Percentage Increase and Decrease | 百分比增减

📚 Percentage Increase and Decrease | 百分比增减

Percentage change helps us describe how much a quantity has grown or shrunk compared to its original value. Whether we are calculating a price rise, a population drop, or a test score improvement, understanding percentage increase and decrease turns raw differences into meaningful comparisons.

百分比变化帮助我们描述某个量相对于原始值增长或减少的程度。无论是计算价格上涨、人口下降还是考试成绩提高，理解百分比增减能将原始差异转化为有意义的比较。

1. Original Value and New Value | 原值与新值

Before we can find a percentage change, we must identify two amounts: the original value (where we started) and the new value (where we ended). The change is the difference between them, calculated as new value minus original value.

在计算百分比变化之前，我们必须确定两个数值：原值（初始数值）和新值（最终数值）。变化量是两者的差，即新值减去原值。

For example, if a jumper originally cost £25 and is now £30, the price has increased by £5. If a phone originally cost £400 and now costs £340, the price has decreased by £60.

例如，一件套头衫原价25英镑，现价30英镑，价格上涨了5英镑。如果一部手机原价400英镑，现价340英镑，价格下降了60英镑。

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2. The Formula for Percentage Increase | 百分比增加的公式

Percentage increase measures growth as a fraction of the original amount. The formula is: Percentage Increase = (Increase ÷ Original Value) × 100%. The increase is the new value minus the original value.

百分比增加衡量的是相对于原值的增长部分。公式为：百分比增加 = (增加量 ÷ 原值) × 100%。增加量等于新值减去原值。

Using the jumper example: increase = £30 – £25 = £5. Original value = £25. So percentage increase = (5 ÷ 25) × 100% = 0.2 × 100% = 20%.

套用套头衫的例子：增加量 = 30英镑 – 25英镑 = 5英镑。原值 = 25英镑。因此百分比增加 = (5 ÷ 25) × 100% = 0.2 × 100% = 20%。

Always remember to divide by the original amount, not the new amount.

务必记住除以原值，而非新值。

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3. The Formula for Percentage Decrease | 百分比减少的公式

When a value falls, we use a similar formula: Percentage Decrease = (Decrease ÷ Original Value) × 100%. The decrease is the original value minus the new value.

当数值下降时，我们使用类似公式：百分比减少 = (减少量 ÷ 原值) × 100%。减少量等于原值减去新值。

For the phone: decrease = £400 – £340 = £60. Original value = £400. Percentage decrease = (60 ÷ 400) × 100% = 0.15 × 100% = 15%.

以手机为例：减少量 = 400英镑 – 340英镑 = 60英镑。原值 = 400英镑。百分比减少 = (60 ÷ 400) × 100% = 0.15 × 100% = 15%。

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4. Step-by-Step Method | 分步解题方法

A good habit is to follow four steps: (1) Find the difference between new and original values. (2) Decide if it is an increase or a decrease. (3) Divide the difference by the original value. (4) Multiply by 100 and add the % symbol.

一个好的习惯是遵循四个步骤：(1) 求出新值与原值之差。(2) 判断是增加还是减少。(3) 将差值除以原值。(4) 乘以100并加上百分号。

This method works for prices, scores, lengths, masses, and any other measurable quantity.

该方法适用于价格、分数、长度、质量以及任何其他可测量的量。

For instance, a tree grew from 1.2 m to 1.5 m: difference = 0.3 m, increase, (0.3 ÷ 1.2) × 100% = 25% increase.

例如，一棵树从1.2米长到1.5米：差值 = 0.3米，增加，(0.3 ÷ 1.2) × 100% = 25% 增加。

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5. Using a Multiplier: Increase | 使用乘数：增加

Another way to handle percentage increase is to use a multiplier. If a quantity increases by 7%, the new value is 107% of the original, or 1.07 times the original. So new value = original value × 1.07.

处理百分比增加的另一种方法是使用乘数。如果某个量增加了7%，那么新值就是原值的107%，即原值的1.07倍。所以新值 = 原值 × 1.07。

To find the multiplier for an increase of p%, use 1 + p/100. For example, a 30% increase gives a multiplier of 1 + 0.30 = 1.30.

要求出增加p%的乘数，使用 1 + p/100。例如，30%的增加得到的乘数是1 + 0.30 = 1.30。

This is extremely useful when calculating sale prices after a markup or working out population growth.

这在计算加价后的售价或计算人口增长时非常有用。

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6. Using a Multiplier: Decrease | 使用乘数：减少

For a percentage decrease, the multiplier is 1 – p/100. A 15% decrease leaves 85% of the original, so the multiplier is 0.85. The new value = original value × 0.85.

对于百分比减少，乘数为 1 – p/100。减少15%后剩下原值的85%，因此乘数为0.85。新值 = 原值 × 0.85。

If a bicycle originally costs £200 and is reduced by 20%, the sale price = 200 × 0.80 = £160.

如果一辆自行车原价200英镑，降价20%，则销售价 = 200 × 0.80 = 160英镑。

Multipliers help us avoid two-step calculations and reduce errors in exams.

乘数帮助我们避免两步计算，并减少考试中的错误。

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7. Finding the Original Value After a Change | 已知变化后求原值

Sometimes we know the new value and the percentage change but need to find the original. We reverse the multiplier. If after a 10% increase the price is £44, the multiplier was 1.10, so original = £44 ÷ 1.10 = £40.

有时我们知道新值和百分比变化，需要求出原值。我们逆向使用乘数。如果价格上涨10%后为44英镑，乘数原为1.10，因此原值 = 44英镑 ÷ 1.10 = 40英镑。

For a decrease: if a discounted price is £72 after a 20% reduction, the multiplier was 0.80, so original = £72 ÷ 0.80 = £90.

对于减少：如果打折后的价格是72英镑，折扣为20%，乘数原为0.80，所以原值 = 72英镑 ÷ 0.80 = 90英镑。

Students often mistake this step and multiply instead of divide; always check whether your answer makes sense in the context.

学生常会误用乘法而不是除法；一定要检查答案在情境中是否合理。

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8. Percentage Change with Negative Numbers | 涉及负数的百分比变化

When original values are negative, percentage change formulas need extra care, but at KS3 we usually work with positive amounts. However, interpreting a negative change tells us if the quantity went down or reflects a loss.

当原值为负数时，百分比变化公式需要格外小心，但在KS3阶段我们通常处理正数。不过，理解负变化可以告诉我们数量是下降了还是反映损失。

For example, if a company’s profit changes from –£2 million to £1 million, the change is an increase of £3 million, but percentage change is usually not calculated for sign changes in KS3.

例如，如果一家公司的利润从–200万英镑变为100万英镑，变化是增加了300万英镑，但在KS3通常不计算符号变化的百分比变化。

Focus on everyday contexts like temperature or bank balances where increases and decreases are clear in size.

重点关注日常情境，比如温度或银行余额，其中增减的大小是明确的。

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9. Repeated Percentage Changes | 连续的百分比变化

When a value undergoes several percentage changes one after another, we multiply the multipliers. For example, a £100 investment rises 10% in Year 1 and 5% in Year 2.

当一个数值连续经历多次百分比变化时，我们将乘数相乘。例如，一项100英镑的投资第一年增长10%，第二年增长5%。

After Year 1: 100 × 1.10 = £110. After Year 2: 110 × 1.05 = £115.50. The combined effect is 100 × 1.10 × 1.05 = £115.50, so the overall percentage increase is 15.5%, not 15%.

第一年后：100 × 1.10 = 110英镑。第二年后：110 × 1.05 = 115.50英镑。综合效果为100 × 1.10 × 1.05 = 115.50英镑，因此总体百分比增加为15.5%，而非15%。

This demonstrates that percentage changes are not simply added together.

这表明百分比变化并非简单相加。

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10. Reverse Percentage Problems in Real Life | 实际生活中的逆向百分比问题

Shops often advertise prices “excluding VAT” or after a discount. Reverse percentage calculations are needed to find the pre-tax or pre-sale price. If an item costs £120 including 20% VAT, the price before VAT is £120 ÷ 1.20 = £100.

商店常以“不含增值税”或打折后价格做广告。需要逆向百分比计算来求出税前或折前价格。如果一件商品含20%增值税的价格为120英镑，税前价格为120 ÷ 1.20 = 100英镑。

This skill is vital for understanding receipts and comparing deals.

这项技能对于理解收据和比较优惠至关重要。

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11. Common Mistakes to Avoid | 常见错误避免

Mistake 1: Using the new value as the denominator when calculating percentage change. Always divide by the original value.

错误1：在计算百分比变化时使用新值作为分母。务必除以原值。

Mistake 2: Adding or subtracting percentages directly when changes are applied sequentially. Use multipliers and multiply them.

错误2：在连续应用变化时直接加减百分比。使用乘数并将它们相乘。

Mistake 3: Forgetting to convert the decimal to a percentage by multiplying by 100.

错误3：忘记将小数乘以100以转换为百分比。

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12. Practice Questions to Test Understanding | 练习题目巩固理解

1. A shirt’s price is reduced from £35 to £28. Find the percentage decrease.

1. 一件衬衫的价格从35英镑降至28英镑。求百分比减少。

2. After a 12% increase, a bus fare is £2.24. What was the original fare?

2. 公交车费上涨12%后为2.24英镑。原车费是多少？

3. A population of 50,000 increases by 4% one year and then 6% the next. What is the final population?

3. 一个5万人口的小镇第一年增长4%，第二年增长6%。最终人口是多少？

Answers: 1) Decrease = £7, (7 ÷ 35) × 100% = 20%. 2) Multiplier = 1.12, original = £2.24 ÷ 1.12 = £2.00. 3) 50,000 × 1.04 × 1.06 = 55,120.

答案：1) 减少 = 7英镑，(7 ÷ 35) × 100% = 20%。2) 乘数 = 1.12，原值 = 2.24 ÷ 1.12 = 2.00英镑。3) 50,000 × 1.04 × 1.06 = 55,120。

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