📚 Fractions, Decimals and Percentages: IGCSE Core Skills | IGCSE 核心技能：分数、小数与百分数
Fractions, decimals and percentages are three different ways of describing the same idea: a part of a whole. In IGCSE Mathematics, you must be able to convert between them, compare them, and apply percentage change and interest formulas confidently.
分数、小数和百分数是描述同一个概念的三种不同形式：整体的一部分。在 IGCSE 数学中，你必须能够在它们之间熟练转换、比较大小，并自信地运用百分数增减和利息公式。
1. Understanding the Three Forms | 理解三种表示形式
A fraction has a numerator and a denominator. A decimal uses a decimal point to show tenths, hundredths, and so on. A percentage is a number out of 100.
分数有分子和分母。小数用小数点表示十分位、百分位等。百分数是以 100 为基准的数。
For example, one half can be written as 1/2, 0.5, or 50%. These three expressions are equal.
例如，二分之一可以写成 1/2、0.5 或 50%。这三种表达是相等的。
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
These common conversions are worth memorising because they appear frequently in both non-calculator and calculator papers.
这些常见转换值得熟记，因为它们在非计算器和计算器试卷中都经常出现。
2. Converting Fractions to Decimals and Percentages | 分数转小数与百分数
To convert a fraction to a decimal, divide the numerator by the denominator. To convert the decimal to a percentage, multiply by 100.
将分数转换为小数时，用分子除以分母。将小数转换为百分数时，乘以 100。
Decimal = Numerator ÷ Denominator
Percentage = Decimal × 100%
Example: 3/8 = 3 ÷ 8 = 0.375, then 0.375 × 100% = 37.5%.
示例：3/8 = 3 ÷ 8 = 0.375，然后 0.375 × 100% = 37.5%。
.333... and 2/9 = 0.222... These are exact values even though the decimal repeats forever.
有些分数会转换为循环小数，例如 1/3 = 0.333... 和 2/9 = 0.222...。即使小数无限循环，它们仍然是精确值。
3. Converting Decimals to Fractions and Percentages | 小数转分数与百分数
To convert a terminating decimal to a fraction, write the decimal over its place value and simplify. To convert to a percentage, multiply by 100.
将有限小数转换为分数时，根据小数位数写成分母为 10、100、1000 等的分数并化简。转换为百分数时乘以 100。
Example: 0.35 = 35/100 = 7/20, and 0.35 × 100% = 35%.
示例：0.35 = 35/100 = 7/20，且 0.35 × 100% = 35%。
. For example, 0.625 = 625/1000 = 5/8 after dividing the numerator and denominator by 125.
对于两位以上小数的数，规则相同。例如，0.625 = 625/1000，将分子和分母同时除以 125 后得到 5/8。
4. Converting Percentages to Fractions and Decimals | 百分数转分数与小数
To convert a percentage to a fraction, write it over 100 and simplify. To convert to a decimal, divide by 100.
将百分数转换为分数时，先写成分母为 100 的分数并化简。转换为小数时除以 100。
Decimal = Percentage ÷ 100
Example: 62.5% = 62.5/100 = 625/1000 = 5/8, and 62.5% = 0.625.
示例：62.5% = 62.5/100 = 625/1000 = 5/8，且 62.5% = 0.625。
If a percentage is above 100, the value is greater than one. For example, 150% = 1.5 = 3/2.
如果百分数大于 100，则数值大于 1。例如，150% = 1.5 = 3/2。
5. Ordering and Comparing Rational Numbers | 有理数的排序与比较
A common exam task is to arrange fractions, decimals, and percentages in ascending or descending order. The safest method is to convert all values to the same form, usually decimals.
常见的考试题型是将分数、小数和百分数按升序或降序排列。最稳妥的方法是先将所有数值转换为同一种形式，通常转换为小数。
Example: Order 3/5, 0.58, 61% from smallest to largest. Convert: 3/5 = 0.60, 0.58 = 0.58, 61% = 0.61. So the order is 0.58, 3/5, 61%.
示例：将 3/5、0.58、61% 从小到大排列。转换为小数：3/5 = 0.60，0.58 = 0.58，61% = 0.61。因此顺序为 0.58、3/5、61%。
When comparing recurring decimals, write them with enough decimal places to see the difference. For example, 0.666... is greater than 0.66 because 0.666... continues with sixes, while 0.66 stops at two decimal places.
比较循环小数时，写出足够多的小数位以看出差异。例如，0.666... 大于 0.66，因为 0.666... 后面仍然是 6，而 0.66 只到两位小数。
6. Recurring Decimals and Fraction Conversion | 循环小数与分数互化
or group of digits forever. Common examples include 0.333... = 1/3 and 0.272727... = 3/11.
循环小数会无限重复一个或一组数字。常见例子包括 0.333... = 1/3 和 0.272727... = 3/11。
To convert a recurring decimal to a fraction, set up an equation, multiply by a power of 10, and subtract to eliminate the repeating part.
将循环小数转换为分数时，设方程、乘以 10 的幂，再相减以消去循环部分。
on. Let x = 0.272727..., then 100x = 27.272727... Subtract: 99x = 27, so x = 27/99 = 3/11.
示例：将 0.272727... 转换为分数。设 x = 0.272
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