📚 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
To convert a fraction to a decimal, divide the numerator by the denominator. For example, 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
To compare fractions, rewrite them with a common denominator. For example, to compare 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
To add or subtract fractions, they must have the same denominator. Add or 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。
For different denominators, find the 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。
Mixed numbers can be converted to improper 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
To multiply fractions, multiply the numerators together and the 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’: keep the first fraction, 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。
For 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
When adding or subtracting decimals, align the decimal points vertically. For example, 3.45 + 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。
To multiply decimals, ignore the decimal points, multiply as whole numbers, then count the total decimal places in the question and put the decimal point in the answer. Example: 0.4 × 0.2 = 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。
To divide by a decimal, multiply both dividend and divisor 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
To find a percentage of 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。
Reverse 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%
Recurring decimals can be written with a dot notation. For example, 1/3 = 0.333… and 2/11 = 0.1818… In IGCSE, 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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