throughout IGCSE word problems: recipe scaling (using fractions), currency conversions (decimals), percentage profit and loss, simple and compound interest, and interpreting statistical data. Being able to switch between forms quickly saves time and reduces mistakes. Always read the question carefully to decide which form of the number is most appropriate for the calculation. Show each step of your conversion method clearly for full marks.
分数、小数和百分比贯穿IGCSE各类应用题：食谱调配（分数）、货币换算（小数）、盈亏百分比、单利和复利，以及统计数据的解读。能够在不同形式之间快速切换，可以节省时间并减少错误。解题时需仔细读题，判断哪种数字形式最适合计算。清晰地写出每一步转换过程，才能拿到满分。
Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图