Understanding Percentages | 理解百分比

📚 Understanding Percentages | 理解百分比

Percentages are all around us – from shop discounts and test scores to bank interest and statistical reports. The word ‘percent’ comes from the Latin per centum, meaning ‘by the hundred’. When we talk about a percentage, we are simply describing a quantity as a fraction of 100. This makes percentages a powerful tool for comparing different quantities on the same scale. In Key Stage 3 Mathematics, you will learn how to convert between fractions, decimals and percentages, calculate percentage increases and decreases, solve real-life problems involving percentages, and understand more advanced applications such as compound interest and reverse percentages.

百分比在我们生活中随处可见——从商店折扣、考试分数到银行利息和统计报告。“percent” 这个词源自拉丁语 per centum，意思是“每一百”。当我们谈论百分比时，其实就是把一个数量描述为 100 的几分之几。这样，百分比就成为了一个在同一尺度上比较不同数量的有力工具。在 KS3 数学阶段，你将学习如何将分数、小数和百分比相互转换，计算百分比的增加与减少，解决涉及百分比的实际问题，并理解复利和反向百分比等更高级的应用。

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1. What is a Percentage? | 什么是百分比？

A percentage represents a part out of 100. For example, 45% means 45 out of 100, which can be written as the fraction 45/100 or the decimal 0.45. Because percentages always relate to 100, they provide a standard way to express proportions, making it much easier to compare results from different situations. In mathematics, the symbol % is used to denote ‘per cent’.

百分比表示 100 份中的一部分。例如，45% 表示每 100 份中的 45 份，可以写作分数 45/100 或小数 0.45。由于百分比总是与 100 相关，它提供了一种表达比例的标准方法，使不同情况下的结果更容易进行比较。在数学中，符号 % 用于表示“每一百”。

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2. Converting Fractions to Percentages | 分数转换为百分比

To convert a fraction into a percentage, multiply the fraction by 100 and add the % sign. For instance, 3/5 as a percentage is (3 ÷ 5) × 100 = 60%. If the denominator is not a factor of 100, you can first change the fraction into a decimal by dividing the numerator by the denominator, then multiply by 100. This method works for any fraction.

要把分数转换成百分比，只需将分数乘以 100 并加上 % 符号。例如，3/5 作为百分比就是 (3 ÷ 5) × 100 = 60%。如果分母不是 100 的因数，你可以先将分子除以分母化成小数，再乘以 100。这个方法对于任何分数都适用。

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3. Converting Decimals to Percentages | 小数转换为百分比

Converting a decimal to a percentage is straightforward: multiply the decimal by 100 and add the % sign. For example, 0.28 becomes 28%, and 1.5 becomes 150%. This works because decimals are based on place value, and multiplying by 100 shifts the decimal point two places to the right. A decimal greater than 1 will give a percentage larger than 100%, which is often used to represent growth or profit beyond the original amount.

将小数转换为百分比非常简单：将小数乘以 100 并加上 % 符号。例如，0.28 变成 28%，1.5 变成 150%。因为小数是按位值计算的，乘以 100 相当于把小数点向右移动两位。大于 1 的小数会产生大于 100% 的百分比，这通常用来表示超过原有数量的增长或利润。

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4. Converting Percentages to Fractions and Decimals | 百分比转换为分数和小数

To change a percentage into a fraction, write the percentage over 100 and then simplify if possible. For example, 75% = 75/100 = 3/4. For a decimal, simply divide the percentage by 100, which means moving the decimal point two places to the left. 8% becomes 0.08. Mixed number percentages like 12½% can first be turned into an improper fraction (25/2 %), then divided by 100 to give (25/2) × (1/100) = 25/200 = 1/8 as a fraction, or 0.125 as a decimal.

要将百分比转换为分数，把百分比写在 100 之上，然后尽可能地化简。例如，75% = 75/100 = 3/4。若要转换为小数，只需将百分比除以 100，也就是把小数点向左移动两位。8% 变成 0.08。像 12½% 这样的带分数百分比可以先化成假分数 (25/2 %)，再除以 100 得到 (25/2) × (1/100) = 25/200 = 1/8 的分数，或者 0.125 的小数。

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5. Finding a Percentage of an Amount | 求一个数的百分之几

To find a percentage of a quantity, you can either use the fraction method or the decimal method. With the fraction method, write the percentage as a fraction over 100 and multiply by the amount. For example, 30% of 200 = (30/100) × 200 = 60. Using the decimal method, convert the percentage to a decimal and multiply: 0.30 × 200 = 60. Both methods are equivalent, and you can choose whichever is more convenient for the numbers involved. For non-calculator work, breaking the percentage into easier parts (e.g. 10% and 5%) often saves time.

求一个数量的百分之几，你可以使用分数法或小数法。用分数法时，把百分比写成分母为 100 的分数，然后乘以该数量。例如，30% 的 200 = (30/100) × 200 = 60。用小数法时，将百分比转换成小数再相乘：0.30 × 200 = 60。两种方法是等价的，你可以根据具体数字选择更方便的一种。不用计算器时，把百分比拆成容易计算的部分（如 10% 和 5%）常常能节省时间。

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6. Percentage Increase and Decrease | 百分比的增加与减少

When a quantity increases or decreases by a certain percentage, the new value can be found by calculating the change and then adding or subtracting it from the original. For a percentage increase, find the increase amount and add it: new value = original + (percentage × original). Alternatively, you can use a multiplier. A 15% increase means the new value is 100% + 15% = 115% of the original, so the multiplier is 1.15. Similarly, a 20% decrease gives a multiplier of 100% – 20% = 80%, or 0.80. Using multipliers is often faster and reduces mistakes in multi-step problems.

当一个数量增加或减少某个百分比时，新值可以通过先计算变化量，然后从原值中加上或减去来求得。对于百分比的增加，先求出增加量再加上：新值 = 原值 + (百分比 × 原值)。另一种方法是使用乘数。增加 15% 意味着新值是原值的 100% + 15% = 115%，因此乘数是 1.15。类似地，减少 20% 得到的乘数是 100% – 20% = 80%，即 0.80。使用乘数通常更快捷，并能减少多步骤题目中的错误。

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7. Percentage Change | 百分比变化

Percentage change tells us how much a value has increased or decreased compared to its original size. The formula is:

Percentage change = (change in value ÷ original value) × 100%

If the result is positive, it is a percentage increase; if negative, it is a decrease. For example, if a price rises from £50 to £65, the change is £15, so the percentage increase = (15 ÷ 50) × 100 = 30%. This concept is widely used in financial calculations, growth rates, and data analysis.

百分比变化告诉我们一个值相对于其原始大小增加或减少了多少。公式为：

百分比变化 = (变化值 ÷ 原始值) × 100%

如果结果为正，就是百分比增加；如果为负，则是减少。例如，价格从 50 英镑涨到 65 英镑，变化了 15 英镑，因此百分比增加 = (15 ÷ 50) × 100 = 30%。这个概念在财务计算、增长率和数据分析中被广泛使用。

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8. Reverse Percentages | 反向百分比

Reverse percentage problems ask you to find the original amount before a percentage increase or decrease took place. If you know that a value after a 10% increase is 110, the original must be 100. Mathematically, you divide the final amount by the multiplier. For an increase of 10%, the multiplier is 1.10, so original = final ÷ 1.10. For a decrease, you divide by a multiplier less than 1. It is essential to identify whether the final value represents more or less than 100% of the original before choosing the right multiplier.

反向百分比问题要求你找出在百分比增减发生之前的原始数值。如果你知道某数值在增加 10% 后是 110，那么原值必定是 100。数学上，你需要用最终数值除以乘数。增加 10% 的乘数是 1.10，所以原值 = 最终值 ÷ 1.10。对于减少，则是除以小于 1 的乘数。关键是要先判断最终数值代表原值的 100% 以上还是以下，才能选择正确的乘数。

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9. Percentage Profit and Loss | 百分比利润与亏损

Businesses often express profit or loss as a percentage of the cost price. Profit percentage = (profit ÷ cost price) × 100%. Loss percentage = (loss ÷ cost price) × 100%. For example, if a shop buys a toy for £12 and sells it for £15, the profit is £3, so the percentage profit is (3 ÷ 12) × 100 = 25%. This formula helps compare the performance of different products, as it shows the return relative to the investment rather than the absolute monetary gain.

商业中经常将利润或亏损表示为成本价格的百分比。利润百分比 = (利润 ÷ 成本价) × 100%。亏损百分比 = (亏损 ÷ 成本价) × 100%。例如，一家商店以 12 英镑买入一个玩具，以 15 英镑卖出，利润为 3 英镑，那么百分比利润为 (3 ÷ 12) × 100 = 25%。这个公式有助于比较不同产品的表现，因为它显示的是相对于投资的回报，而不是绝对的货币收益。

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10. Simple Interest | 单利

Simple interest is calculated only on the original principal amount. The formula is:

Simple interest = P × r × t

where P is the principal (starting amount), r is the annual interest rate as a decimal, and t is the time in years. For example, if you deposit £500 at a rate of 4% per year for 3 years, the interest is 500 × 0.04 × 3 = £60. The total amount after 3 years is £560. Simple interest is linear and does not consider interest on previously earned interest. It is commonly used in short-term loans and some savings accounts.

单利只根据原始本金计算利息。公式为：

单利 = P × r × t

其中 P 是本金（起始金额），r 是以小数表示的年利率，t 是时间（以年为单位）。例如，如果你存入 500 英镑，年利率 4%，存 3 年，利息就是 500 × 0.04 × 3 = 60 英镑。3 年后的总额为 560 英镑。单利是线性的，不会在之前赚取的利息上再产生利息。它常用于短期贷款和部分储蓄账户。

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11. Discounts and Sale Prices | 折扣和售价

Shops frequently use percentages to advertise discounts. To find the sale price after a discount, calculate the discount amount and subtract it from the original price. A 30% discount on an item costing £80 means the discount is 0.30 × 80 = £24, so the sale price is £80 – £24 = £56. Alternatively, you can use the multiplier method: a 30% discount means you pay 70% of the original price, so 0.70 × 80 = £56. This method is efficient when multiple discounts or additional taxes are applied in the same problem.

商店经常用百分比来宣传折扣。要计算折扣后的售价，先算出折扣金额，再从原价中减去。一件 80 英镑的商品打 30% 折扣，意味着折扣为 0.30 × 80 = 24 英镑，售价就是 80 – 24 = 56 英镑。另一种方法是使用乘数：打 30% 折扣意味着你只需支付原价的 70%，因此 0.70 × 80 = 56 英镑。当同一个问题中出现多次折扣或附加税费时，这种方法非常高效。

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12. Percentages in Real-Life Contexts | 现实生活中的百分比应用

Beyond the classroom, percentages appear in many everyday situations: VAT is a percentage added to the price of goods, tips in restaurants are often calculated as a percentage of the bill, test scores are expressed as percentages, and population growth or inflation rates are reported as percentages. Understanding how to manipulate percentages allows you to make informed decisions, whether you are budgeting, shopping, or interpreting news. As you progress through KS3 and beyond, you will see percentages integrated into statistics, probability, algebra, and financial mathematics, making them one of the most practical and recurrent topics in the curriculum.

课堂之外，百分比出现在许多日常场景中：增值税是按百分比加到商品价格上的，餐厅的小费往往按账单的百分比计算，考试分数以百分比表示，人口增长或通货膨胀率也以百分比报道。理解如何运用百分比可以帮助你在预算、购物或解读新闻时做出明智的决策。随着 KS3 及更高阶段的学习，你会看到百分比融入统计、概率、代数和金融数学中，使其成为课程中最实用、最常见的话题之一。

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Published by TutorHao | Mathematics Revision Series | aleveler.com

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