📚 Fractions, Decimals and Percentages: Conversion, Calculation and Exam Skills | 分数、小数和百分数：转换、计算与考试技巧
Fractions, decimals and percentages (FDP) are among the most frequently tested numerical skills in IGCSE Mathematics. They appear in arithmetic, algebra, ratio, probability and real-life financial contexts, so mastering conversions and calculations builds fluency and confidence.
分数、小数和百分数（FDP）是 IGCSE 数学中最常考查的数字技能之一。它们出现在算术、代数、比例、概率以及现实生活财务情境中，因此掌握转换与计算能提升熟练度与信心。
1. Why FDP Matters in IGCSE Maths | 为什么 FDP 在 IGCSE 数学中重要
IGCSE exam questions often mix fractions, decimals and percentages within the same problem, especially in best-buy, interest, proportion and data-handling contexts. You need to move flexibly between forms and choose the most efficient method for each calculation.
IGCSE 考题经常在同一个问题中混合使用分数、小数和百分数，尤其在最佳购买、利息、比例和数据处理情境中。你需要灵活地在不同形式之间切换，并为每种计算选择最有效的方法。
A common exam task is to place several values such as 2/5, 0.35 and 38% in ascending order. To compare them, convert all values to the same form, usually decimals or percentages.
常见的考试任务是给多个数值排序，例如 2/5、0.35 和 38%。为了比较，需要把所有数值转换成同一种形式，通常是小数的形式或百分数的形式。
2. Converting Between Fractions, Decimals and Percentages | 分数、小数和百分数之间的转换
To convert a fraction to a decimal, divide the numerator by the denominator. To convert a decimal to a percentage, multiply by 100%. To convert a percentage to a fraction, write the percentage over 100 and simplify if possible.
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%；将百分数转换为分数时写成百分数除以 100 并化简。
Fraction → Decimal: numerator ÷ denominator
Decimal → Percentage: decimal × 100%
Percentage → Fraction: percentage ÷ 100, then simplify
| Fraction | 分数 | Decimal | 小数 | Percentage | 百分数 |
|---|---|---|
| 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% |
3. Equivalent Fractions and Simplification | 等价分数与化简
Equivalent fractions represent the same value even though they look different. You can multiply or divide the numerator and denominator by the same non-zero integer without changing the value of the fraction.
等价分数表示相同的值，尽管它们看起来不同。你可以将分子和分母同时乘以或除以同一个非零整数，而不会改变分数的值。
a/b = (a × k) ÷ (b × k), for k ≠ 0
To simplify a fraction fully, divide the numerator and denominator by their highest common factor (HCF). For example, 18/24 can be simplified by dividing both terms by 6 to give 3/4.
要将分数完全化简，需要用分子和分母的最大公因数（HCF）约分。例如，18/24 可以将分子和分母同时除以 6，得到 3/4。
4. Adding and Subtracting Fractions | 分数的加法与减法
Before adding or subtracting fractions, convert them to equivalent fractions with a common denominator. If mixed numbers are involved, convert them to improper fractions first to make the process clearer.
在加减分数之前，需要先将它们通分为具有相同分母的等价分数。如果涉及带分数，先转换为假分数会让过程更清晰。
a/b + c/d = (a × d + b × c) / (b × d)
a/b − c/d = (a × d − b × c) / (b × d)
/3 + 1/4, use 12 as the common denominator: 1/3 = 4/12 and 1/4 = 3/12, so the sum is 7/12.
例如，计算 1/3 + 1/4 时，用 12 作为公分母：1/3 = 4/12，1/4 = 3/12，所以和为 7/12。
5. Multiplying and Dividing Fractions | 分数的乘法与除法
To multiply fractions, multiply the numerators together and multiply the denominators together. To divide fractions, multiply by the reciprocal of the divisor, which means swapping its numerator and denominator.
分数相乘时，将分子相乘作为新分子，将分母相乘作为新分母；分数相除时，乘以除数的倒数，也就是交换除数的分子和分母。
(a/b) × (c/d) = (a × c) / (b × d)
(a/b) ÷ (c/d) = (a/b) × (d/c) = (a × d) / (b × c)
r can be simplified. For example, (2/3) × (9/4) = 18/12, which simplifies to 3/2 or 1 1/2.
始终检查答案是否可以化简。例如，(2/3) × (9/4) = 18/12，化简后为 3/2 或 1 1/2。
6. Decimal Operations and Place Value | 小数运算与数位值
When adding or subtracting decimals, align the decimal points and keep place values in columns. When multiplying decimals, perform the whole-number multiplication first, then count the total number of decimal places in the factors.
小数加减时，将小数点对齐并保持数位对齐；小数乘法时，先按整数乘法计算，然后数出因数中小数位数的总和。
tiplication, 0.3 × 0.2 = 0.06 because 3 × 2 = 6 and there are two decimal places in total.
例如，0.35 + 0.4 = 0.75，因为十分位上的 4 与十分位上的 3 相加。对于乘法，0.3 × 0.2 = 0.06，因为 3 × 2 = 6，且两个因数一共有两位小数。
When dividing by a decimal, move the decimal point in both the divisor and dividend to make the divisor a whole number. The value of the quotient does not change.
当除数是一位小数时，将被除数和除数的小数点同时向右移动，使除数成为整数。商的值不会改变。
7. Finding a Percentage of an Amount | 求一个数量的百分数
To find a percentage of an amount, convert the percentage to a decimal multiplier or a fraction and multiply. For calculator questions, the decimal multiplier method is usually fastest.
求一个数量的百分数时，将百分数转换为小数乘数或分数，再进行乘法。对于允许使用计算器的题目，小数乘数法通常最快。
P% of X = (P ÷ 100) × X
For example, 15% 15 × 240 = 36. For non-calculator work, it is often easier to split the percentage into 10% and 5% components and add them.
例如，240 的 15% = 0.15 × 240 = 36。在不使用计算器时，通常把百分数拆成 10% 和 5% 再相加更容易。
8. Percentage Increase and Decrease | 百分数的增加与减少
Percentage change problems are best handled by using decimal multipliers. To increase an amount by r%, multiply by (1 + r/100); to decrease an amount by r%, multiply by (1 − r/100).
百分数变化问题最好使用小数乘数处理。要增加 r%，乘以 (1 + r/100)；要减少 r%，乘以 (1 − r/100)。
Increase: New = Original × (1 + r/100)
Decrease: New = Original × (1 − r/100)
For example, increasing £80 by 25% gives 80 × 1.25 = £100. Decreasing £80 by 25% gives 80 × 0.75 = £60. For repeated changes, apply the multipliers one after another.
例如，将 80 英镑增加 25%，得到 80 × 1.25 = 100 英镑。将 80 英镑减少 25%，得到 80 × 0.75 = 60 英镑。对于连续变化，依次使用相应的乘数。
9. Reverse Percentages | 逆百分数问题
If you know the final amount after a percentage change and need to find the original amount, divide by the relevant multiplier. This is the reverse percentage method.
如果已知百分数变化后的最终数量，需要求原数量，则除以相应的乘数。这就是逆百分数方法。
Original = Final ÷ (1 + r/100) for an increase
Original = Final ÷ (1 − r/100) for a decrease
For example, if a price has increased by 20% and is now £48, the original price is 48 ÷ 1.20 = £40. If a sale price of £54 represents a 10% decrease, the original price is 54 ÷ 0.90 = £60.
例如，如果价格增加了 20%，现在是 48 英镑，则原价为 48 ÷ 1.20 = 40 英镑。如果售价 54 英镑表示减少 10%，则原价为 54 ÷ 0.90 = 60 英镑。
10. Converting Recurring Decimals to Fractions | 将循环小数转换为分数
Recurring decimals can be converted into exact fractions using a simple subtraction method. Set the recurring decimal equal to x, multiply by a power of 10 that matches the repeating block, then subtract the original equation.
循环小数可以通过简单的相减法转换为精确分数。设循环小数等于 x，乘以与循环块匹配的 10 的幂，然后减去原方程。
ing digit such as 0.777..., set x = 0.777..., then 10x = 7.777... . Subtracting gives 9x =
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