📚 Mastering Fractions, Decimals and Percentages | 掌握分数、小数和百分比
Fractions, decimals and percentages are three different ways of expressing the same idea: a part of a whole. In the Cambridge KS3 Mathematics curriculum, you need to move fluently between these forms and use them to solve real problems, from discounts to data comparisons. This revision guide takes you through the key methods step by step.
分数、小数和百分比是表达同一个概念的三种不同方法：整体的一部分。在剑桥 KS3 数学课程中，你需要在这三种形式之间灵活转换，并运用它们解决实际问题，从折扣到数据比较。本复习指南将带你一步一步掌握核心方法。
1. Understanding Equivalent Fractions | 理解等价分数
50/100. This idea is essential when comparing fractions or converting them to percentages.
等价分数表示相同的数值，尽管分子和分母不同。要找出等价分数，可以把分子和分母同时乘以或除以同一个非零数。例如，1/2 = 2/4 = 3/6 = 50/100。这个思路在比较分数或将分数转换为百分比时至关重要。
To simplify a fraction, divide both the numerator and the denominator by their highest common factor (HCF). For example, 18/24 simplifies to 3/4 because the HCF of 18 and 24 is 6. Always check whether your answer can be simplified further.
要化简分数，将分子和分母同时除以它们的最大公因数（HCF）。例如，18/24 化简为 3/4，因为 18 和 24 的最大公因数是 6。始终检查答案是否可以继续化简。
2. Converting Fractions to Decimals | 分数转化为小数
ring decimals, such as 1/3 = 0.333..., can be written with a dot over the repeating digit.
要将分数转化为小数，用分子除以分母。对于简单的分数，你可以使用已知的等价关系：1/4 = 0.25，1/5 = 0.2，3/8 = 0.375。对于任意分数，可使用短除法或计算器。循环小数，如 1/3 = 0.333...，可以在重复数字上方加点表示。
- Change the fraction to an equivalent fraction over 10, 100 or 1000 if possible.
如果可能，先将分数转化为分母为 10、100 或 1000 的等价分数。 - Otherwise, divide the numerator by the denominator.
否则，用分子除以分母。 - Write the decimal point carefully and keep place values aligned.
仔细书写小数点，并保持数位对齐。
3. Converting Decimals to Percentages | 小数转化为百分比
A percentage means 'out of 100'. To convert a decimal to a percentage, multiply by 100 and add the % sign. For example, 0.45 × 100 = 45%. To convert a percentage to a decimal, divide by 100: 75% r that moving the decimal point two places to the right is the same as multiplying by 100.
百分比表示“每一百份中的多少”。要将小数转化为百分比，乘以 100 并加上百分号。例如，0.45 × 100 = 45%。要将百分比转化为小数，除以 100：75% = 0.75。记住，将小数点向右移动两位等同于乘以 100。
| Decimal | 小数 | Percentage | 百分比 |
| 0.1 | 10% |
| 0.25 | 25% |
| 0.5 | 50% |
| 0.75 | 75% |
| 1.0 | 100% |
4. Converting Fractions to Percentages | 分数转化为百分比
uivalent fraction with denominator 100. For example, 3/5 = (3 × 20)/(5 × 20) = 60/100 = 60%. This method is useful for simple fractions and builds number sense.
首先通过除法将分数转化为小数，然后乘以 100。或者，找出分母为 100 的等价分数。例如，3/5 = (3 × 20)/(5 × 20) = 60/100 = 60%。这种方法对简单分数很有用，还能培养数感。
ons that do not have an easy equivalent over 100, use division: 5/8 = 5 ÷ 8 = 0.625 = 62.5%. Keep the decimal exact until the final step to avoid rounding errors.
对于不容易转化为分母 100 的分数，使用除法：5/8 = 5 ÷ 8 = 0.625 = 62.5%。在最后一步之前保持小数精确，以避免舍入误差。
5. Finding a Percentage of a Quantity | 求一个数量的百分比
f a quantity, change the percentage to a decimal or fraction and multiply. For example, 30% of 80 = 0.30 × 80 = 24. For non-calculator work, split the percentage into easy parts: 10%, 5% and 1% are often enough.
要求一个数量的百分比，将百分比转化为小数或分数，然后相乘。例如，80 的 30% = 0.30 × 80 = 24。在不使用计算器的题目中，将百分比分解为简单的部分：10%、5% 和 1% 通常就足够了。
Example: Find 15% of 60.
10% of 60 = 6, 5% of 60 = 3, so 15% of 60 = 6 + 3 = 9.
示例：求 60 的 15%。
60 的 10% = 6，60 的 5% = 3，因此 60 的 15% = 6 + 3 = 9。
6. Percentage Increase and Decrease | 百分比增减
nt by (1 + p/100). To decrease an amount, multiply by (1 - p/100). For example, increasing £80 by 5% gives £80 × 1.05 = £84. Decreasing £80 by 5% gives £80 × 0.95 = £76.
要将一个量增加某个百分比，将原量乘以（1 + p/100）。要减少一个量，乘以（1 - p/100）。例如，将 80 英镑增加 5%，得到 80 × 1.05 = 84 英镑。将 80 英镑减少 5%，得到 80 × 0.95 = 76 英镑。
nal amount is unknown, reverse the process by dividing. For example, if a price after a 20% decrease is £40, then the original price is £40 ÷ 0.80 = £50.
当原量未知时，用除法反向求解。例如，如果价格下降 20% 后为 40 英镑，则原价为 40 ÷ 0.80 = 50 英镑。
7. Comparing and Ordering FDP | 分数、小数和百分比的比较与排序
e to the same form. Decimals are usually the easiest common form. Convert 3/5, 0.58 and 55% to decimals: 3/5 = 0.6, 0.58 = 0.58, 55% = 0.55. Therefore 3/5 > 0.58 > 55%.
要比较分数、小数和百分比，将三者都转化为同一种形式。小数通常是最容易的共同形式。将 3/5、0.58 和 55% 都转化为小数：3/5 = 0.6，0.58 = 0.58，55% = 0.55。因此 3/5 > 0.58 > 55%。
o order 2/3, 3/4 and 5/6, use 12: 2/3 = 8/12, 3/4 = 9/12, 5/6 = 10/12, so 2/3 < 3/4 < 5/6.
对于分母不同的分数，先找到公分母。要排序 2/3、3/4 和 5/6，使用 12：2/3 = 8/12，3/4 = 9/12，5/6 = 10/12，所以 2/3 < 3/4 < 5/6。
8. Word Problems and Real-Life Applications | 应用题与实际应用
Cambridge KS3 exam questions often set FDP problems in real contexts: discounts, test scores, VAT, interest and probability. Read the problem carefully, identify the original amount and the percentage change, and decide whether to multiply or divide. Always state the units in your final answer.
剑桥 KS3 考试题目经常将分数、小数和百分比问题放在真实情境中：折扣、考试成绩、增值税、利息和概率。仔细阅读题目，确定原量和百分比变化，并决定使用乘法还是除法。最终答案中始终标明单位。
Example: A jacket costs £45 before a 20% discount. Amount saved = 0.20 × 45 = £9. Sale price = £45 - £9 = £36. If a student scores 36 out of 48, the percentage score is 36 ÷ 48 × 100 = 75%.
示例：一件夹克原价 45 英镑，打八折。节省金额 = 0.20 × 45 = 9 英镑。售价 = 45 - 9 = 36 英镑。如果一名学生满分 48 分得到 36 分，则百分比成绩为 36 ÷ 48 × 100 = 75%。
9. Common Errors and How to Avoid Them | 常见错误与避免方法
One common mistake is adding percentages to different bases. For example, a 10% increase followed by a 10% decrease does not return to the original amount. Starting with £100, a 10% increase gives £110; then a 10% decrease gives £99. Always check which amount the percentage is applied to.
一个常见错误是将百分比加在不同的基数上。例如，先增加 10% 再减少 10% 并不能回到原量。从 100 英镑开始，增加 10% 得到 110 英镑；再减少 10% 得到 99 英镑。始终检查百分比作用于哪个数量。
- Do not confuse percentages with percentage points.
不要将百分比与百分点混淆。 - Do not forget to divide by 100 before multiplying.
不要忘记在相乘前除以 100。 - Do not round too early in multi-step problems.
在多步问题中不要过早舍入。
10. Exam Tips and Checklist | 考试技巧与检查清单
age at the end. Memorise common equivalents such as 1/8 = 0.125, 1/3 ≈ 0.333, 2/3 ≈ 0.667.
展示每一步计算过程，即使最终答案错误，也能获得方法分。计算过程中将百分比保持为小数或分数，只在最后转化为百分比。熟记常见等价关系，如 1/8 = 0.125，1/3 ≈ 0.333，2/3 ≈ 0.667。
Use estimation to spot unreasonable answers. For example, 40% of 90 should be around 36, not 3.6 or 360. Finally, check that your percentage increase has a multiplier greater than 1, and a percentage decrease has a multiplier less than 1.
使用估算发现不合理的答案。例如，90 的 40% 应约为 36，而不是 3.6 或 360。最后，检查百分比增加的乘数是否大于 1，百分比减少的乘数是否小于 1。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图