📚 Fractions, Decimals and Percentages | 分数、小数和百分数
In IGCSE Mathematics, fractions, decimals and percentages form a core number skill set that appears in almost every paper. You will need to convert between these three forms quickly and accurately, use them in real-life problems such as discounts, interest and measurement, and apply them to algebra and data questions. This article covers the key methods, typical exam traps and strategies to build confidence.
在IGCSE数学中，分数、小数和百分数构成核心数字技能，几乎出现在每份试卷中。你需要快速准确地在三种形式之间转换，在折扣、利息和测量等实际问题中使用它们，并将其应用于代数和数据题。本文涵盖关键方法、常见考试陷阱以及建立信心的策略。
1. Understanding Fractions | 理解分数
A fraction represents a part of a whole and is written in the form a/b, where a is the numerator and b is the denominator. The denominator cannot be zero because division by zero is undefined.
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。分母不能为零，因为除以零是未定义的。
Fractions can be proper, improper or mixed. A proper fraction has a numerator smaller than the denominator, such as 3/5. An improper fraction has a numerator greater than or equal to the denominator, such as 9/4. A mixed number combines a whole number and a proper fraction, such as 2 1/3.
分数可以是真分数、假分数或带分数。真分数的分子小于分母，例如 3/5。假分数的分子大于或等于分母，例如 9/4。带分数由整数和真分数组合而成，例如 2 1/3。
On a number line, fractions divide the interval between whole numbers into equal parts. Knowing this helps you estimate and compare fractions.
在数轴上，分数将整数之间的区间分成相等的部分。了解这一点有助于你估算和比较分数。
2. Equivalent Fractions and Simplifying | 等值分数与化简
Equivalent fractions have the same value even though they look different. They are obtained by multiplying or dividing both the numerator and the denominator by the same non-zero number.
等值分数虽然看起来不同，但值相同。它们通过将分子和分母同时乘以或除以同一个非零数得到。
For example, 2/3 is equivalent to 4/6, 6/9 and 20/30. Simplifying a fraction means dividing both numerator and denominator by their highest common factor, so 20/30 simplifies to 2/3.
例如，2/3 等价于 4/6、6/9 和 20/30。化简分数意味着将分子和分母同时除以它们的最大公因数，因此 20/30 化简为 2/3。
Always give fractions in their simplest form in final answers unless the question asks otherwise. This is a frequent source of lost marks in IGCSE papers.
除非题目另有要求，最终答案中的分数始终要化为最简形式。这是IGCSE试卷中常见的失分点。
3. Converting Fractions to Decimals | 分数化为小数
To convert a fraction to a decimal, divide the numerator by the denominator. You can use short division or a calculator, but IGCSE non-calculator papers expect you to know common conversions by heart.
要将分数化为小数，用分子除以分母。你可以使用短除法或计算器，但IGCSE非计算器试卷要求你熟记常见换算。
For example, 3/8 means 3 ÷ 8 = 0.375. Some fractions produce recurring decimals, such as 1/3 = 0.333… which can be written as 0.3 with a dot above the 3.
例如，3/8 表示 3 ÷ 8 = 0.375。一些分数会产生循环小数，例如 1/3 = 0.333… 可以写作 0.3，在3上方加一个点。
It is useful to remember that 1/2 = 0.5, 1/4 = 0.25, 3/4 = 0.75 and 1/10 = 0.1. These common decimals help you check your work quickly.
记住 1/2 = 0.5、1/4 = 0.25、3/4 = 0.75 和 1/10 = 0.1 很有用。这些常见小数能帮助你快速检查答案。
4. Converting Fractions to Percentages | 分数化为百分数
To convert a fraction to a percentage, first convert it to a decimal by dividing the numerator by the denominator, then multiply by 100 and add the % symbol.
要将分数化为百分数，首先通过分子除以分母将其化为小数，然后乘以100并加上%符号。
For example, 3/5 = 0.6, and 0.6 × 100 = 60%. Alternatively, you can make the denominator 100 if possible: 3/5 = 60/100 = 60%.
例如，3/5 = 0.6，0.6 × 100 = 60%。或者，如果可能，可以将分母化为100：3/5 = 60/100 = 60%。
Some fractions give repeating percentages, such as 1/3 = 33 1/3%. In IGCSE exams, you should be familiar with common fractions and their percentage equivalents.
一些分数会得到重复的百分数，例如 1/3 = 33 1/3%。在IGCSE考试中，你应该熟悉常见分数及其百分数等价形式。
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/5 | 0.2 | 20% |
| 1/10 | 0.1 | 10% |
5. Ordering Mixed Forms | 混合形式的排序
To compare and order fractions, decimals and percentages, convert all of them into the same form, usually decimals. This removes confusion and makes the order obvious.
要比较和排序分数、小数和百分数，通常将它们都化为同一种形式，一般是小数。这样可以消除混淆，使顺序清晰。
For example, to order 3/5, 0.55 and 58%, first convert: 3/5 = 0.6, 0.55 is already a decimal, and 58% = 0.58. In ascending order: 0.55, 0.58, 0.6, so 0.55 < 58% < 3/5.
例如，要排序 3/5、0.55 和 58%，首先转换：3/5 = 0.6，0.55 已经是小数，58% = 0.58。按升序排列：0.55、0.58、0.6，因此 0.55 < 58% < 3/5。
When ordering, use the original forms in your final answer, not the decimals, unless the question asks for decimals. Show your conversion steps to earn method marks.
排序时，最终答案中使用原始形式，而不要使用小数，除非题目要求用小数。展示转换步骤可获得方法分。
6. Adding and Subtracting Fractions | 分数的加减法
To add or subtract fractions, first find a common denominator. Then rewrite each fraction with that denominator, and add or subtract the numerators. The denominator stays the same.
要加减分数，首先找到公分母。然后将每个分数改写为具有该分母的形式，再加或减分子。分母保持不变。
For example, to calculate 1/3 + 1/4, the lowest common denominator is 12. So 1/3 = 4/12 and 1/4 = 3/12. Adding gives 7/12.
例如，计算 1/3 + 1/4，最小公分母是12。因此 1/3 = 4/12，1/4 = 3/12。相加得到 7/12。
For mixed numbers, convert them to improper fractions first, then add or subtract, and finally simplify and convert back to a mixed number if needed.
对于带分数，先将它们化为假分数，然后进行加减，最后化简并在需要时转换回带分数。
a/b + c/d = (a × d + c × b)/(b × d)
7. Multiplying and Dividing Fractions | 分数的乘除法
To multiply fractions, multiply the numerators together and the denominators together. Then simplify the result if possible.
要乘分数，将分子相乘、分母相乘。然后如果可能，化简结果。
a/b × c/d = (a × c)/(b × d)
For example, 2/5 × 3/4 = (2 × 3)/(5 × 4) = 6/20 = 3/10. You can cancel common factors before multiplying to make the work easier.
例如，2/5 × 3/4 = (2 × 3)/(5 × 4) = 6/20 = 3/10。你可以在乘法之前约去公因数，使计算更简单。
To divide fractions, multiply by the reciprocal of the second fraction. The reciprocal of c/d is d/c.
要除分数，乘以第二个分数的倒数。c/d 的倒数是 d/c。
a/b ÷ c/d = a/b × d/c
For example, 3/8 ÷ 2/5 = 3/8 × 5/2 = 15/16. Simplify your answer and check that it is reasonable.
例如，3/8 ÷ 2/5 = 3/8 × 5/2 = 15/16。化简答案并检查其是否合理。
8. Percentage Increase and Decrease | 百分数的增减
A percentage change compares the change in a quantity to the original quantity. The formula for percentage change is:
百分数变化将某量的变化与原量进行比较。百分数变化的公式为：
Percentage change = (change ÷ original) × 100
For an increase, the new amount = original + increase. For a decrease, the new amount = original – decrease. Alternatively, use multipliers: for a 15% increase, multiply by 1.15; for a 15% decrease, multiply by 0.85.
对于增加，新量 = 原量 + 增加量。对于减少，新量 = 原量 – 减少量。或者使用乘数：增加15%，乘以1.15；减少15%，乘以0.85。
Example: a shop increases the price of a $40 shirt by 12%. The new price is 40 × 1.12 = $44.80. Study this method carefully because it is tested frequently in IGCSE number and real-life context questions.
示例：一家商店将一件40美元的衬衫提价12%。新价格为 40 × 1.12 = 44.80美元。仔细学习这种方法，因为它在IGCSE数字和实际情境题中经常考查。
9. Reverse Percentages | 逆百分数问题
Reverse percentage problems ask you to find the original amount before a percentage change when you know the final amount. Do not simply subtract the percentage from the final amount; instead, use division by the multiplier.
逆百分数问题要求你在已知最终量时，求百分数变化之前的原始量。不要简单地从最终量中减去百分数；而应该除以乘数。
For example, after a 20% decrease, a jacket costs $64. The multiplier for a 20% decrease is 0.8, so original price = 64 ÷ 0.8 = $80.
例如，一件夹克降价20%后售价64美元。降价20%的乘数是0.8，因此原价 = 64 ÷ 0.8 = 80美元。
Similarly, if a price includes 15% tax and the total is $230, the pre-tax price is 230 ÷ 1.15 = $200. Always identify the correct multiplier before dividing.
类似地，如果价格包含15%的税，总额是230美元，则税前价格为 230 ÷ 1.15 = 200美元。在除法之前，务必确定正确的乘数。
10. Exam-Style Strategies | 考试型策略
In IGCSE exams, show full working for every conversion and calculation. Even if your final answer is wrong, method marks can be awarded for correct steps.
在IGCSE考试中，对每一个转换和计算都要展示完整步骤。即使最终答案错误，正确的步骤仍可获得方法分。
Use common sense to check answers: a percentage greater than 100% means an increase beyond the original, while a decimal less than 1 means a proper fraction. Rounding should be done only at the final answer unless the question states otherwise.
用常识检查答案：大于100%的百分数表示超过原量，小于1的小数表示真分数。除非题目另有说明，四舍五入应在最终答案时进行。
Practise mixed exercises where you must switch between fractions, decimals and percentages in a single problem. This mirrors real IGCSE questions that combine topics such as ratio, probability and algebra with number skills.
练习混合题型，在单个问题中你需要在分数、小数和百分数之间切换。这模拟了真实的IGCSE题目，这些题目常将比例、概率和代数与数字技能结合起来。
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