Fractions, Decimals and Percentages | 分数、小数与百分数

📚 Fractions, Decimals and Percentages | 分数、小数与百分数

Fractions, decimals and percentages are three different ways of expressing parts of a whole. They appear in nearly every IGCSE Mathematics topic, from algebra to statistics, so mastering their interconversion is essential for exam success.

分数、小数和百分数是表示“整体的一部分”的三种不同方式。它们几乎出现在IGCSE数学的所有章节中，从代数到统计，因此掌握它们之间的转换是考试取得高分的关键。

1. Understanding Fractions | 理解分数

A fraction is written as a/b, where a is the numerator (the number of parts we have) and b is the denominator (the total number of equal parts). For example, ¾ means we have 3 parts out of 4 equal parts.

分数写作 a/b 的形式，其中 a 是分子（表示我们取了多少份），b 是分母（表示总共分成的等份数）。例如，¾ 表示从4个等份中取3份。

Proper fractions: numerator < denominator, e.g. ⅔.
真分数：分子小于分母，如 ⅔。
Improper fractions: numerator ≥ denominator, e.g. 7/5.
假分数：分子大于或等于分母，如 7/5。
Mixed numbers: a whole number and a proper fraction, e.g. 1¾.
带分数：一个整数加一个真分数，如 1¾。

To change an improper fraction to a mixed number, divide the numerator by the denominator. The quotient becomes the whole number, the remainder becomes the new numerator.

将假分数化为带分数时，用分子除以分母。商作为整数部分，余数作为新的分子。

7/5 = 1 with remainder 2 → 1⅖

2. Understanding Decimals | 理解小数

A decimal uses place value to show tenths, hundredths, thousandths and so on. The decimal point separates whole numbers from fractional parts. For example, 0.7 means 7 tenths, and 0.23 means 23 hundredths.

小数利用位值表示十分位、百分位、千分位等。小数点将整数部分和小数部分分开。例如，0.7 表示 7 个十分之一，0.23 表示 23 个百分之一。

Decimal	Place value	Fraction
0.1	1 tenth	1/10
0.01	1 hundredth	1/100
0.001	1 thousandth	1/1000

When reading decimals, say “point” for the decimal point, then read each digit separately: 0.36 is “zero point three six”.

读小数时，小数点读作“点”，后面的数字依次读出：0.36 读作“零点三六”。

3. Understanding Percentages | 理解百分数

The word “percent” means “per hundred” (from Latin per centum). A percentage is a fraction with denominator 100. For instance, 45% means 45 out of 100, which is 45/100.

“百分之”一词来自拉丁语 per centum，意为“每一百”。百分数就是分母为100的分数。例如，45% 表示100份中的45份，即 45/100。

The symbol % is read as “percent”.
符号 % 读作“百分之”。
100% represents the whole amount.
100% 表示整体数量。
Percentages can exceed 100%, such as 150%, when comparing against a base value.
百分数可以超过100%，如150%，用于与基准值比较时。

75% = 75/100 = ¾

4. Converting Fractions to Decimals | 分数化小数

To convert a fraction to a decimal, divide the numerator by the denominator. You may use long division or a calculator.

将分数化为小数时，用分子除以分母。可以笔算或使用计算器。

¾ = 3 ÷ 4 = 0.75

Some fractions produce recurring decimals, such as ⅓ = 0.3333… which is written as 0.3̇. For IGCSE, you should recognize common recurring patterns.

有些分数会产生循环小数，如 ⅓ = 0.3333…，记作 0.3̇。IGCSE考试中需要识别常见的循环模式。

½ = 0.5
¼ = 0.25
⅕ = 0.2
⅛ = 0.125

5. Converting Decimals to Fractions | 小数化分数

Write the decimal over a power of ten based on its number of decimal places, then simplify. For example, 0.6 has one decimal place, so it is 6/10, which simplifies to 3/5.

将小数写成分母为10的幂的分数，分母的小数位数和10的指数相同，然后约分。例如，0.6有一位小数，所以是 6/10，约分得 3/5。

0.75 = 75/100 = ¾

For recurring decimals, a longer method is needed, but for non‑recurring decimals this place‑value method always works.

对于循环小数，需要更复杂的方法；但对于有限小数，上述位值法始终有效。

Decimal	Fraction (simplified)
0.2	1/5
0.375	3/8
2.5	5/2 = 2½

6. Converting Fractions to Percentages | 分数化百分数

To convert a fraction to a percentage, multiply the fraction by 100%. Because multiplying by 100% is the same as multiplying by 1, the value does not change.

将分数化为百分数时，把分数乘以 100%。因为乘以100%相当于乘以1，所以数值不变。

⅗ × 100% = 300/5 % = 60%

Alternatively, convert the fraction to a decimal first, then multiply by 100%: ⅗ = 0.6, and 0.6 × 100% = 60%.

也可以先把分数化成小数，再乘以100%：⅗ = 0.6，0.6 × 100% = 60%。

½ = 50%
⅓ ≈ 33.3%
⅔ ≈ 66.7%
⅕ = 20%

7. Converting Percentages to Fractions | 百分数化分数

To convert a percentage to a fraction, write it over 100 and simplify. Remember to remove the % symbol.

将百分数化为分数时，把百分数写成以100为分母的分数，然后约分。注意去掉 % 符号。

32% = 32/100 = 8/25

If the percentage is a decimal, such as 12.5%, first write it as 12.5/100, then multiply top and bottom by 10 to get 125/1000 = 1/8.

如果百分数是小数，如12.5%，先写成 12.5/100，然后分子分母同乘10得 125/1000 = 1/8。

Percentages greater than 100% also follow the same rule: 150% = 150/100 = 3/2.

大于100%的百分数同样适用：150% = 150/100 = 3/2。

8. Converting Decimals to Percentages | 小数化百分数

To convert a decimal to a percentage, multiply by 100 and add the % symbol. For example, 0.47 × 100% = 47%.

将小数化为百分数时，乘以100并加上 % 符号。例如，0.47 × 100% = 47%。

0.08 = 8%

Moving the decimal point two places to the right is the same as multiplying by 100. So 0.125 becomes 12.5%.

把小数点向右移动两位就等于乘以100。所以 0.125 变成 12.5%。

0.3 = 30%
1.4 = 140%
0.05 = 5%
0.625 = 62.5%

9. Converting Percentages to Decimals | 百分数化小数

To convert a percentage to a decimal, divide by 100. That means moving the decimal point two places to the left and removing the % symbol.

将百分数化为小数时，除以100，即把小数点向左移动两位并去掉 % 符号。

25% = 25 ÷ 100 = 0.25

For example: 3.2% becomes 0.032, because 3.2 ÷ 100 = 0.032.

例如：3.2% 变成 0.032，因为 3.2 ÷ 100 = 0.032。

Percentage	Decimal
10%	0.1
66.6%	0.666
0.5%	0.005

10. Ordering and Comparing | 排序与比较

To compare fractions, decimals, and percentages, first convert all of them to the same form (usually decimals) and then arrange them.

比较分数、小数和百分数的大小时，先将它们都化成同一种形式（通常化成小数），再排序。

Example: Arrange 0.4, 3/8, and 35% in ascending order.

例：将 0.4、3/8 和 35% 从小到大排列。

0.4 = 0.400, 3/8 = 0.375, 35% = 0.350

Order: 35% < 3/8 < 0.4

Always include the original form in your final answer, do not leave answers as decimals if the question asked for percentages.

最终答案应保留题目要求的原始形式。如果题目要求百分数，就不要以小数作答。

11. Real‑world Applications | 实际应用

Percentages are used in discounts, interest rates, exam scores, and data interpretation. In IGCSE exams, you may need to calculate a percentage of a quantity, or find a percentage increase or decrease.

百分数广泛应用于折扣、利率、考试分数和数据分析。IGCSE考试中，你可能需要计算一个量的百分之几，或求增加/减少的百分数。

Simple interest: Interest = Principal × Rate × Time
单利：利息 = 本金 × 利率 × 时间
Percentage change: (New value − Old value) / Old value × 100%
百分变化率：(新值 − 旧值) / 旧值 × 100%
Discount: sale price = original price − (original price × discount %)
折扣：售价 = 原价 − 原价 × 折扣率

A jacket costs £80 and is reduced by 25%. Sale price = 80 − (80 × 0.25) = £60.

一件夹克80英镑，降价25%。售价 = 80 − (80 × 0.25) = 60英镑。

12. Practice Problems | 练习题

Try these questions to test your understanding:

尝试以下问题，测试你的理解：

Convert 7/20 to a decimal and a percentage.
将 7/20 化为小数和百分数。
Write 0.625 as a fraction in its simplest form.
把 0.625 写成最简分数。
Calculate 15% of 240.
计算 240 的 15%。
Arrange in ascending order: 0.09, 1/10, 8%.
将 0.09、1/10、8% 从小到大排列。

Answers: 1) 0.35, 35% 2) 5/8 3) 36 4) 8% < 0.09 < 1/10

答案：1) 0.35, 35% 2) 5/8 3) 36 4) 8% < 0.09 < 1/10

Mastering these conversions will give you a strong foundation for more advanced topics. Keep practising and always check your answers.

熟练掌握这些转换，将为学习更高阶的数学内容打下坚实基础。继续练习，并养成检查答案的好习惯。

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Fractions, Decimals and Percentages | 分数、小数与百分数