📚 Mastering Percentage Increase and Decrease: p178_1 Practice | 掌握百分比增减：p178_1 练习
. You will learn how to calculate percentage increases and decreases, work backwards to find original values, and apply these methods to real-world contexts. By the end, you will be confident in tackling mixed percentage change questions, interpreting percentage multipliers, and avoiding common pitfalls.
本文拆解了KS3剑桥数学中常见的百分比问题核心技能，这类题目常出现在教材第178页练习1（p178_1.pdf）。你将学会如何计算百分比的增加与减少、逆向求解原值，并将这些方法应用于真实情境。学完本文后，你将能够自信地处理混合百分比变化问题、理解百分比乘数，并避开常见错误。
1. Understanding the Basics of Percentage Change | 理解百分比变化的基础
A percentage change describes how much a quantity has gone up or down relative to its original amount, expressed as a number out of 100. When we say a price has increased by 15%, it means for every 100 units of the original price, 15 units are added. The key formula for percentage change is:
百分比变化描述了一个量相对于其原始值增加或减少了多少，以百分数的形式表示。当我们说价格上涨15%，意味着原始价格每100个单位增加了15个单位。百分比变化的关键公式为：
Percentage Change = (Change in Value ÷ Original Value) × 100
百分比变化 = (值的变化量 ÷ 原始值) × 100
In many KS3 exercises, you are given the original amount and the new amount, and are asked to find the percentage increase or decrease. Remember that an increase gives a positive percentage, while a decrease gives a negative percentage, which we usually refer to as a 'percentage decrease'.
在KS3练习题中，经常给出原始量和新量，要求你求出百分比增加或减少。记住，增加产生正百分比，而减少产生负百分比，通常我们称之为“百分比减少”。
2. Using Percentage Multipliers for Increases | 用百分比乘数求增加
A more powerful method than calculating the increase separately is to use a single multiplier. For an increase of r%, the multiplier is 1 + r/100. For example, to increase by 20%, you multiply by 1.20. This directly gives the new value after the increase.
比单独计算增加量更高效的方法是使用一个乘数。对于增加r%，乘数为1 + r/100。例如，要增加20%，你乘以1.20，直接得到增加后新值。
New Value = Original Value × (1 + r/100)
新值 = 原始值 × (1 + r/100)
If a jumper originally costs £35 and its price is increased by 12%, the new price is 35 × 1.12 = £ specially when working with several successive changes or when doing reverse calculations.
如果一件毛衣原价35英镑，价格上涨12%，那么新价格为35 × 1.12 = 39.20英镑。乘数法节省时间且减少错误，特别是在处理多次连续变化或逆向计算时非常有用。
3. Percentage Decrease Multipliers | 百分比减少乘数
For a decrease of r% he multiplier is 1 − r/100. This multiplier is less than 1. For instance, a decrease of 30% means you multiply by 0.70 (because 1 − 0.30 = 0.70).
对于减少r%，乘数为1 − r/100。这个乘数小于1。例如，减少30%意味着你乘以0.70（因为1 − 0.30 = 0.70）。
New Value = Original Value × (1 − r/100)
新值 = 原始值 × (1 − r/100)
If a laptop priced at £450 is reduced by 15% in a sale, the sale price is 450 × 0.85 = £ n p178_1 style questions where you must quickly find a sale price after a single discount.
如果一台笔记本电脑原价450英镑，在促销中降价15%，那么促销价为450 × 0.85 = 382.50英镑。该方法广泛用于p178_1类题目中，你需要快速求出单次折扣后的售价。
4. Finding the Original Value Before a Percentage Change | 逆向求解百分比变化前的原值
because the new value equals original × multiplier, so original = new value ÷ multiplier.
KS3百分比学习中最具挑战性的任务之一是逆向计算百分比变化。如果你知道百分比增加后的新值，原值可以用除以乘数的方法求得。因为新值等于原值 × 乘数，所以原值 = 新值 ÷ 乘数。
Original Value = New Value ÷ (1 + r/100) [for an increase]
原值 = 新值 ÷ (1 + r/100) [针对增加]
For a decrease, use the multiplier 1 − r/100. For example, if a phone costs £272 after a 20% increase, the original price was 272 ÷ 1.20 = £226.67 (to two decimal places). Note that you cannot simply subtract 20% w value; that would give an incorrect answer. Always divide by the appropriate multiplier.
对于减少，使用乘数1 − r/100。例如，如果一部手机在涨价20%后售价为272英镑，那么原价为272 ÷ 1.20 ≈ 226.67英镑（保留两位小数）。注意，不能简单地将新值的20%减去，那样会得到错误答案。始终除以相应的乘数。
5. Successive Percentage Changes | 连续百分比变化
Sometimes a quantity is increased by one percentage and then by another, or first increased and then decreased. The overall effect is found by multiplying the successive multipliers. The order of the changes matters because the base amount changes each time.
有时一个量先增加一个百分比，然后再次增加，或先增加后减少。整体效果可以通过将连续乘数相乘得到。变化的顺序很重要，因为每次的基准量都发生了变化。
Suppose a salary of £2000 is increased by 10% and later by 5%. The overall multiplier is 1.10 × 1.05 = 1.155, giving a final salary of 2000 × 1.155 = £2310. This represents an overall increase of 15.5%, not simply 10% + 5% = 15%. The compounded growth is slightly higher.
假设一份2000英镑的薪水先增加10%，后又增加5%。整体乘数为1.10 × 1.05 = 1.155，最终薪水为2000 × 1.155 = 2310英镑。这表示整体增长了15.5%，而非简单的10% + 5% = 15%。复合增长略高。
When a value is first increased then decreased, the net change can be surprising. A 20% increase followed by a 20% riginal value. The multiplier would be 1.20 × 0.80 = 0.96, giving an overall decrease of 4%.
当数值先增加后减少时，净变化可能出乎意料。上涨20%再下跌20%，并不会回到原值。乘数为1.20 × 0.80 = 0.96，整体下降了4%。
6. Comparing Percentage Increases and Decreases | 比较百分比增加与减少
KS3 exam questions often ask you to compare two offers or two different percentage changes. You need to calculate and contrast the final amounts or the percentage profit/loss. Always bring both scenarios down to a common comparable value, such as the final price or the percentage change relative to the original.
KS3考试常要求你比较两种优惠或两种不同的百分比变化。你需要计算并对比最终数额或盈亏百分比。始终将两种情境归结到一个可比的共同值上，比如相对于原价的终价或百分比变化。
For example, which is a better deal: A 30% discount or a 'pay 65% of the original price' offer? A 30% discount means you pay 70% of the original. Paying 65% is cheaper. Here you compare the multipliers 0.70 and 0.65. Simple multiplier comparison is a very efficient strategy.
例如，哪种优惠更划算：打七折（30%折扣）还是“仅需支付原价的65%”？30%折扣意味着你支付原价的70%。支付65%更便宜。这里你比较乘数0.70和0.65。简单的乘数比较是一个非常高效的策略。
7. Reverse Percentage Problems: Finding the Original Price | 逆向百分比问题：求原价
Reverse percentages are vital for p178_1 type tasks. A classic setup is: 'A TV is sold for £468 after a 20% discount. What was the original price?' Since the sale price is 80% of the original, we can write:
逆向百分比对p178_1类题目至关重要。一个经典设置是：“一台电视机打折20%后售价468英镑。原价是多少？”因为售价是原价的80%，我们可以写出：
Original Price = 468 ÷ 0.80 = £585
原价 = 468 ÷ 0.80 = 585英镑
Always check: 585 × 0.80 = 468, which matches. Some pupils mistakenly find 20% of £ centage change, so the original is larger after a decrease, and smaller after an increase.
一定要验证：585 × 0.80 = 468，匹配成功。有些学生会错误地求出468的20%然后加上去，这是不对的。记住，给定值是百分比变化之后剩余的值，所以减少后原值更大，增加后原值更小。
8. Using a Bar Model or Ratio Table for Percentage Problems | 用条形模型或比率表解百分比问题
tackling p178_1 questions. Divide a bar into 100 equal parts to represent the original 100%. If a decrease of 15% to show the reduction, leaving 85 parts as the remaining value. If the remaining value is £102, then 85 parts = £102, so 1 part = 102 ÷ 85 = £1.20, and 100 parts = £120. This builds a solid conceptual understanding.
对于视觉型学习者，条形模型在解决p178_1问题时非常有用。将一条方形分成100等份表示原始的100%。如果下降15%，给15份涂色表示减少的部分，剩下85份为剩余值。如果剩余值是102英镑，那么85份 = 102英镑，1份 = 102 ÷ 85 = 1.20英镑，100份 = 120英镑。这能建立扎实的概念理解。
a decrease. This translates directly to dividing or multiplying as needed and is found in many Cambridge Checkpoint Maths resources.
另外，比率表可以将相同的过程结构化。对于减少，按比率 原值 : 新值 = 100 : (100 − r) 来计算。这直接转化为所需的除法或乘法，并出现在许多剑桥Checkpoint数学学习资源中。
9. Common Errors When Dealing with Percentages | 处理百分比时的常见错误
Understanding typical mistakes helps you avoid them in an exam. One frequent error is adding or subtracting the percentage directly to the new value to find the original. Another is confusing a 50% increase with a 100% increase—multiplying by 1.50 versus 2.00. Students also often forget that a percentage change formula requires division by the original value, not the new one.
理解典型错误有助于在考试中避开它们。一个常见错误是直接将百分比加到或减到新值上来找原值。另一个错误是混淆50%增加与100%增加——分别对应乘1.50和2.00。学生也常忘记百分比变化公式需要除以原值，而非新值。
Additionally, when encountering a loss of r%, some mistakenly use the multiplier 1 + r/100. Always double-check your multiplier: for a loss or decrease, the multiplier must be less than 1. For profit or increase, it must be greater than 1.
此外，当遇到亏损r%时，有些人错误地使用乘数1 + r/100。务必检查你的乘数：对于亏损或减少，乘数必须小于1；对于盈利或增加，乘数必须大于1。
10. Real-Life Application: Discounts, Tax, and Profit | 实际应用：折扣、税收与利润
xercises, you are asked to calculate the final price after a discount, or the original price before tax. For instance, if a meal costs £56 including a 12% service charge, what was the price before the charge? You divide by 1.12.
百分比变化无处不在：购物折扣、增值税计算、银行利息、人口增长等等。在许多p178_1类型的练习中，会要求你计算折扣后的最终价格，或税前原价。例如，一顿饭含12%服务费共56英镑，那么不含服务费的价格是多少？除以1.12即可。
Another context is profit and loss. If a shop buys a product for £25 and sells it for £40, the percentage profit based on the cost price is ((40 − 25) ÷ 25) × 100 = 60% original, sale, and profit percentages fluidly is a key skill tested at Checkpoint level.
另一个情境是盈亏。如果一家商店以25英镑买进一件商品并以40英镑卖出，基于成本价的利润百分比为 ((40 − 25) ÷ 25) × 100 = 60%。能够在原价、售价和利润百分比之间灵活切换，是Checkpoint考试考查的关键技能。
11. Practising with Mixed Percentage Questions | 混合百分比练习题
should practise with mixed questions where you identify whether you need to find the final value, the original value, or the percentage change itself. Start by underlining key words: “increased by”, “reduced to”, “original price”, “percentage profit”. Then decide on your strategy: multiplier (forward) or division (reverse).
要真正掌握p178_1的内容，你需要练习混合题型，判断你是需要求最终值、原值还是百分比变化本身。先划出关键词：“增加”、“降至”、“原价”、“利润百分比”。然后决定策略：乘数（正向）或除法（逆向）。
Try questions like: “After a 15% rise, a train ticket costs £34.50. What was the cost before the rise?” (Answer: £30.00). Or “A population of 48,000 decreased by 5% then increased by 8%. Find the final population.” (Answer: 48,000 × 0.95 × 1.08 = 49,248). Such exercises build fluency.
试做类似题目：“一张火车票上涨15%后售价为34.50英镑，上涨前的价格是多少？”（答案：30.00英镑）。或者：“一个48,000的人口先减少5%，再增加8%，求最终人口。”（答案：48,000 × 0.95 × 1.08 = 49,248）。此类练习能提升熟练度。
12. Summary and Key Takeaways for p178_1 Success | 总结与p178_1成功要点
es to visualise the relationship, and never confuse the original and new values. With consistent practice using the strategies outlined here, you will approach any p178_1 style question with clarity and confidence.
处理百分比增减是KS3剑桥数学课程的基石。始终从判断变化是增加还是减少入手，写下正确的乘数。对于逆向问题，用乘数去除。使用条形模型或比率表来形象化关系，切勿混淆原值和新值。通过反复运用本文所述策略进行练习，你将能清晰而自信地应对任何p178_1类型的题目。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图