📚 Percentage Calculations (Exercise 116.1) | 百分比计算（练习 116.1）
In KS3 Cambridge Mathematics, percentages appear everywhere – from shopping discounts to test scores. Exercise 116.1 focuses on finding a percentage of a quantity, a skill that forms the backbone of many real-life problems. This article breaks down the methods step by step, ensuring you can tackle any percentage question with confidence.
在剑桥 KS3 数学中，百分比无处不在——从购物折扣到考试分数。练习 116.1 专注于求一个数量的百分比，这项技能是许多实际问题的支柱。本文将逐步拆解各种方法，确保你能自信应对任何百分比题目。
1. Understanding Percentages | 理解百分比
Percent means ‘out of 100’. The symbol % is used to represent a fraction with denominator 100. For example, 37% means 37 out of 100, or 37/100. This simple idea allows us to compare different quantities easily, no matter how large or small the original numbers are.
百分比的意思是“每一百”。符号 % 用于表示分母为 100 的分数。例如，37% 表示 100 份中的 37 份，即 37/100。这个简单的概念让我们可以轻松比较不同的数量，无论原始数字是大是小。
2. Converting Percentages to Fractions and Decimals | 百分比与分数、小数的转换
To find a percentage of a quantity, we usually need to change the percentage into a decimal or a fraction. Dividing the percentage by 100 gives the decimal form: 42% = 42 ÷ 100 = 0.42. As a fraction, 42% = 42/100, which can often be simplified. For example, 42/100 simplifies to 21/50 by dividing numerator and denominator by 2.
要计算一个数量的百分之几，我们通常需要把百分比转换为小数或分数。把百分数除以 100 就得到小数形式：42% = 42 ÷ 100 = 0.42。作为分数，42% = 42/100，通常还可以化简。例如，42/100 分子分母同时除以 2 就得到 21/50。
42% = 42/100 = 0.42
42% = 42/100 = 0.42
3. The Multiplier Method | 乘数法
One of the fastest ways to find a percentage of a quantity is to use a multiplier. First, write the percentage as a decimal (the multiplier). Then multiply the original quantity by this decimal. For example, to find 35% of 200, the multiplier is 35 ÷ 100 = 0.35. Multiply: 200 × 0.35 = 70. So 35% of 200 is 70.
求一个数量的百分比最快的方法之一是使用乘数。首先，将百分比写成小数（即乘数）。然后，用原始数量乘以这个小数。例如，求 200 的 35%，乘数是 35 ÷ 100 = 0.35。相乘：200 × 0.35 = 70。因此，200 的 35% 是 70。
The table below shows common percentages and their decimal multipliers:
下表展示了一些常见百分比及其小数乘数：
| Percentage | Decimal Multiplier |
|---|---|
| 100% | 1.0 |
| 50% | 0.5 |
| 25% | 0.25 |
| 10% | 0.1 |
| 5% | 0.05 |
| 1% | 0.01 |
4. Using Benchmarks: 10%, 5%, 1% | 利用基准值：10%、5%、1%
If you prefer mental calculation, you can build up a percentage from 10%, 5% and 1% blocks. To find 10% of a number, simply divide by 10. Then 5% is half of 10%, and 1% is one-tenth of 10%. By combining these, you can work out many percentages without a calculator. For example, to find 23% of 180: 10% = 18, so 20% = 36, then 1% = 1.8, so 3% = 5.4. Add them: 36 + 5.4 = 41.4.
如果你更喜欢心算，可以用 10%、5% 和 1% 作为积木来搭建百分比。求一个数的 10%，只需除以 10。然后 5% 是 10% 的一半，1% 是 10% 的十分之一。通过组合这些值，你可以在不使用计算器的情况下算出很多百分比。例如，求 180 的 23%：10% = 18，所以 20% = 36；然后 1% = 1.8，所以 3% = 5.4。加起来：36 + 5.4 = 41.4。
23% of 180 = 41.4
180 的 23% = 41.4
5. Percentage Increase and Decrease | 百分比的增加与减少
When a quantity is increased by a percentage, we can find the increase and then add it to the original. Alternatively, adding the percentage to 100% gives the new total percentage. For an increase of 15%, the new amount is 115% of the original. Using a multiplier of 1.15 is often quicker. For example, to increase £240 by 15%, multiply: 240 × 1.15 = 276. The new amount is £276.
当一个数量增加某个百分比时，我们可以先求出增加量，然后加到原数上。或者，用 100% 加上增加的百分比，得到新的总百分比。对于增加 15%，新数量是原来的 115%。使用乘数 1.15 往往更快。例如，要把 £240 增加 15%，相乘：240 × 1.15 = 276。新金额为 £276。
For a percentage decrease, subtract the percentage from 100% to get the remaining percentage. A decrease of 20% leaves 80% of the original, so the multiplier is 0.8. Decreasing £240 by 20% gives 240 × 0.8 = 192.
对于百分比减少，从 100% 中减去减少的百分比，得到剩余百分比。减少 20% 后剩下原来的 80%，所以乘数是 0.8。£240 减少 20% 为 240 × 0.8 = 192。
Increase by 15%: ×1.15
增加 15%：×1.15
Decrease by 20%: ×0.8
减少 20%：×0.8
6. Reverse Percentages – Finding the Original Amount | 逆向百分比——求原值
Sometimes we know the amount after a percentage increase or decrease, and we need to find the original amount. This is called a reverse percentage. If 115% of the original is known, the original is found by dividing by 1.15. For example, a jacket’s price after a 15% increase is £57.50. The original price is 57.50 ÷ 1.15 = 50. So the jacket cost £50 before the increase.
有时我们知道一个数量经过百分比增加或减少后的值，需要求原值。这叫做逆向百分比。如果已知原值的 115%，那么原值就是除以 1.15。例如，一件夹克涨价 15% 后价格为 £57.50。原价是 57.50 ÷ 1.15 = 50。所以涨价前夹克的价格是 £50。
Similarly, after a 20% decrease, the remaining amount is 80% of the original. If a sale price is £64, then the original was 64 ÷ 0.8 = 80. Always identify whether the given amount represents more than or less than 100% and choose the correct divisor.
同样，减少 20% 后，剩余金额是原价的 80%。如果折后价为 £64，原价就是 64 ÷ 0.8 = 80。一定要先判断给出的数值代表多于还是少于 100%，然后选择合适的除数。
7. Expressing One Quantity as a Percentage of Another | 求一个量占另一个量的百分比
To express one number as a percentage of another, write the first number divided by the second, then multiply by 100%. For example, if a student scores 18 out of 25 in a test, the percentage is (18/25) × 100% = 72%. This method works whenever you need to compare a part to a whole.
要把一个数表示为另一个数的百分之几，就用第一个数除以第二个数，再乘以 100%。例如，一名学生在测试中得了 25 分中的 18 分，百分比是 (18/25) × 100% = 72%。每当需要将部分与整体进行比较时，都可以使用这种方法。
It is also useful to simplify the fraction first if possible. For instance, 16 out of 40 can be simplified to 2/5, and then (2/5) × 100% = 40%. This often makes mental arithmetic easier.
在可能的情况下，先化简分数也很有用。例如，40 中的 16 可以化简为 2/5，然后 (2/5) × 100% = 40%。这常常使心算更加容易。
8. Real-Life Applications – Discounts and Interest | 实际应用——折扣与利息
Percentages are everywhere in money situations. A shop may offer a 25% discount on a £120 item. The discount amount is 25% of 120 = 0.25 × 120 = £30, so the sale price is 120 − 30 = £90. Using a multiplier, you can directly calculate 75% of 120 = 0.75 × 120 = £90.
百分比在金钱场景中随处可见。一家商店可能对一件 £120 的商品打 25% 的折扣。折扣金额是 120 的 25% = 0.25 × 120 = £30，所以折后价为 120 − 30 = £90。使用乘数，可以直接计算 120 的 75% = 0.75 × 120 = £90。
Simple interest is another common application. If you deposit £200 in a bank account earning 4% simple interest per year, the interest for one year is 0.04 × 200 = £8. After three years, the interest is 3 × 8 = £24, and the total amount is £224.
单利是另一个常见应用。如果你在银行存入 £200，年单利率为 4%，一年的利息是 0.04 × 200 = £8。三年后利息为 3 × 8 = £24，总额为 £224。
9. Common Mistakes and How to Avoid Them | 常见错误与避免方法
Many learners forget to divide the percentage by 100 before multiplying. For example, they might calculate ‘35% of 200’ as 35 × 200 = 7000, which is clearly wrong. Always convert the percentage to a decimal first. Another common error is mixing up ‘increase by 20%’ with ‘increase to 120%’. Although both refer to the same multiplier (1.2), it’s important to read questions carefully. In reverse percentages, students often multiply instead of divide. If a price is after a 20% reduction, don’t multiply by 0.8 again; divide by 0.8 to find the original.
许多学习者会忘记在乘法前将百分比除以 100。例如，他们可能会把“200 的 35%”算成 35 × 200 = 7000，这显然是错的。一定要先把百分比转化为小数。另一个常见错误是混淆“增加 20%”和“增加到 120%”。虽然两者引用相同的乘数 (1.2)，但仔细读题很重要。在逆向百分比中，学生经常该除的时候用了乘法。如果价格是减少 20% 之后的，不要再次乘以 0.8；要除以 0.8 才能得到原值。
To avoid mistakes, always check the units and whether the final answer makes sense. For example, 10% of a number should be much smaller than the number itself, and a price after a discount should be lower than the original.
为了避免错误，请始终检查单位和最终答案是否合理。例如，一个数的 10% 应该比这个数小很多，折扣后的价格应该低于原价。
10. Practice Questions with Step-by-Step Solutions | 练习题与分步解答
Test your understanding with these questions modelled on Exercise 116.1:
用这些模拟练习 116.1 的题目来检验你的理解：
- Find 18% of 350.
- 求 350 的 18%。
- A dress is reduced by 30% in a sale. If the original price was £85, what is the sale price?
- 一条裙子在原价 £85 的基础上打七折，折后价是多少？
- After a 12% increase, a train ticket costs £67.20. What was the original cost?
- 一张火车票涨价 12% 后票价是 £67.20，原票价是多少？
- Out of 60 pupils, 45 passed a test. What percentage passed?
- 60 名学生中有 45 人通过了一项测验，通过率是多少？
Solutions / 解答：
1. Multiplier for 18% = 0.18. 0.18 × 350 = 63.
18% 的乘数是 0.18。0.18 × 350 = 63。
2. A 30% reduction leaves 70%. Multiply by 0.7: 85 × 0.7 = 59.50. The sale price is £59.50.
减少 30% 后剩下 70%。乘以 0.7：85 × 0.7 = 59.50。折后价是 £59.50。
3. The ticket after increase is 112% of the original, so divide by 1.12: 67.20 ÷ 1.12 = 60. The original cost was £60.
涨价后的票价是原价的 112%，所以要除以 1.12：67.20 ÷ 1.12 = 60。原票价是 £60。
4. (45/60) × 100% = 75%.
(45/60) × 100% = 75%。
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