📚 Mastering Fractions, Decimals and Percentages | 掌握分数、小数和百分比
Fractions, decimals and percentages are three different ways of representing parts of a whole. At KS3 Cambridge level, you need to move fluently between these forms, compare quantities, and solve problems involving proportional reasoning. This article will guide you through the key methods, common pitfalls, and exam-style strategies.
分数、小数和百分比是表示整体一部分的三种不同方式。在剑桥 KS3 阶段，你需要熟练地在这些形式之间转换、比较数量，并解决涉及比例推理的问题。本文将带你掌握关键方法、常见误区和考试策略。
1. Understanding the Link Between the Three Forms | 理解三种形式之间的联系
A fraction such as ¾ means ‘3 parts out of 4’. The same value can be written as the decimal 0.75 or as the percentage 75%. All three forms describe the same proportion of a whole.
分数 ¾ 表示“4 份中的 3 份”。同一个数值也可以写成小数 0.75 或百分比 75%。三种形式描述的都是整体所占的同一比例。
The key is to remember that the fraction bar acts as a division sign. To convert a fraction to a decimal, divide the numerator by the denominator. For example, 3 ÷ 4 = 0.75.
关键是要记住分数线相当于除号。要将分数转换为小数，用分子除以分母。例如，3 ÷ 4 = 0.75。
numerator ÷ denominator = decimal
2. Converting Fractions to Decimals | 将分数转换为小数
To convert a fraction to a decimal, carry out the division. Some fractions give a terminating decimal, such as ⅛ = 0.125, while others give a recurring decimal, such as ⅓ = 0.333… with the 3 repeating forever.
要将分数转换为小数，进行除法运算。有些分数得到有限小数，例如 ⅛ = 0.125；有些分数则得到循环小数，例如 ⅓ = 0.333…，数字 3 无限重复。
For fractions with denominators that are powers of 10 or factors of 100, you can use equivalent fractions. For example, 7/25 = 28/100 = 0.28. Common equivalents include ½ = 0.5, ¼ = 0.25, ¾ = 0.75 and ⅕ = 0.2.
对于分母是 10 的幂或 100 的因数的分数，可以使用等值分数。例如，7/25 = 28/100 = 0.28。常见的等值关系包括 ½ = 0.5、¼ = 0.25、¾ = 0.75 和 ⅕ = 0.2。
3. Converting Decimals to Percentages | 将小数转换为百分比
To convert a decimal to a percentage, multiply by 100 and add the % sign. This is the same as moving the decimal point two places to the right.
要将小数转换为百分比，乘以 100 并加上百分号。这相当于将小数点向右移动两位。
0.85 × 100 = 85%
Therefore, 0.07 = 7% and 1.2 = 120%. Be careful with decimals less than 0.01. For example, 0.005 = 0.5%, not 5%, because 0.005 × 100 = 0.5.
因此，0.07 = 7%，1.2 = 120%。注意小于 0.01 的小数。例如，0.005 = 0.5%，而不是 5%，因为 0.005 × 100 = 0.5。
4. Converting Percentages to Fractions | 将百分比转换为分数
To convert a percentage to a fraction, write the percentage over 100 and simplify if possible. For example, 45% = 45/100 = 9/20.
要将百分比转换为分数，将百分数写在 100 上面，然后尽可能约分。例如，45% = 45/100 = 9/20。
For percentages with a decimal, such as 12.5%, write 12.5/100 and multiply numerator and denominator by 10 to remove the decimal: 125/1000 = 1/8. Always simplify to the lowest terms unless the question asks otherwise.
对于带小数的百分比，例如 12.5%，写成 12.5/100，然后分子和分母同时乘以 10 去掉小数点：125/1000 = 1/8。除非题目另有要求，否则始终化简到最简分数。
5. Ordering Fractions, Decimals and Percentages | 比较分数、小数和百分比的大小
To put a mixed list in order, convert all values to the same form. Decimals are often the easiest for comparison. For example, order ⅖, 0.45, 38%, ⅝ from smallest to largest.
要对混合列表排序，先把所有数值转换成同一种形式。小数通常最容易比较。例如，将 ⅖、0.45、38%、⅝ 从小到大排列。
Convert: ⅖ = 0.4, 38% = 0.38, ⅝ = 0.625. Now the order is clear: 38%, ⅖, 0.45, ⅝. In KS3 exams, converting all values to decimals is a reliable method for comparison questions.
转换后：⅖ = 0.4，38% = 0.38，⅝ = 0.625。现在顺序就很清楚了：38%、⅖、0.45、⅝。在 KS3 考试中，将所有数值转换为小数是比较类题目的可靠方法。
6. Adding and Subtracting Fractions | 分数的加减法
Before adding or subtracting fractions, they must have the same denominator. Find the lowest common denominator (LCD) of the fractions.
在加减分数之前，它们必须有相同的分母。找到这些分数的最小公分母（LCD）。
Example: ⅓ + ¼. The LCD of 3 and 4 is 12. Rewrite: ⅓ = 4/12 and ¼ = 3/12. Then add: 4/12 + 3/12 = 7/12.
示例：⅓ + ¼。3 和 4 的最小公分母是 12。重写为：⅓ = 4/12，¼ = 3/12。然后相加：4/12 + 3/12 = 7/12。
For mixed numbers, convert to improper fractions first, perform the operation, then convert back if needed. Never add or subtract denominators directly.
对于带分数，先转换为假分数，进行运算，然后根据需要转换回来。切勿直接加减分母。
7. Multiplying and Dividing Fractions | 分数的乘除法
To multiply fractions, multiply the numerators together and the denominators together. Simplify your answer if possible. Example: ⅔ × ¾ = (2×3)/(3×4) = 6/12 = ½.
乘法分数时，分子相乘，分母相乘。如果可能，化简答案。示例：⅔ × ¾ = (2×3)/(3×4) = 6/12 = ½。
a/b × c/d = (a×c)/(b×d)
To divide by a fraction, multiply by its reciprocal. Example: ⅖ ÷ ¾ = ⅖ × 4/3 = 8/15. Always flip the second fraction and change the division sign to multiplication.
除以一个分数，等于乘以它的倒数。示例：⅖ ÷ ¾ = ⅖ × 4/3 = 8/15。始终翻转第二个分数，并将除号改为乘号。
8. Percentage of an Amount | 求一个数量的百分比
To find a percentage of an amount, convert the percentage to a decimal or fraction, then multiply. For example, 30% of 250 = 0.30 × 250 = 75.
要求一个数量的百分比，先将百分比转换为小数或分数，然后相乘。例如，250 的 30% = 0.30 × 250 = 75。
You can also use the 10% method: find 10% by dividing by 10, then scale up or down. For 15% of 80, 10% is 8 and 5% is 4, so 15% = 12. This method is especially useful for mental calculations.
也可以使用 10% 方法：除以 10 求出 10%，然后按比例放大或缩小。求 80 的 15%，10% 是 8，5% 是 4，所以 15% = 12。这种方法对于心算特别有用。
9. Percentage Increase and Decrease | 百分比增减
To increase an amount by a percentage, add the percentage to 100%, convert to a decimal, and multiply. For example, increase 40 by 15%: 100% + 15% = 115% = 1.15; 1.15 × 40 = 46.
要将一个数增加某个百分比，将百分比加到 100%，转换为小数，然后相乘。例如，将 40 增加 15%：100% + 15% = 115% = 1.15；1.15 × 40 = 46。
To decrease, subtract the percentage from 100%. For example, decrease 80 by 25%: 100% – 25% = 75% = 0.75; 0.75 × 80 = 60. Using a multiplier is much faster than finding the change and then adding or subtracting.
减少时，从 100% 中减去百分比。例如，将 80 减少 25%：100% – 25% = 75% = 0.75；0.75 × 80 = 60。使用乘数比先求出变化量再加减要快得多。
10. Word Problems and Real-Life Applications | 应用题与实际应用
KS3 exam questions often combine fractions, decimals and percentages in real-life contexts, such as discounts, test scores, or recipes. Start by identifying which form is given and which form is needed.
KS3 考试题目经常在生活场景中综合考查分数、小数和百分比，例如折扣、考试成绩或食谱。首先要判断题目给出的是哪种形式，以及需要哪种形式。
Example: A jacket costs £80. In a sale, it is reduced by 15%. What is the sale price? The multiplier is 85% = 0.85, so sale price = 0.85 × 80 = £68.
示例：一件夹克售价 80 英镑。打折时降价 15%。售价是多少？乘数是 85% = 0.85，所以售价 = 0.85 × 80 = 68 英镑。
Always show your working clearly, including the multiplier or equivalent fraction, to gain method marks even if a calculation error is made.
始终清晰地展示解题步骤，包括乘数或等值分数，这样即使计算错误也能获得方法分。
11. Common Misconceptions and Exam Tips | 常见误区与考试技巧
A common error is to think 0.5% is the same as 0.5. Remember, 0.5% = 0.005, which is much smaller. Another error is adding fractions by adding numerators and denominators directly, such as ½ + ⅓ = 2/5, which is wrong.
一个常见误区是以为 0.5% 等于 0.5。要记住，0.5% = 0.005，小得多。另一个误区是直接将分子和分母相加来加分数，例如 ½ + ⅓ = 2/5，这是错误的。
When comparing values, convert all to the same form. In percentage increase or decrease questions, do not forget to use 100% plus or minus the given percentage for the multiplier. Check whether an answer should be a fraction, decimal or percentage, and present it in the requested form.
比较大小时，将所有数值转换成同一种形式。在百分比增减问题中，不要忘记用 100% 加上或减去给定的百分比来得到乘数。检查答案应该以分数、小数还是百分比形式呈现，并按要求的形式作答。
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