📚 Mastering Percentage Change | 掌握百分比变化
Percentage change is a fundamental concept in mathematics that we use in everyday life, from calculating discounts and tax to understanding profit and loss. In this article, you will learn how to calculate percentage increase and decrease, apply the percentage change formula, and solve real-world problems. Mastering percentage change is essential for KS3 mathematics and beyond.
百分比变化是数学中的一个基本概念，我们在日常生活中经常用到它，比如计算折扣、税款，以及理解盈利和亏损。在本文中，你将学会如何计算百分比增加和减少，应用百分比变化公式，并解决实际问题。掌握百分比变化对于KS3数学及后续学习至关重要。
1. What is Percentage Change? | 什么是百分比变化？
Percentage change tells us how much a quantity has increased or decreased compared to its original value, expressed as a percentage. It is a way to compare the difference between an old value and a new value relative to the original amount. If the new value is greater than the original, we have a percentage increase. If it is smaller, we have a percentage decrease.
百分比变化告诉我们一个量相对于其原值增加或减少了多少，以百分比的形式表示。这是一种比较旧值和新值之间差异相对于原来数值的方式。如果新值大于原值，我们就得到了百分比增加；如果新值更小，则得到百分比减少。
- Percentage Increase: New value is larger than the original value (New > Original).
- Percentage Decrease: New value is smaller than the original value (New < Original).
• 百分比增加：新值大于原值（新值 > 原值）。
• 百分比减少：新值小于原值（新值 < 原值）。
2. Calculating Percentage Increase | 计算百分比增加
To calculate the percentage increase, first find the actual increase by subtracting the original value from the new value. Then divide the increase by the original value and multiply by 100%.
要计算百分比增加，首先用新值减去原值求出实际增加量。然后将该增加量除以原值，再乘以100%。
Formula for percentage increase:
Percentage Increase = (Increase ÷ Original Value) × 100%
百分比增加的公式：
百分比增加 = (增加量 ÷ 原值) × 100%
Steps to follow:
- Step 1: Calculate the increase = New value − Original value
- Step 2: Divide the increase by the original value.
- Step 3: Multiply the result by 100% to get the percentage increase.
遵循以下步骤：
第一步：计算增加量 = 新值 − 原值
第二步：将增加量除以原值。
第三步：将结果乘以100%得出百分比增加。
3. Calculating Percentage Decrease | 计算百分比减少
For a percentage decrease, the actual decrease is found by subtracting the new value from the original value. Then divide the decrease by the original value and multiply by 100%.
对于百分比减少，实际减少量是通过用原值减去新值求得的。然后将减少量除以原值，再乘以100%。
Percentage Decrease = (Decrease ÷ Original Value) × 100%
Percentage Decrease = (Decrease ÷ Original Value) × 100%
步骤：
- Step 1: Calculate the decrease = Original value − New value
- Step 2: Divide the decrease by the original value.
- Step 3: Multiply by 100%.
第一步：计算减少量 = 原值 − 新值
第二步：将减少量除以原值。
第三步：乘以100%。
4. The Percentage Change Formula | 百分比变化公式
The general formula for percentage change works for both increase and decrease. It measures the difference between the new and original values relative to the original value.
百分比变化的通用公式对于增加和减少都适用。它衡量的是新值与原值之间的差异相对于原值的比例。
Percentage Change = [(New Value − Original Value) ÷ Original Value] × 100%
百分比变化 = [(新值 − 原值) ÷ 原值] × 100%
If the result is positive, it indicates a percentage increase. If negative, it represents a percentage decrease. You can ignore the negative sign and call it a decrease.
如果结果为正，表明是百分比增加；如果为负，则代表百分比减少。你可以忽略负号，将其称为减少。
| Type of Change | Calculation | Interpretation |
|---|---|---|
| Increase | (New − Original) ÷ Original × 100% | Positive result |
| Decrease | (Original − New) ÷ Original × 100% | Negative result (taken as positive decrease) |
下表总结了变化类型和相应的计算与解释：
变化类型：增加 – 计算式为 (新值 − 原值) ÷ 原值 × 100%，结果为正。减少 – 计算式为 (原值 − 新值) ÷ 原值 × 100%，结果为负，视为正数减少。
5. Worked Example 1: Price Rise | 例题1：价格上涨
A pair of shoes originally cost £40. The price increases to £50. What is the percentage increase?
一双鞋原价为40英镑，价格上涨到50英镑。百分比增加是多少？
Step-by-step solution:
- Increase = £50 − £40 = £10
- Divide by original: £10 ÷ £40 = 0.25
- Multiply by 100%: 0.25 × 100% = 25%
逐步求解：
增加量 = 50 − 40 = 10英镑
除以原值：10 ÷ 40 = 0.25
乘以100%：0.25 × 100% = 25%
The percentage increase is 25%.
百分比增加为25%。
6. Worked Example 2: Discount | 例题2：折扣
A bicycle is reduced from £200 to £170. Find the percentage decrease.
一辆自行车从200英镑降价到170英镑。求百分比减少。
Solution:
- Decrease = £200 − £170 = £30
- Decrease ÷ Original = £30 ÷ £200 = 0.15
- 0.15 × 100% = 15%
求解：
减少量 = 200 − 170 = 30英镑
减少量 ÷ 原值 = 30 ÷ 200 = 0.15
0.15 × 100% = 15%
The bicycle has been reduced by 15%.
这辆自行车降价了15%。
7. Finding the Original Amount | 求原值
Sometimes you know the new value and the percentage change, and you need to find the original amount. This involves reversing the percentage change.
有时你知道新值和百分比变化，需要求出原值。这需要逆向计算百分比变化。
If the new value represents a percentage increase, the original amount can be found by dividing the new value by (1 + percentage increase as a decimal).
如果新值代表经过百分比增加后的结果，可以通过将新值除以（1 + 转换为小数的百分比增加）来求得原值。
Original Value = New Value ÷ (1 + Percentage Increase in decimal form)
原值 = 新值 ÷ (1 + 百分比增加的小数形式)
For example, after a 20% increase, a jacket costs £60. The decimal form of 20% is 0.20. Original price = £60 ÷ 1.20 = £50.
例如，一件外套涨价20%后售价60英镑。20%的小数形式是0.20。原价 = 60 ÷ 1.20 = 50英镑。
For a decrease, divide by (1 − percentage decrease as a decimal).
对于减少的情况，则除以（1 − 转换为小数的百分比减少）。
Original Value = New Value ÷ (1 − Percentage Decrease in decimal)
原值 = 新值 ÷ (1 − 百分比减少的小数形式)
If a laptop is sold at £450 after a 10% discount, the original price = £450 ÷ 0.90 = £500.
如果一台笔记本电脑在打九折后售价450英镑，原价 = 450 ÷ 0.90 = 500英镑。
8. Percentage Change in Real-Life Contexts | 现实生活中的百分比变化
Percentage change appears in many everyday situations: salary raises, population growth, inflation, store discounts, and investment returns. Understanding the concept helps make informed decisions.
百分比变化出现在许多日常情境中：加薪、人口增长、通货膨胀、商店折扣以及投资回报。理解这一概念有助于做出明智的决策。
For instance, a 5% pay rise means your salary increases by 5% of your original salary. If you earned £24,000, the new salary would be £24,000 + (5% of £24,000) = £24,000 + £1,200 = £25,200.
例如，加薪5%意味着你的工资在原工资的基础上增加5%。如果你的原工资是24,000英镑，新工资将为24,000 +（24,000的5%）= 24,000 + 1,200 = 25,200英镑。
When shopping, a '30% off' tag means you pay 70% of the original price. If a game console originally cost £300, the sale price = 70% × £300 = £210.
购物时，标签上写着“打七折”意味着你只需支付原价的70%。如果一台游戏机原价300英镑，折扣价 = 70% × 300 = 210英镑。
Careful: successive percentage changes do not simply add up. A 10% increase followed by a 10% decrease does not bring you back to the original value. This is a common pitfall.
注意：连续的百分比变化不能简单地相加。先增加10%再减少10%并不会使你回到原值。这是一个常见的陷阱。
9. Common Mistakes | 常见错误
Students often confuse the original value with the new value when calculating percentage change. Always identify which amount is the starting point (original) and which is the final value.
学生们在计算百分比变化时常常混淆原值和新值。一定要先判断哪个量是起始点（原值），哪个是最终值。
Mistake 1: Using the new value as the denominator instead of the original. The denominator must always be the original amount.
错误1：用新值代替原值做分母。分母必须始终是原值。
Mistake 2: Forgetting to multiply by 100%. The ratio of change to original gives a decimal; multiplying by 100% converts it to a percentage.
错误2：忘记乘以100%。变化量与原值的比值得出的是小数；乘以100%才能转换成百分比。
Mistake 3: Adding percentage changes directly. For example, a 50% increase followed by a 50% ease results in an overall decrease, not the original amount. (Starting with 100, after 50% increase → 150, then 50% decrease → 75.)
错误3：直接将百分比变化相加。例如，先增加50%再减少50%最终结果是总体减少，而不是回到原值。（从100开始，增加50%后得150，再减少50%得75。）
- Correct: Percentage change is always based on the original value.
- Always express as a percentage of the starting amount.
• 正确做法：百分比变化始终基于原值。
• 始终表示为原始数量的百分比。
10. Practice Questions | 练习题
Try these problems to test your understanding. Solutions are provided below each question.
尝试解决以下问题来检验你的理解。解答见每道题下方。
Q1: A phone's price rises from £150 to £180. Find the percentage increase.
问题1：一部手机的价格从150英镑涨到180英镑。求百分比增加。
Solution: Increase = £30, 30 ÷ 150 = 0.2, 0.2 × 100% = 20%.
解答：增加量 = 30英镑，30 ÷ 150 = 0.2，0.2 × 100% = 20%。
Q2: A town's population drops from 25,000 to 23,500. Calculate the percentage decrease.
问题2：一个小镇的人口从25,000人下降到23,500人。计算百分比减少。
Solution: Decrease = 1,500, 1,500 ÷ 25,000 = 0.06, 0.06 × 100% = 6%.
解答：减少量 = 1,500，1,500 ÷ 25,000 = 0.06，0.06 × 100% = 6%。
Q3: After a 15% reduction, a tablet costs £340. What was the original price?
问题3：一台平板电脑降价15%后售价为340英镑。原价是多少？
Solution: £340 ÷ 0.85 = £400. (Because 100% − 15% = 85% = 0.85.)
解答：340 ÷ 0.85 = 400英镑。（因为100% − 15% = 85% = 0.85。）
Q4: A jacket is priced at £80. The store applies a 25% increase then offers a 20% discount on the new price. Find the final price.
问题4：一件夹克定价80英镑。商店先提价25%，然后在新价格基础上打八折。求最终价格。
Solution: After increase: £80 × 1.25 = £100. After discount: £100 × 0.80 = £80. Final price is £80, but note this is a coincidence; it is not always the original.
解答：提价后：80 × 1.25 = 100英镑。打折后：100 × 0.80 = 80英镑。最终价格为80英镑，但这是巧合；并不总是回到原值。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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