📚 Number Skills: Fractions, Decimals and Percentages | IGCSE 数学：分数、小数与百分数
In IGCSE Mathematics, fractions, decimals and percentages are not separate topics; they are three ways of representing the same value. You will meet them in algebra, probability, ratio, statistics and real-life problems. Understanding how to convert, compare and calculate with these forms is a core skill for the Cambridge IGCSE exam.
在 IGCSE 数学中，分数、小数和百分数并不是孤立的主题；它们是同一个值的三种表示方式。你会在代数、概率、比、统计和现实生活问题中遇到它们。掌握这些形式之间的转换、比较和计算是剑桥 IGCSE 考试的核心技能。
1. The Three Forms of a Part-Whole Relationship | 三种表示“部分与整体”的形式
A fraction is written as a/b, where a is the numerator and b is the non-zero denominator. It shows a parts out of b equal parts. For example, 3/4 means 3 parts out of 4 equal parts.
分数写作 a/b，其中 a 是分子，b 是不为零的分母。它表示 b 等份中的 a 份。例如，3/4 表示 4 等份中的 3 份。
A decimal uses a decimal point and place value. For example, 0.75 means 7 tenths and 5 hundredths, or 75 hundredths. The digits after the decimal point represent tenths, hundredths, thousandths and so on.
小数使用小数点和位值。例如，0.75 表示 7 个十分之一和 5 个百分之一，或者说 75 个百分之一。小数点后的数字依次表示十分位、百分位、千分位等。
A percentage means 'per hundred'. The symbol % represents division by 100. Therefore, 45% = 45/100 = 0.45. Percentages are useful for comparing proportions quickly.
百分数表示“每一百份”。符号 % 表示除以 100。因此，45% = 45/100 = 0.45。百分数便于快速比较比例。
2. Converting Between Forms | 三种形式之间的转换
le, 3/8 = 3 ÷ 8 = 0.375. Some fractions produce recurring decimals, such as 1/3 = 0.333...
将分数转换为小数，用分子除以分母。例如，3/8 = 3 ÷ 8 = 0.375。有些分数会产生循环小数，例如 1/3 = 0.333...
Fraction → Decimal: a/b = a ÷ b
Decimal → Percent: 0.375 × 100 = 37.5%
To convert a decimal to a percentage, multiply by 100. For example, 0.375 × 100 = 37.5%. To convert a percentage to a fraction, write it over 100 and simplify. For example, 65% = 65/100 = 13/20.
将小数转换为百分数，乘以 100。例如，0.375 × 100 = 37.5%。将百分数转换为分数，把它写在 100 之上并化简。例如，65% = 65/100 = 13/20。
The table below shows common conversions that appear regularly in IGCSE questions.
下表列出了 IGCSE 题目中经常出现的常见转换。
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/5 | 0.6 | 60% |
| 7/10 | 0.7 | 70% |
3. Ordering and Comparing | 排序与比较
e 2/3 and 3/5, use 15 as the common denominator: 2/3 = 10/15 and 3/5 = 9/15, so 2/3 > 3/5.
比较分数时，先将它们化为同分母。例如，比较 2/3 和 3/5，以 15 为公分母：2/3 = 10/15，3/5 = 9/15，所以 2/3 > 3/5。
Decimals are easier to compare when aligned by place value: compare tenths first, then hundredths, then thousandths. So 0.61 > 0.599 because 61 hundredths is greater than 59.9 hundredths.
小数按照位值对齐后更容易比较：先比较十分位，再比较百分位，然后比较千分位。因此 0.61 > 0.599，因为 61 个百分之一大于 59.9 个百分之一。
When mixed forms appear, convert all values to decimals or percentages first before ordering. This avoids mistakes caused by different denominators.
当题目中同时出现多种形式时，先把所有数值都转换为小数或百分数，再排序。这样可以避免因分母不同而产生的错误。
4. Adding and Subtracting Fractions | 分数的加法与减法
subtract the numerators and keep the denominator unchanged. For example, 2/5 + 1/5 = 3/5.
分数相加或相减时，它们必须有相同的分母。只需将分子相加或相减，分母保持不变。例如，2/5 + 1/5 = 3/5。
lowest common multiple (LCM) of the denominators. Example: 1/3 + 1/4 = 4/12 + 3/12 = 7/12.
如果分母不同，先求分母的最小公倍数（LCM）。例如：1/3 + 1/4 = 4/12 + 3/12 = 7/12。
roper fractions first. For instance, 1 1/2 + 2 1/4 = 3/2 + 9/4 = 6/4 + 9/4 = 15/4 = 3 3/4.
带分数可以先转换为假分数。例如，1 1/2 + 2 1/4 = 3/2 + 9/4 = 6/4 + 9/4 = 15/4 = 3 3/4。
5. Multiplying and Dividing Fractions | 分数的乘法与除法
he denominators together. Simplify before or after multiplying. Example: 2/3 × 4/5 = 8/15.
分数相乘时，分子乘以分子，分母乘以分母。可以在乘法之前或之后进行化简。例如：2/3 × 4/5 = 8/15。
To divide by a fraction, multiply by its reciprocal. This is often remembered as 'keep-change-flip' on, change ÷ to ×, flip the second fraction. Example: 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6.
除以一个分数，等于乘以它的倒数。这常被记作“保持不变-改变符号-翻转分数”：保留第一个分数，将 ÷ 改为 ×，翻转第二个分数。例如：2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6。
r mixed numbers, change to improper fractions first. Example: 1 1/2 ÷ 2 = 3/2 × 1/2 = 3/4.
对于带分数，先转换为假分数。例如：1 1/2 ÷ 2 = 3/2 × 1/2 = 3/4。
6. Calculating with Decimals | 小数的运算
2.3 = 5.75, not 3.68. Writing one number below the other helps keep place values correct.
加减小数时，小数点要对齐。例如，3.45 + 2.3 = 5.75，而不是 3.68。把一个数写在另一个数下方，有助于保持位值正确。
2 = 0.08, because each number has one decimal place, so the answer has two decimal places.
小数相乘时，先忽略小数点，按整数相乘，然后数出题目中小数的总位数，在答案中点上小数点。例如：0.4 × 0.2 = 0.08，因为每个数有一位小数，所以答案有两位小数。
by a power of 10 to make the divisor a whole number. Example: 4.5 ÷ 0.15 = 450 ÷ 15 = 30.
除以小数时，将被除数和除数同时乘以 10 的幂，使除数变成整数。例如：4.5 ÷ 0.15 = 450 ÷ 15 = 30。
7. Percentage of a Quantity | 求一个数的百分之几
a quantity, convert the percentage to a decimal or fraction and multiply. For example, 30% of 240 = 0.30 × 240 = 72.
求一个数的百分之几，先把百分数转换为小数或分数，再进行乘法。例如，240 的 30% = 0.30 × 240 = 72。
Useful decimal multipliers include: 10% = 0.1, 25% = 0.25, 75% = 0.75, 1% = 0.01. These are often faster than writing a fraction over 100.
常用的小数倍乘因子包括：10% = 0.1，25% = 0.25，75% = 0.75，1% = 0.01。这些通常比先写 100 分之几再算更快。
For a quick method, find 10% first by dividing by 10, then scale. Example: 15% of 80 = 10% (8) + 5% (4) = 12.
一种快速方法是先除以 10 求出 10%，再进行缩放。例如，80 的 15% = 10%（8）+ 5%（4）= 12。
8. Percentage Increase and Decrease | 百分数的增加与减少
For an increase of p%, multiply by (1 + p/100). For example, a 15% increase on £60 is 60 × 1.15 = £69.
增加 p% 时，乘以 (1 + p/100)。例如，£60 增加 15% 为 60 × 1.15 = £69。
Increase: New value = Original × (1 + p/100)
Decrease: New value = Original × (1 − p/100)
For a decrease of p%, multiply by (1 − p/100). For example, a 20% decrease on 45 kg is 45 × 0.80 = 36 kg.
减少 p% 时，乘以 (1 − p/100)。例如，45 kg 减少 20% 为 45 × 0.80 = 36 kg。
se percentage problems find the original quantity before a percentage change. If 80 is 125% of the original, then the original is 80 ÷ 1.25 = 64.
反向百分数问题用于求百分数变化之前的原始量。如果 80 是原始值的 125%，那么原始值为 80 ÷ 1.25 = 64。
9. Common Conversions You Must Memorise | 必须记住的常见转换
Some fraction-decimal-percentage conversions appear constantly in IGCSE exams. Memorising them saves time and reduces arithmetic errors.
一些分数-小数-百分数转换在 IGCSE 考试中经常出现。记住它们可以节省时间并减少计算错误。
- 1/2 = 0.5 = 50%
- 1/3 = 0.333... = 33.3%
- 1/4 = 0.25 = 25%
- 1/5 = 0.2 = 20%
- 1/8 = 0.125 = 12.5%
- 3/4 = 0.75 = 75%
- 1/10 = 0.1 = 10%
, you may need to recognise and compare recurring decimals, especially when ordering numbers.
循环小数可以用点号表示。例如，1/3 = 0.333...，2/11 = 0.1818...。在 IGCSE 中，你可能需要识别并比较循环小数，尤其是在数字排序时。
10. Exam Tips and Common Errors | 考试技巧与常见错误
Common mistake: adding fractions without a common denominator. Always check denominators before adding. For example, 1/2 + 1/3 is not 2/5; the correct answer is 5/6.
常见错误：没有通分就直接相加分数。相加前一定要检查分母。例如，1/2 + 1/3 不等于 2/5；正确答案是 5/6。
Common mistake: confusing 0.5% with 50%. Note that 0.5% = 0.005, while 50% = 0.5. Always write the percent symbol clearly and divide by 100 when converting.
常见错误：混淆 0.5% 和 50%。注意 0.5% = 0.005，而 50% = 0.5。转换时一定要写清楚百分号并除以 100。
In multi-step problems, show each conversion clearly. Use a calculator only for final arithmetic, but show your method to gain method marks. This is especially important when a question asks you to give an exact fraction or simplify fully.
在多步骤问题中，要清楚地写出每一步转换。只在最后计算时使用计算器，但要展示解题方法，以获得方法分。当题目要求给出精确分数或完全化简时，这一点尤其重要。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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