📚 Cambridge KS3 Maths: Percentages, Ratios and Proportion | 剑桥KS3数学：百分数、比例与比
Percentages, ratios and proportion appear throughout the Cambridge KS3 mathematics curriculum, from shopping discounts and recipe changes to scale drawings and speed calculations. This revision guide brings the key ideas together so you can move from simple conversions to more demanding problem solving with confidence.
百分数、比和比例贯穿剑桥KS3数学课程，从购物折扣、配方调整到比例尺作图和速度计算都会用到。这份复习指南把核心概念集中起来，帮助你从简单转换逐步过渡到更有挑战性的问题求解。
1. Understanding Percentages | 理解百分数
A percentage is simply a number out of 100. The word 'per cent' means 'per hundred', so 37% means 37 out of every 100, and 100% means the whole amount.
百分数就是一个以100为分母的数。“per cent”表示“每一百”，所以37%表示每一百中有37，100%表示整体。
Percentages are useful because they give a standard way of comparing parts of different wholes. For example, 18 out of 25 and 70 out of 100 are hard to compare directly, but both can be written as percentages.
百分数很有用，因为它提供了比较不同整体部分的标准方法。例如，25人中的18人和100人中的70人直接比较比较困难，但两者都可以写成百分数。
percentage = (part ÷ whole) × 100
2. Converting Between Fractions, Decimals and Percentages | 分数、小数与百分数之间的转换
You need to be able to switch quickly between fractions, decimals and percentages. To change a fraction to a percentage, divide the numerator by the denominator and then multiply by 100.
你需要能够在分数、小数和百分数之间快速转换。要把分数转换为百分数，先用分子除以分母，再乘以100。
ivide by 100. To change a decimal to a percentage, multiply by 100. For example, 0.45 = 45% and 8% = 0.08.
要把百分数转换为小数，除以100；要把小数转换为百分数，乘以100。例如，0.45 = 45%，8% = 0.08。
Common equivalents such as 1/2 = 0.5 = 50%, 1/4 = 0.25 = 25% and 3/4 = 0.75 = 75% should be memorised because they speed up calculations in exams.
常见的等值关系如 1/2 = 0.5 = 50%、1/4 = 0.25 = 25% 和 3/4 = 0.75 = 75% 应该记住，因为它们在考试中能加快计算速度。
3. Finding a Percentage of a Quantity | 求一个数量的百分数
Without a calculator, build up the answer using 10%, 5% and 1%. For example, to find 35% of 80, first find 10% = 8, then 5% = 4, so 30% = 24 and 5% = 4, giving 28.
在没有计算器的情况下，可以利用10%、5%和1%来逐步求出答案。例如，求80的35%，先求10% = 8，再求5% = 4，所以30% = 24，5% = 4，得到28。
With a calculator, convert the percentage to a decimal and multiply. For 35% of 80, calculate 0.35 × 80 = 28.
使用计算器时，把百分数转换为小数再相乘。求80的35%，计算 0.35 × 80 = 28。
ge is greater than 100, the answer is larger than the original quantity. For instance, 120% of 50 = 1.2 × 50 = 60.
当百分数大于100时，结果比原数量大。例如，50的120%等于 1.2 × 50 = 60。
4. Percentage Increase and Decrease | 百分数增减
To increase a quantity by a percentage, add the percentage to 100% and then use the new percentage. For example, increasing 60 by 15% means finding 115% of 60, which is 1.15 × 60 = 69.
要把一个数量增加某个百分数，先把该百分数加到100%上，再求新的百分数。例如，把60增加15%就是求60的115%，即 1.15 × 60 = 69。
To decrease a quantity, subtract the percentage from 100%. Decreasing 80 by 20% means finding 80% of 80, which is 0.8 × 80 = 64.
要减少某个百分数，从100%中减去该百分数。把80减少20%就是求80的80%，即 0.8 × 80 = 64。
Always pay attention to whether the question asks for the increase only or the new amount. The increase on 60 at 15% is 9, but the new amount is 69.
始终注意题目要求的是只求增加量还是新数量。60增加15%，增加量是9，但新数量是69。
5. Reverse Percentages | 逆向百分数
Reverse percentage problems give the value after a percentage change and ask for the original amount. Write the new amount as a percentage of the original, then divide to find 1% and multiply to find 100%.
逆向百分数问题给出百分数变化后的值，要求原数量。把新数量写成原数量的百分数，然后除以该百分数求1%，再乘以100求100%。
For example, if a jacket costs £84 after a 30% reduction, £84 represents 70% of the original price. Divide by 70 to find 1% = £1.20, then multiply by 100 to get the original price £120.
例如，一件夹克降价30%后售价为84英镑，那么84英镑代表原价的70%。除以70得1% = 1.20英镑，再乘以100得原价120英镑。
Do not simply increase £84 by 30%, because 30% of £84 is not the same as 30% of the original price. The original amount is always the base for the percentage change.
不要简单地把84英镑增加30%，因为84英镑的30%与原价的30%并不相同。原数量才是百分数变化的基础。
6. Introduction to Ratio | 比的基础
A ratio compares quantities of the same kind. For example, if a fruit bowl has 3 apples and 5 oranges, the ratio of apples to oranges is 3 : 5.
比用来比较同类数量。例如，一个水果碗里有3个苹果和5个橙子，苹果与橙子的比就是 3 : 5。
The order of a ratio matters. A ratio of 2 : 1 is not the same as 1 : 2. Always write ratios in the order requested by the question.
比的顺序很重要。2 : 1 和 1 : 2 并不相同。一定要按照题目要求的顺序写出比。
ng each part by the same non-zero number gives an equivalent ratio, such as 2 : 3 = 4 : 6.
比可以写成不同形式但仍表示相同关系。把每一部分同时乘以或除以同一个非零数，会得到相等的比，例如 2 : 3 = 4 : 6。
7. Simplifying and Sharing in a Ratio | 简化比与按比分配
To simplify a ratio, divide all parts by their highest common factor. The ratio 12 : 18 : 24 simplifies by dividing each part by 6 to give 2 : 3 : 4.
要简化比，把所有部分同时除以它们的最大公因数。比 12 : 18 : 24 的每部分除以6，得到 2 : 3 : 4。
To share an amount in a ratio, add the parts to find the total number of shares, divide the amount by this total, then multiply by each part. For example, £90 shared in the ratio 2 : 3 gives 5 parts, so one part is £18; the shares are £36 and £54.
按比分配时，先把比的各部分相加得到总份数，用总量除以总份数，再分别乘以各部分。例如，90英镑按 2 : 3 分配，总份数为5，一份是18英镑，因此两份为36英镑，三份为54英镑。
When a ratio includes more than two parts, the same method applies. Adding all the parts gives the total number of shares used in the division.
当比包含两个以上的部分时，方法相同。把所有部分相加得到用于分配的总份数。
8. Proportion and Unitary Method | 比例与单位法
Two quantities are in direct proportion when they increase or decrease at the same rate. For example, if 5 pens cost £3, then 15 pens cost three times as much, £9.
当两个数量以相同速率增加或减少时，它们成正比。例如，5支笔3英镑，那么15支笔就是三倍价格，9英镑。
The unitary method finds the value of one unit first. If 8 kg of flour costs £6, then 1 kg costs £6 ÷ 8 = £0.75, so 12 kg costs £0.75 × 12 = £9.
单位法先求出1个单位的值。如果8千克面粉6英镑，那么1千克就是 6 ÷ 8 = 0.75英镑，所以12千克为 0.75 × 12 = 9英镑。
Use the unitary method for recipe problems, currency conversions, speeds and maps. It helps you show clear working and reduces mistakes when quantities are not multiples of each other.
单位法可用于配方问题、货币换算、速度和地图问题。它能展示清晰的计算过程，并在数量不是倍数关系时减少错误。
9. Problem Solving with Percentages and Ratios | 百分数与比的问题求解
Many KS3 exam questions combine several skills in one problem. Read the question carefully and underline key information such as 'of', 'increase', 'decrease', 'share', and 'in the ratio'.
许多KS3考试题把多项技能结合在一个问题中。仔细读题，并在“的”“增加”“减少”“分配”“按比例”等关键信息下划线。
For example, a bike worth £240 is reduced by 25% in a sale, then a further 10% is taken off the sale price. The first reduction gives £240 × 0.75 = £180, and the second gives £180 × 0.90 = £162. The final price is not 65% of £240 because the second reduction applies to a smaller amount.
例如，一辆价值240英镑的自行车在促销中降价25%，然后在促销价上再减10%。第一次降价后为 240 × 0.75 = 180英镑，第二次为 180 × 0.90 = 162英镑。最终价格不是240英镑的65%，因为第二次降价是在较小金额上计算的。
comes 3 : 2, let the original numbers be 3x and 5x, then solve 5x − 12 = 2x to find x = 4.
当添加或移除数量后比值发生变化时，使用共同乘数建立变化前后的比。例如，红蓝计数器的比为 3 : 5，拿走12个蓝色计数器后比变为 3 : 2，设原来的数量分别为 3x 和 5x，解方程 5x − 12 = 2x，得 x = 4。
10. Common Misconceptions and Exam Tips | 常见误区与应试技巧
A common mistake is confusing percentages with absolute amounts. A 50% increase on a small number can be a much smaller actual change than a 10% increase on a large number.
一个常见误区是把百分数和绝对数量混淆。对较小数字增加50%，其实际变化可能远小于对较大数字增加10%。
Another mistake is adding ratios incorrectly. When a ratio is 2 : 3 : 5, the total number of shares is 10, not 3. Always add the parts before dividing the quantity.
另一个错误是错误地相加比。当比为 2 : 3 : 5 时，总份数是10，不是3。在分配数量之前，一定要先相加各部分。
In exams, show all working even when using a calculator. Marks are given for method, and writing the multiplier or division step helps you avoid hidden errors.
考试中，即使使用计算器也要写出所有步骤。方法可以得分，写出乘数或除法步骤有助于避免隐藏错误。
Finally, check that your answer makes sense in the context of the question. A percentage decrease should not give a negative amount unless the question allows it, and a shared amount should always add back to the original total.
最后，检查答案在题目情境中是否合理。百分数减少不应得到负值，除非题目允许；按比分配的各个部分相加后应等于原来的总量。
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