📚 Mastering Percentages for Cambridge KS3 Maths | 掌握剑桥 KS3 数学百分数
A percentage is a fraction with denominator 100. The symbol % means 'out of 100', so 35% ink fractions, decimals, ratio and real-life problems such as discounts, tax and interest.
百分数是一个分母为 100 的分数。符号 % 表示“每一百”，因此 35% = 35/100 = 0.35。在 KS3 阶段，百分数将分数、小数、比与实际生活问题（如折扣、税费和利息）联系起来。
1. What Is a Percentage? | 什么是百分数？
t of 100. It allows different quantities to be compared on the same scale. For example, 25% of every 100, which can also be written as the fraction 25/100 = 1/4 or the decimal 0.25.
百分数是一种把数字表示为每一百中的多少的方式。它使不同的数量能够在同一尺度上进行比较。例如，25% 表示每一百中有 25，也可以写成分数 25/100 = 1/4 或小数 0.25。
Every percentage can be converted into a fraction or decimal, and these conversions are the foundation for many KS3 calculations.
每个百分数都可以转换为分数或小数，这些转换是许多 KS3 计算的基础。
2. Converting Between Fractions, Decimals and Percentages | 分数、小数与百分数互化
To convert a decimal to a percentage, multiply by 100. To convert a percentage to a decimal, divide by 100. To convert a fraction to a percentage, multiply the fraction by 100. To convert a percentage to a fraction, write it over 100 and simplify.
将小数化为百分数时乘以 100；将百分数化为小数时除以 100。将分数化为百分数时用分数乘以 100；将百分数化为分数时写成分母为 100 的分数并化简。
- 0.75 × 100 = 75%
- 40% ÷ 100 = 0.40
- 3/5 × 100 = 60%
- 65% = 65/100 = 13/20
These conversions are useful when you need to compare a test score, a discount or a data proportion quickly.
当需要快速比较考试成绩、折扣或数据比例时，这些转换非常有用。
3. Finding a Percentage of a Quantity | 求一个量的百分数
amount, use the rule: percentage of amount = (percentage / 100) × amount. For example, 15% of £80 is calculated as 0.15 × 80 = £12.
要求一个量的百分数，使用规则：量的百分数 = (百分数 / 100) × 总量。例如，80 英镑的 15% 计算为 0.15 × 80 = 12 英镑。
15% of 80 = (15 / 100) × 80 = £12
This method is tested in questions about discounts, test marks, tax and commission.
该方法常用于折扣、考试分数、税费和佣金类问题。
4. Using Non-Calculator Methods: 10%, 5%, 1% and Multiples | 非计算器方法：10%、5%、1%及倍数
You can build harder percentages from simple benchmarks. 10% = amount ÷ 10, 5% = 10% ÷ 2, and 1% = amount ÷ 100. For example, to find 35% of 240, first find 10% = 24, then 5% = 12, so 30% = 3 × 24 = 72 and 35% = 72 + 12 = 84.
你可以从简单的基准构造更复杂的百分数。10% = 总量 ÷ 10，5% = 10% ÷ 2，1% = 总量 ÷ 100。例如，求 240 的 35%：先求 10% = 24，再求 5% = 12，因此 30% = 3 × 24 = 72，35% = 72 + 12 = 84。
- 10% of 240 = 24
- 5% of 240 = 12
- 35% of 240 = 72 + 12 = 84
This non-calculator approach is often required in Cambridge KS3 mental and written papers.
这种非计算器方法在剑桥 KS3 的口算和笔试中经常要求使用。
5. Percentage Increase and Decrease | 百分数增加与减少
To increase an amount by r%, multiply by (1 + r / 100). To decrease an amount by r%, multiply by (1 - r / 100). Example: £60 increased by 15% = 60 × 1.15 = £69. £80 decreased by 25% = 80 × 0.75 = £60.
将数量增加 r%，乘以 (1 + r / 100)；减少 r%，乘以 (1 - r / 100)。例如，60 英镑增加 15% = 60 × 1.15 = 69 英镑；80 英镑减少 25% = 80 × 0.75 = 60 英镑。
New value = Original × (1 ± r / 100)
Using multipliers is faster and less prone to error than finding the change and then adding or subtracting.
使用乘数比先求出变化量再加减更快，也更不容易出错。
6. Finding the Original Value After a Percentage Change | 百分数变化后求原值
If a final amount includes a known percentage change, divide by the multiplier. After a 20% increase, final value = original × 1.20, so original = final ÷ 1.20. After a 15% decrease, final value = original × 0.85, so original = final ÷ 0.85.
如果最终数量包含已知的百分数变化，除以乘数即可。增加 20% 后，最终值 = 原值 × 1.20，因此原值 = 最终值 ÷ 1.20；减少 15% 后，最终值 = 原值 × 0.85，因此原值 = 最终值 ÷ 0.85。
Example: A coat costs £69 after a 15% increase. Original price = 69 ÷ 1.15 = £60.
例如：一件外套涨价 15% 后为 69 英镑。原价 = 69 ÷ 1.15 = 60 英镑。
This reverse percentage skill is common in KS3 word problems and checkpoints.
这种反向百分数技能在 KS3 应用题和 checkpoint 考试中很常见。
7. Percentage Change Calculations | 百分数变化计算
Percentage change = (new value - original value) / original value × 100%. If a price rises from £40 to £50, the change is £10, so percentage change = 10/40 × 100 = 25%. If it falls from £50 to £45, percentage change = -5/50 × 100 = -10%, a 10% decrease.
百分数变化 = (新值 - 原值) / 原值 × 100%。如果价格从 40 英镑涨到 50 英镑，变化量为 10 英镑，因此百分数变化 = 10/40 × 100 = 25%；如果从 50 英镑降到 45 英镑，百分数变化 = -5/50 × 100 = -10%，即减少 10%。
Percentage change = (New - Original) / Original × 100%
Remember that the denominator is always the original value, not the new value.
注意分母始终是原值，而不是新值。
8. Percentages in Real-Life Contexts | 百分数在实际生活中的应用
est, use I = P × r × t / 100. For a discount, multiply the original price by (1 - discount% / 100).
百分数出现在折扣、增值税、利润与亏损、单利、考试成绩和调查数据中。单利公式为 I = P × r × t / 100。计算折扣时，用原价乘以 (1 - 折扣率 / 100)。
| Context | Example | Calculation |
| Discount | 20% off £45 | 45 × 0.80 = £36 |
| Simple interest | £200 at 5% for 2 years | I = 200 × 5 × 2 / 100 = £20 |
Always read the question carefully to decide whether you need a percentage of an amount, a percentage change or an original value.
仔细审题，判断需要求的是量的百分数、百分数变化还是原值。
9. Comparing Quantities Using Percentages | 用百分数比较数量
Percentages standardise comparisons. For example, 17 out of 25 is 68%, while 19 out of 30 is about 63.3%, so the first is a better score. Always write both as percentages before comparing to avoid misleading by different totals.
百分数使比较标准化。例如，25 题中对 17 题是 68%，30 题中对 19 题约为 63.3%，因此前者表现更好。比较前应先将两者写成百分数，以免因总数不同而产生误导。
This technique is especially useful in data and statistics questions where different sample sizes appear.
当数据与统计问题中出现不同的样本量时，这种技巧尤其有用。
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
Do not confuse percentage increase with percentage of amount. Do not add or subtract percentages directly when the bases are different. For example, a 20% increase followed by a 10% decrease is not a net 10% increase.
不要混淆“百分数增加”与“求一个量的百分数”。当基数不同时，不要直接相加或相减百分数。例如，先增加 20% 再减少 10%，净变化并不是增加 10%。
Show clear working, use multipliers, and check that the final answer is sensible. In exams, look for wording such as 'of', 'increased by', 'decreased by', 'original value' and 'percentage change'.
展示清晰步骤，使用乘数，并检查最终答案是否合理。考试中注意“的”、“增加”、“减少”、“原值”和“百分数变化”等关键词。
Practising these question types from Cambridge KS3 past papers will help you avoid common traps and gain confidence.
练习剑桥 KS3 历年真题中的这些题型，有助于避开常见陷阱并增强信心。
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