Mastering Percentages | 掌握百分比

📚 Mastering Percentages | 掌握百分比

Percentages are a fundamental concept in IGCSE Mathematics, bridging arithmetic, ratio, and real-world applications like finance and data interpretation. Mastering percentages is essential for exam success.

百分比是 IGCSE 数学中的一个基础概念，它连接了算术、比率以及金融和数据解读等现实应用。掌握百分比对于考试成功至关重要。

1. Percent Basics | 百分比基础

A percentage is a number or ratio expressed as a fraction of 100. The word “percent” comes from the Latin “per centum,” meaning “by the hundred.”

百分比是一个以100为分母的分数或比率。”Percent” 一词源自拉丁语 “per centum”，意为”每一百”。

The symbol is ‘%’. For example, 45% means 45 out of every 100, which is equivalent to the fraction 45/100.

其符号为 ‘%’。例如，45% 表示每100个中的45个，相当于分数 45/100。

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2. Converting Percentages, Fractions, and Decimals | 转换百分比、分数和小数

To convert a percentage to a decimal, divide by 100. For example, 35% becomes 35 ÷ 100 = 0.35. To convert a decimal to a percentage, multiply by 100, so 0.35 becomes 0.35 × 100 = 35%.

将百分比转换为小数，需除以100。例如，35% 变为 35 ÷ 100 = 0.35。将小数转换为百分比，需乘以100，因此 0.35 变为 0.35 × 100 = 35%。

To convert a percentage to a fraction, write it over 100 and simplify. For instance, 25% = 25/100 = 1/4. As a decimal, 1/4 = 0.25.

将百分比转换为分数，需将其写为分母为100的分数并化简。例如，25% = 25/100 = 1/4。作为小数，1/4 = 0.25。

The following conversions are frequently tested in IGCSE exams and are worth memorising:

以下转换在 IGCSE 考试中经常出现，值得记忆：

Fraction	Percentage	Decimal
1/2	50%	0.5
1/4	25%	0.25
3/4	75%	0.75
1/5	20%	0.2
1/10	10%	0.1

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3. Finding a Percentage of a Quantity | 求一个数的百分比

To find X% of Y, multiply Y by X/100. For example, 15% of 80 = (15/100) × 80 = 0.15 × 80 = 12.

要求 Y 的 X%，需将 Y 乘以 X/100。例如，80 的 15% = (15/100) × 80 = 0.15 × 80 = 12。

Using a calculator, you can directly key in ’80 × 15 %’ or ’80 × 0.15′. Without a calculator, break the percentage into manageable parts, such as 10%, 5%, or 1%.

使用计算器时，你可以直接输入 ’80 × 15 %’ 或 ’80 × 0.15’。不使用计算器时，可将百分比分解为易于计算的部分，如10%、5%或1%。

Example: Find 30% of 50. Since 10% of 50 is 5, then 30% is 5 × 3 = 15.

例如：求 50 的 30%。因为 50 的 10% 是5，所以 30% 为 5 × 3 = 15。

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4. Percentage Change | 百分比变化

Percentage change measures the extent of change relative to the original value. The core formula is:

百分比变化衡量的是相对于原始值的变化程度。核心公式为：

Percentage Change = (Actual Change ÷ Original Value) × 100%

If the actual change is positive, it is a percentage increase. If the actual change is negative, it is a percentage decrease.

如果实际变化为正，则为百分比增加。如果实际变化为负，则为百分比减少。

Example: A bag’s price rises from $20 to $25. The actual change is $5. Therefore, the percentage change is (5 ÷ 20) × 100% = 25% increase.

例如：一个手提包的价格从20美元涨到25美元。实际变化为5美元。因此，百分比变化为 (5 ÷ 20) × 100% = 增加25%。

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5. Reverse Percentages | 逆运算百分比

A reverse percentage question asks you to find the original amount before a percentage change, given the final amount. This is a common high-mark question in IGCSE papers.

逆运算百分比问题是在已知最终金额的情况下，让你求出百分比变化之前的原始金额。这是 IGCSE 试卷中常见的高分题。

If a value increased by 15% to become $115, the original value is $115 ÷ 1.15 = $100. The multiplier 1.15 represents the original (100%) plus the 15% increase.

如果一个值增加了15%后变为115美元，那么原始值为 115 ÷ 1.15 = 100美元。乘数1.15表示原始值（100%）加上15%的增加量。

Key rule: Divide the final amount by the decimal multiplier representing the change. For an increase, use (1 + percentage/100). For a decrease, use (1 – percentage/100).

关键规则：用最终金额除以代表该变化的十进制乘数。对于增加，使用 (1 + 百分比/100)。对于减少，使用 (1 – 百分比/100)。

Example: After a 20% discount, a laptop costs $400. Original price = 400 ÷ (1 – 0.20) = 400 ÷ 0.80 = $500.

例如：一台笔记本电脑在享受20%的折扣后售价为400美元。原价 = 400 ÷ (1 – 0.20) = 400 ÷ 0.80 = 500美元。

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6. Simple Interest | 单利

Simple interest is calculated only on the original principal amount. It does not accumulate on previous interest. The formula is:

单利仅根据原始本金计算。它不会在之前的利息上累积。公式为：

Simple Interest (I) = Principal (P) × Rate (R) × Time (T)

Where R is the annual interest rate expressed as a decimal (R/100), and T is the time in years.

其中 R 是以小数表示的年利率（R/100），T 是以年为单位的时间。

Example: Invest $1000 at 5% per annum for 3 years. Interest = 1000 × 0.05 × 3 = $150. The total amount at the end is $1000 + $150 = $1150.

例如：以每年5%的利率投资1000美元，为期3年。利息 = 1000 × 0.05 × 3 = 150美元。期末总金额为 1000 + 150 = 1150美元。

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7. Compound Interest | 复利

Compound interest is calculated on the original principal plus previously accumulated interest. In each period, interest is earned on the new total. The formula for the total amount (A) is:

复利是根据原始本金加上之前累积的利息来计算的。在每一期中，利息都是根据新的总额来计算的。总金额（A）的公式为：

Total Amount (A) = P × (1 + r/100)ⁿ

Where P is the principal, r is the annual interest rate, and n is the number of years the money is invested or borrowed.

其中 P 是本金，r 是年利率，n 是资金被投资或借贷的年数。

Example: Invest $1000 at 5% per annum compounded annually for 3 years. Amount = 1000 × (1.05)³ = 1000 × 1.157625 = $1157.63 (rounded to 2 decimal places). The interest earned is $1157.63 – $1000 = $157.63.

例如：以每年5%的利率按年复利投资1000美元，为期3年。总金额 = 1000 × (1.05)³ = 1000 × 1.157625 = 1157.63美元（四舍五入到两位小数）。赚取的利息为 1157.63 – 1000 = 157.63美元。

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8. Percentage Increase and Decrease | 百分比增减

To increase a number by X%, multiply it by (1 + X/100). To decrease a number by X%, multiply it by (1 – X/100). This direct multiplication method is efficient and reduces calculation errors.

要将一个数增加 X%，需将其乘以 (1 + X/100)。要将一个数减少 X%，需将其乘以 (1 – X/100)。这种直接乘法运算效率高，并能减少计算错误。

Example: Increase 200 by 12%. New value = 200 × (1 + 0.12) = 200 × 1.12 = 224. This is equivalent to finding 12% of 200 (which is 24) and adding it to 200.

例如：将200增加12%。新值 = 200 × (1 + 0.12) = 200 × 1.12 = 224。这等同于先求出200的12%（即24），然后将其加到200上。

Example: Decrease 150 by 30%. New value = 150 × (1 – 0.30) = 150 × 0.70 = 105.

例如：将150减少30%。新值 = 150 × (1 – 0.30) = 150 × 0.70 = 105。

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9. Word Problems and Real-World Contexts | 应用题与现实情境

IGCSE exams often frame percentage problems in real-world contexts such as sales tax, discounts, profit and loss, and population changes. Understanding the context is key to selecting the correct method.

IGCSE 考试经常将百分比问题放在现实情境中，如销售税、折扣、盈亏和人口变化。理解情境是选择正确方法的关键。

Example: A shopkeeper buys an item for $80 and sells it for $100. What is the profit percentage?

例如：一位店主以80美元购进一件商品，并以100美元售出。利润率是多少？

Profit = Selling Price – Cost Price = $100 – $80 = $20. Profit percentage = (Profit ÷ Cost Price) × 100% = (20 ÷ 80) × 100% = 25%. It is crucial to use the cost price, not the selling price, as the denominator.

利润 = 售价 – 成本价 = 100美元 – 80美元 = 20美元。利润率 = (利润 ÷ 成本价) × 100% = (20 ÷ 80) × 100% = 25%。关键在于使用成本价，而不是售价，作为分母。

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10. Common Pitfalls and Exam Tips | 常见陷阱与考试技巧

Pitfall 1: Applying a percentage increase and then a percentage decrease of the same value does NOT return you to the original number. Example: 100 increased by 10% is 110. Then decreased by 10% is 110 × 0.9 = 99, not 100.

误区1：先增加一个百分比，再减少相同百分比，并不会让你回到原始数值。例如：100增加10%是110。再减少10%是 110 × 0.9 = 99，而不是100。

Pitfall 2: Forgetting to simplify fractions or misplacing decimal points when converting between percentages, fractions, and decimals can lead to errors in subsequent calculations.

误区2：在百分比、分数和小数之间转换时，忘记化简分数或点错小数点，会导致后续计算出现错误。

Tip 1: Always identify the “original value” in a percentage change problem. This value serves as the denominator in the percentage change formula.

技巧1：在百分比变化问题中，始终找准”原始值”。该值作为百分比变化公式中的分母。

Tip 2: When calculating compound interest over multiple years, use the power key (xʸ or ^) on your calculator correctly. Remember to apply the law of indices: (1.05)² × (1.05)³ = (1.05)⁵.

技巧2：在计算多年的复利时，正确使用计算器上的乘幂键（xʸ 或 ^）。记住指数法则：(1.05)² × (1.05)³ = (1.05)⁵。

Tip 3: Read the question carefully to determine whether it is asking for the final amount, the percentage change, the original amount, or simply the interest earned.

技巧3：仔细审题，明确题目要求的是最终金额、百分比变化、原始金额，还是仅要求所赚取的利息。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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