📚 Percentage Change and Reverse Percentages | 百分比增减与逆向计算
Percentages are everywhere – from sale discounts in shops to interest rates in banks. In KS3 Cambridge Mathematics, you need to go beyond finding a simple percentage of a quantity. You must master percentage increase, percentage decrease, and reverse percentage calculations. This article explains each type step by step, with careful translations into Chinese, so you can confidently tackle any percentage problem in a real-world context.
百分比无处不在——从商店的打折到银行的利率。在KS3剑桥数学中，你需要掌握的不只是求一个量的简单百分比。你必须精通百分比增加、百分比减少以及逆向百分比计算。本文逐步解释每种类型，并提供对应的中文翻译，帮助你自信地解决现实情境中的任何百分比问题。
1. What Is a Percentage? | 什么是百分比？
A percentage is simply a fraction out of 100. It tells you how many parts you have per hundred. The word comes from the Latin 'per centum', meaning 'by the hundred'. We use the symbol % to represent a percentage. For example, 30% means 30 out of 100, which can be written as the fraction 30/100 or the decimal 0.30.
百分比就是一个分母为100的分数。它表示每一百份中有多少份。这个词来自拉丁语“per centum”，意思是“每一百”。我们用符号%来表示百分比。例如，30%意味着100份中的30份，可以写成分数30/100或小数0.30。
To convert a percentage into a decimal, divide by 100 – simply move the decimal point two places to the left. So 7% becomes 0.07, and 125% becomes 1.25. This is essential for all percentage calculations.
要把百分比转换为小数，除以100——只需要把小数点向左移动两位。因此，7%变成0.07，125%变成1.25。这对所有百分比计算都至关重要。
2. The Decimal Multiplier Method | 小数乘数法
The most efficient way to handle percentage change is by using a decimal multiplier. For any percentage increase or decrease, you can multiply the original amount by a single number to get the new amount directly. This method avoids the two-step process of finding the percentage then adding or subtracting.
处理百分比变化最有效的方法是使用小数乘数。对于任何百分比的增加或减少，你可以将原量乘以一个单一的数字，直接得到新的量。这种方法避免了先求百分比再加或减的两步过程。
For an increase of r%, the multiplier is 1 + (r ÷ 100). For a decrease of r%, the multiplier is 1 – (r ÷ 100). If the change is less than 100%, the multiplier stays positive. If the decrease is exactly 100%, the multiplier becomes 0. If you have a decrease greater than 100% (which is not common in KS3 but worth noting), the multiplier would be negative, representing a loss beyond the original value.
对于增加r%，乘数是 1 + (r ÷ 100)。对于减少r%，乘数是 1 – (r ÷ 100)。如果变化小于100%，乘数保持正数。如果正好减少100%，乘数变为0。如果减少超过100%（在KS3阶段不常见，但值得注意），乘数将为负数，表示超出原值的损失。
Multiplier for increase: 1 + (r/100)
Multiplier for decrease: 1 - (r/100)
3. Percentage Increase | 百分比增加
Suppose a shop increases the price of a T-shirt by 15%. The original price is £ rst calculate the multiplier: 1 + (15 ÷ 100) = 1.15. Multiply the original amount by 1.15: £20 × 1.15 = £23. The T-shirt now costs £23.
假设一家商店将一件T恤的价格提高了15%。原价是20英镑。要找到新价格，首先计算乘数： 1 + (15 ÷ 100) = 1.15。将原量乘以1.15： 20 × 1.15 = 23英镑。这件T恤现在售价23英镑。
You can also think in steps: find 15% of £20 (which is £3), then add it to the original: £20 + £3 = £ The multiplier method is quicker, especially when dealing with several successive changes.
你也可以分步思考：先求20英镑的15%（即3英镑），然后把它加到原价上： 20 + 3 = 23英镑。当涉及多个连续变化时，乘数法更快。
Example: A population of 4500 grows by 8%. New population = 4500 × 1.08 = 4860.
例子：一个4500的人口增长了8%。新人口数 = 4500 × 1.08 = 4860。
4. Percentage Decrease | 百分比减少
se works in the same way, but the multiplier is less than 1. If a bicycle originally costs £360 and is reduced by 20% in a sale, the multiplier is 1 – 0.20 = 0.80. The sale price = £360 × 0.80 = £288.
百分比减少的方法相同，但乘数小于1。如果一辆自行车原价360英镑，打八折（降价20%），乘数为 1 – 0.20 = 0.80。售价 = 360 × 0.80 = 288英镑。
Do not fall into the trap of multiplying by the percentage number directly. For a 20% decrease, the new value is 80% of the original, not 20% of the original. Always use the correct multiplier.
不要掉入直接用百分比数字去乘的陷阱。对于20%的减少，新值是原值的80%，而不是原值的20%。务必使用正确的乘数。
Example: A car's value depreciates by 30%. If it was worth £12000, its new value = 12000 × 0.70 = £8400.
例子：一辆汽车的价值贬值了30%。如果它原本值12000英镑，新价值 = 12000 × 0.70 = 8400英镑。
5. Calculating the Percentage Change | 计算百分比变化
Sometimes you know the old amount and the new amount, but you need to find the percentage increase or decrease. The formula is:
有时你知道旧量和新量，但需要求出百分比的增加或减少。公式为：
Percentage change = ( (new value - original value) ÷ original value ) × 100%
百分比变化 = ( (新值 - 原值) ÷ 原值 ) × 100%
egative, it is a decrease. For example, a plant grows from 30 cm to 45 cm. The change = 45 – 30 = 15 cm. Percentage increase = (15 ÷ 30) × 100% = 0.5 × 100% = 50%. The plant's height increased by 50%.
如果答案为正值，则是增加；如果是负值，则是减少。例如，一株植物从30厘米长到45厘米。变化值 = 45 – 30 = 15厘米。百分比增加 = (15 ÷ 30) × 100% = 0.5 × 100% = 50%。植物的高度增加了50%。
What if the value drops? A phone's battery charge falls from 80% to 60%. Change = 60 – 80 = –20. Percentage change = (–20 ÷ 80) × 100% = –25%. So the battery charge decreased by 25%.
如果数值下降呢？一部手机的电量从80%降到60%。变化值 = 60 – 80 = –20。百分比变化 = (–20 ÷ 80) × 100% = –25%。所以电量减少了25%。
6. Reverse Percentages: Finding the Original Amount | 逆向百分比：求原值
A classic KS3 problem: a price after a 10% discount is £45. What was the original price? This requires working backwards. While 10% was taken off, the £45 represents 90% of the original (100% – 10%). So 90% of the original = 45. To find 100% (the original), divide by the multiplier 0.90, not multiply. Original price = £45 ÷ 0.90 = £50.
一个经典的KS3问题：一件商品打九折后的价格是45英镑。原价是多少？这需要逆向计算。因为减去了10%，所以45英镑代表原价的90%（100% – 10%）。因此原价的90% = 45。要求出100%（原价），需要除以乘数0.90，而不是乘以它。原价 = 45 ÷ 0.90 = 50英镑。
Always identify what percentage the given amount represents. In a decrease problem, if an amount is after a reduction of r%, it represents (100 – r)% of the original. In an increase problem, an amount after an increase of r% represents (100 + r)% of the original. Then divide by the corresponding decimal multiplier.
始终要确定给定的量代表的是百分之几。在减少问题中，如果某个量是在减少了r%之后的结果，那么它代表原值的(100 – r)%。在增加问题中，增加r%之后的量代表原值的(100 + r)%。然后除以相应的小数乘数。
Example: After a 20% tax, a final bill is £180. What was the pre-tax amount? £180 represents 120% of the original. Multiplier = 1.20. Original = 180 ÷ 1.20 = £150.
例子：加上20%的税后，最终账单是180英镑。税前金额是多少？180英镑代表原值的120%。乘数 = 1.20。原值 = 180 ÷ 1.20 = 150英镑。
7. Common Misconceptions with Reverse Percentages | 逆向百分比的常见误解
A very common mistake is to find r% of the given amount and simply subtract (for a decrease) or add (for an increase) to get the original. For instance, finding 10% of £45 is £4.50, then adding to £45 gives £49.50 – wrong! This method fails because the 10% you add must relate to the unknown original, not the given reduced amount.
一个非常常见的错误是求出给定量的r%，然后简单地减去（对于减少）或加上（对于增加）来得到原值。例如，求45英镑的10%得到4.50英镑，然后加到45英镑上得到49.50英镑——这是错误的！这种方法之所以失效，是因为你加上的10%必须与未知的原值相关，而不是给定的减少后的量。
The correct approach always involves division by the multiplier. Always ask yourself: 'What percentage of the original does this known quantity represent?' e equation: multiplier × original = known amount, so original = known amount ÷ multiplier.
正确的方法始终涉及除以乘数。永远要问自己：“这个已知的量代表原值的百分之几？”然后建立等式：乘数 × 原值 = 已知量，所以原值 = 已知量 ÷ 乘数。
8. Dealing with Multiple Percentage Changes | 处理多次百分比变化
Things often change more than once. A price might increase by 10%, then decrease by 10%. Does it go back to the original? No! Suppose initial is £100. After 10% increase: £110. Then 10% decrease: 10% of £110 is £11, so new price = £99. The final amount is lower because the second percentage is taken on a larger base.
事物常常变化不止一次。一个价格可能先增加10%，然后再减少10%。它会回到原值吗？不会！假设初始值是100英镑。增加10%后：110英镑。然后减少10%：110英镑的10%是11英镑，所以新价格 = 99英镑。最终金额变少了，因为第二次的百分比是在更大的基数上计算的。
s, just multiply the original amount by each multiplier in sequence. For an increase of r1% followed by a decrease of r2%, overall multiplier = (1 + r1/100) × (1 – ). You can then find the overall percentage change by comparing the final multiplier to 1.
要处理多次变化，只需将原量依次乘以每个乘数。对于先增加r1%再减少r2%的情况，总体乘数 = (1 + r1/100) × (1 – r2/100)。然后你可以通过比较最终乘数与1来求出总体百分比变化。
Example: A salary rises by 5% then by 4%. The combined multiplier = 1.05 × 1.04 = 1.092, which means an overall increase of 9.2%, not simply 9%.
例子：工资先涨了5%，然后又涨了4%。组合乘数 = 1.05 × 1.04 = 1.092，这意味着总体增加了9.2%，而不是简单的9%。
9. Real-World Applications: Discounts and Sales | 实际应用：折扣与促销
Shops often advertise '20% off' or 'Buy one, get one 50% off'. With a simple discount, you can apply the direct multiplier to the marked price. But beware of successive discounts: 'Take an extra 10% off already reduced prices' means you multiply by the second discount multiplier on the first reduced price, not on the original ticket price.
商店经常打广告“打八折”或“买一件，第二件半价”。对于简单的折扣，你可以对标价直接使用乘数。但要小心连续的折扣，“在已降价的基础上再打九折”意味着你要在第一次减价后的价格上乘以第二个折扣的乘数，而不是在最开始的原价上乘。
If an item originally priced £70 is marked down by 20%, then by an extra 10%, the final price is: £70 × 0.80 × 0.90 = £50.40. The total reduction is 100% – (0.80 × 0.90 × 100)% = 100% – 72% = 28% off, not 30%.
如果一件定价70英镑的商品先打八折，再额外打九折，最终价格是：70 × 0.80 × 0.90 = 50.40英镑。总折扣是 100% – (0.80 × 0.90 × 100)% = 100% – 72% = 28%的折扣，而不是30%。
10. Problems Involving Profit and Loss | 涉及利润和损失的问题
ercentage is usually expressed as a percentage of the cost price. If a shop buys goods for £150 and wants a 20% profit, the selling price = cost price × 1.20 = £180. The profit is £30.
当企业盈利时，售价比成本价高。利润百分比通常以成本价的百分比来表示。如果一家商店以150英镑买入商品，并希望获得20%的利润，那么售价 = 成本价 × 1.20 = 180英镑。利润为30英镑。
For a loss, the selling price is lower than the cost price. A loss of 15% means the selling price = cost price × (1 – 0.15) = cost price × 0.85. If a trader sells something for £340 at a 15% loss, what was the cost price? The £340 represents 85% of the cost. So cost = 340 ÷ 0.85 = £400.
对于亏损，售价比成本价低。亏损15%意味着售价 = 成本价 × (1 – 0.15) = 成本价 × 0.85。如果一个商人以340英镑出售某物并亏损了15%，成本价是多少？340英镑代表成本的85%。所以成本 = 340 ÷ 0.85 = 400英镑。
Be careful: profit percentage is based on the cost price unless stated otherwise. In KS3, always use the cost price as the 100% reference in profit and loss calculations.
注意：除非另有说明，利润百分比是以成本价为基础的。在KS3阶段，计算利润和损失时始终以成本价作为100%的基准。
11. Percentage Change with Measurements and Units | 带测量与单位的百分比变化
Percentages do not have units – they are dimensionless numbers. However, you may calculate the percentage change of a quantity that has units, like length, mass, or volume. The steps are identical: subtract the old value from the new, divide by the old value, and multiply by 100%. Make sure the units are the same before subtracting.
百分比没有单位——它们是无量纲的数字。然而，你可以计算具有单位的量的百分比变化，例如长度、质量或体积。步骤是相同的：用新值减去旧值，除以旧值，再乘以100%。确保相减之前单位一致。
bag of flour originally weighs 2 kg. After baking, 300 g is used. What percentage remains? Remaining mass = 2000 g – 300 g = 1700 g. Percentage remaining = (1700 ÷ 2000) × 100% = 85%. The decrease is 15%.
例如，一袋面粉原来重2公斤。烘烤后，用掉了300克。剩下的百分比是多少？剩余质量 = 2000克 – 300克 = 1700克。剩余百分比 = (1700 ÷ 2000) × 100% = 85%。减少了15%。
In science and geography, you often see percentage change in contexts like population growth, rainfall, or temperature variation (though be cautious with negative temperatures). The same formula applies.
在科学和地理中，你经常会在人口增长、降雨量或温度变化（尽管负温度要小心）的情形中看到百分比变化。同样的公式适用。
12. Summary and Key Tips | 总结与关键提示
To master percentages for KS3 Cambridge Mathematics, always remember the decimal multiplier. For an increase of r%, multiply by (1 + r/100). For a decrease of r%, multiply by (1 – r/100). For reverse problems, divide the known amount by the multiplier that corresponds to the percentage it represents. Never add or subtract the percentage of the known amount back to itself to find an original.
要在KS3剑桥数学中掌握百分比，要永远记住小数乘数。对于增加r%，乘以 (1 + r/100)。对于减少r%，乘以 (1 – r/100)。对于逆向问题，用已知量除以代表其百分比的乘数。永远不要将已知量的百分比加回到它自身上来求原值。
Practice regularly with real-life examples: shopping discounts, bank interest, exam scores, and statistical changes. Over time, choosing the correct multiplier will become second nature.
定期用现实生活中的例子进行练习：购物折扣、银行利息、考试分数和统计变化。经过一段时间，选择正确的乘数将成为你的第二天性。
Published by TutorHao | KS3 Mathematics Revision Series | aleveler.com
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