📚 Number Skills: Fractions, Decimals and Percentages | 数字技能：分数、小数与百分数
In IGCSE Mathematics, fractions, decimals and percentages are three equivalent ways of describing parts of a whole. They occur in arithmetic, ratios, money problems, probability and data handling, so confident number skills are essential for both core and extended papers.
在 IGCSE 数学中，分数、小数和百分数是描述整体之部分的三种等价方式。它们出现在算术、比、金钱问题、概率和数据处理中，因此扎实的数字技能对核心卷和扩展卷都至关重要。
1. Representing Parts of a Whole | 表示一个整体的部分
A fraction uses a numerator over a denominator. The denominator shows how many equal parts the whole is split into, and the numerator shows how many of those parts are taken. For example, 3/4 means three parts out of four.
分数使用分子在上、分母在下的形式。分母表示整体被分成多少等份，分子表示取了多少份。例如，3/4 表示四等份中的三份。
A decimal uses place value to show tenths, hundredths, thousandths and so on. A percentage is simply a fraction out of 100, written with the percent symbol. For example, 0.6 means 6 tenths, and 60% means 60 out of 100.
小数使用位值表示十分位、百分位、千分位等。百分数就是分母为 100 的分数，用百分号表示。例如，0.6 表示 6 个十分之一，60% 表示 100 份中的 60 份。
2. Converting Between the Three 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, 3/4 = 3 ÷ 4 = 0.75.
将分数转换为小数，用分子除以分母。例如，3/4 = 3 ÷ 4 = 0.75。
To convert a decimal to a percentage, multiply by 100 and add the percent symbol. For example, 0.625 = 0.625 × 100 = 62.5%.
将小数转换为百分数，乘以 100 并加上百分号。例如，0.625 = 0.625 × 100 = 62.5%。
To convert a percentage to a fraction, write the percentage over 100 and simplify. For example, 40% = 40/100 = 2/5.
将百分数转换为分数，写成百分数除以 100 并化简。例如，40% = 40/100 = 2/5。
| Conversion | Method | 中文方法 |
|---|---|---|
| Fraction → Decimal | Divide the numerator by the denominator | 分子除以分母 |
| Decimal → Percentage | Multiply by 100, add % | 乘以 100，加上 % |
| Percentage → Fraction | Write over 100, then simplify | 写成除以 100，再化简 |
| Fraction → Percentage | Convert to a decimal, then multiply by 100 | 先转为小数，再乘以 100 |
3. Ordering and Comparing Numbers | 数字的排序与比较
When comparing fractions, decimals and percentages, change all of them into the same form. Decimals are usually the easiest form for comparison because place value makes the order obvious.
比较分数、小数和百分数时，把它们都转化为同一种形式。小数通常最容易比较，因为位值使顺序一目了然。
For example, arrange 2/5, 0.45 and 38% in ascending order. Converting gives 2/5 = 0.4, 0.45 = 0.45, and 38% = 0.38. The ascending order is 0.38, 0.4, 0.45, so the original order is 38%, 2/5, 0.45.
例如，将 2/5、0.45 和 38% 按升序排列。转换后得到 2/5 = 0.4，0.45 = 0.45，38% = 0.38。升序为 0.38、0.4、0.45，因此原顺序为 38%、2/5、0.45。
Always compare the digits in the same place value column. It is helpful to write mixed decimals with the same number of decimal places, such as 0.40 instead of 0.4, to avoid common mistakes.
始终比较相同数位上的数字。将位数不同的小数写成相同小数位数会很有帮助，例如用 0.40 代替 0.4，以避免常见错误。
4. Adding and Subtracting Fractions | 分数的加减
Before adding or subtracting fractions, find a common denominator. The lowest common denominator is usually the best choice because it keeps the numbers smaller.
分数加减之前，先找公分母。最小公分母通常是最佳选择，因为它能使数字较小。
For example, 1/2 + 1/3 = 3/6 + 2/6 = 5/6. The common denominator 6 is the lowest common multiple of 2 and 3.
例如，1/2 + 1/3 = 3/6 + 2/6 = 5/6。公分母 6 是 2 和 3 的最小公倍数。
For mixed numbers, convert them to improper fractions first or add the whole number parts and fractional parts separately. For example, 2 3/4 – 1/6 = 11/4 – 1/6 = 33/12 – 2/12 = 31/12 = 2 7/12.
对于带分数，可以先化为假分数，也可以把整数部分和分数部分分别相加。例如，2 3/4 – 1/6 = 11/4 – 1/6 = 33/12 – 2/12 = 31/12 = 2 7/12。
After adding, always simplify the result if possible. A final answer should normally be given as a simplified fraction or mixed number, unless the question states otherwise.
相加后，如果可能，始终化简结果。除非题目另有说明，最终答案通常应写成最简分数或带分数。
5. Multiplying and Dividing Fractions | 分数的乘除
To multiply fractions, multiply the numerators together and the denominators together. You can simplify before or after multiplying to make the calculation easier.
分数相乘时，分子与分子相乘，分母与分母相乘。可以在乘之前或之后化简，使计算更简单。
For example, (2/3) × (9/10) = 18/30 = 3/5. A quicker method is to cancel common factors first: 2/3 × 9/10 = 2/1 × 3/10 = 6/10 = 3/5.
例如，(2/3) × (9/10) = 18/30 = 3/5。更快的方法是先约去公因数：2/3 × 9/10 = 2/1 × 3/10 = 6/10 = 3/5。
To divide by a fraction, multiply by its reciprocal. The reciprocal of a fraction is obtained by swapping the numerator and the denominator.
除以一个分数等于乘以它的倒数。一个分数的倒数是将分子和分母互换得到的。
For example, (5/6) ÷ (2/3) = (5/6) × (3/2) = 15/12 = 5/4 = 1 1/4.
例如，(5/6) ÷ (2/3) = (5/6) × (3/2) = 15/12 = 5/4 = 1 1/4。
Remember that a whole number can be written as a fraction with denominator 1. For example, 4 = 4/1, so 4 × (3/8) = (4/1) × (3/8) = 12/8 = 3/2 = 1 1/2.
记住，整数可以写成分母为 1 的分数。例如，4 = 4/1，所以 4 × (3/8) = (4/1) × (3/8) = 12/8 = 3/2 = 1 1/2。
6. Finding a Percentage of a Quantity | 求一个量的百分数
To find a percentage of a quantity, first convert the percentage into a decimal or a simple fraction, then multiply by the quantity.
要找一个量的百分数，先把百分数转换为小数或简单分数，然后乘以该量。
For example, 15% of 240 = 0.15 × 240 = 36. Equivalently, 15% = 15/100 = 3/20, so 3/20 × 240 = 36.
例如，240 的 15% = 0.15 × 240 = 36。等价地，15% = 15/100 = 3/20，所以 3/20 × 240 = 36。
Recognising common percentage-fraction conversions speeds up calculations. For example, 50% = 1/2, 25% = 1/4, 10% = 1/10, 5% = 1/20 and 12.5% = 1/8.
识别常见的百分数与分数转换可以加快计算速度。例如，50% = 1/2，25% = 1/4，10% = 1/10，5% = 1/20，12.5% = 1/8。
For calculations such as 12.5% of 80, using the fraction 1/8 gives 80 ÷ 8 = 10, which is much quicker than multiplying by 0.125.
对于如 80 的 12.5% 这样的计算，使用分数 1/8 得到 80 ÷ 8 = 10，这比乘以 0.125 快得多。
7. Percentage Increase and Decrease | 百分数增减
For a percentage increase, multiply the original value by (1 + r/100). For a percentage decrease, multiply the original value by (1 – r/100). The number inside the bracket is called the multiplier.
百分数增加时，将原值乘以 (1 + r/100)；百分数减少时，将原值乘以 (1 – r/100)。括号内的数称为乘数。
For example, increasing $80 by 15% gives 80 × 1.15 = $92. Decreasing $120 by 25% gives 120 × 0.75 = $90.
例如，$80 增长 15% 得到 80 × 1.15 = $92。$120 减少 25% 得到 120 × 0.75 = $90。
Be careful with repeated changes: they are multiplied step by step, not added. A 20% increase followed by a 10% increase is not a 30% increase. It is 1.2 × 1.1 = 1.32, which is a 32% increase overall.
注意连续变化：它们逐步相乘，而不是相加。先增长 20% 再增长 10% 并不是增长 30%。而是 1.2 × 1.1 = 1.32，整体增长了 32%。
When a question gives a percentage change and asks for the new value, identify the correct multiplier before doing any subtraction or addition. This reduces the chance of sign errors.
当题目给出百分数变化并要求求新值时，在做任何减法或加法之前先确定正确的乘数。这可以减少符号错误。
8. Reverse Percentages | 逆向百分
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