IGCSE Maths: Mastering Percentages, Ratios and Proportion | IGCSE 数学：掌握百分数、比与比例

📚 IGCSE Maths: Mastering Percentages, Ratios and Proportion | IGCSE 数学：掌握百分数、比与比例

In IGCSE Mathematics, questions on percentages, ratios and proportion appear in both non-calculator and calculator papers. They test your ability to compare quantities, solve everyday financial problems and work with direct and inverse relationships. A secure grasp of these topics will also help you in other areas such as probability, speed and map scales.

在 IGCSE 数学中，百分数、比和比例出现在无计算器和有计算器的试卷中。这些题目考查你比较数量、解决日常财务问题以及处理正比例和反比例关系的能力。扎实掌握这些主题还能帮助你学习概率、速度与地图比例尺等其他内容。

1. What Are Percentages? | 什么是百分数？

A percentage is a fraction with denominator 100. The word ‘per cent’ means ‘out of one hundred’, so 45% means 45 out of 100, written as 45/100 or 0.45. Percentages are useful when comparing different-sized groups.

百分数是分母为 100 的分数。“per cent” 的意思是 “每一百份中”，所以 45% 表示 100 份中的 45 份，写作 45/100 或 0.45。在比较不同大小的组时，百分数非常有用。

Percentage = (part ÷ whole) × 100%

For example, if 18 out of 40 students walk to school, the percentage walking is 18 ÷ 40 × 100% = 45%.

例如，如果 40 名学生中有 18 人步行上学，那么步行的百分比是 18 ÷ 40 × 100% = 45%。

Percentages greater than 100% are possible when a quantity increases beyond its original size, such as 120% of a target.

当一个量超过其原始大小时，百分数可以大于 100%，例如达到目标的 120%。

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2. Converting Between Fractions, Decimals and Percentages | 分数、小数与百分数的互化

To convert a fraction to a percentage, multiply by 100%. To convert a percentage to a decimal, divide by 100. These conversions are essential for using the multiplier method and for comparing different forms.

将分数转化为百分数，乘以 100%；将百分数转化为小数，除以 100。这些互化对于使用乘数法和比较不同形式的数据非常重要。

Fraction → percentage: × 100% | Percentage → decimal: ÷ 100

Example: 3/4 = 0.75 = 75%. The fraction 1/3 converts to 33.333…%, which is often written as 33⅓% or rounded to 33.3%.

例如：3/4 = 0.75 = 75%。分数 1/3 转化为 33.333…%，通常写作 33⅓% 或近似为 33.3%。

For decimals such as 0.125, multiply by 100 to obtain 12.5%; for a percentage such as 8%, divide by 100 to obtain 0.08.

对于小数 0.125，乘以 100 得到 12.5%；对于百分数 8%，除以 100 得到 0.08。

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3. Percentage Increase and Decrease | 百分数增加与减少

The fastest way to handle percentage increase or decrease is to use a multiplier. For an increase of 15%, the new value is 115% of the original, so multiply by 1.15. For a decrease of 15%, multiply by 0.85.

处理百分数增加或减少最快的方法是使用乘数。增加 15% 时，新值是原值的 115%，因此乘以 1.15；减少 15% 时，乘以 0.85。

New value = original × (1 ± percentage as a decimal)

Example: A coat costing £80 is reduced by 25%. The multiplier is 1 − 0.25 = 0.75, so the sale price is 80 × 0.75 = £60.

例如：一件标价 80 英镑的外套减价 25%。乘数为 1 − 0.25 = 0.75，因此售价为 80 × 0.75 = 60 英镑。

To find a percentage change, calculate (change ÷ original) × 100%. If a salary rises from £20,000 to £21,500, the increase is £1,500, so the percentage increase is 1500 ÷ 20000 × 100% = 7.5%.

要计算百分数变化，用 (变化量 ÷ 原值) × 100%。如果工资从 20,000 英镑增加到 21,500 英镑，增加量为 1,500 英镑，所以增长百分比为 1500 ÷ 20000 × 100% = 7.5%。

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4. Finding a Percentage of a Quantity | 求一个数量的百分数

To find a percentage of a quantity, convert the percentage to a decimal or a simple fraction, then multiply. This skill is widely tested in financial and measurement contexts.

要求一个数量的百分数，先将百分数转化为小数或简单分数，再相乘。这项技能在财务和测量问题中经常考查。

Part = percentage as a decimal × whole

Example: Find 15% of 240. Since 15% = 0.15, the answer is 0.15 × 240 = 36.

例如：求 240 的 15%。因为 15% = 0.15，所以答案为 0.15 × 240 = 36。

Using fractions can be quicker for common percentages: 25% is ¼, so 25% of 360 is 90; 10% is 1/10, so 10% of 360 is 36.

对于常见百分数，使用分数可能更快：25% 是 ¼，所以 360 的 25% 是 90；10% 是 1/10，所以 360 的 10% 是 36。

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5. Reverse Percentages | 逆向百分数

Reverse percentage questions give the final value after an increase or decrease and ask for the original value. Do not work out a percentage of the final value and add or subtract; instead divide by the correct multiplier.

逆向百分数问题给出增加或减少后的最终值，要求原值。不要先计算最终值的百分数再加或减；而应除以正确的乘数。

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