📚 Comparative Costs | 比较成本
In the IGCSE Edexcel Mathematics syllabus, comparative costs refer to the ability to compare two or more products, services, or deals to decide which offers the best value for money. This topic involves calculating unit prices, converting currencies, analysing discounts, and interpreting data from advertisements or tables. Mastering comparative costs is essential not only for your exam but also for making informed financial decisions in everyday life.
在 IGCSE 爱德思数学大纲中，比较成本指的是比较两种或多种产品、服务或优惠方案，从而判断哪一种最具性价比的能力。这一主题涉及计算单价、货币换算、分析折扣以及解读广告或表格中的数据。掌握比较成本不仅对考试至关重要，也有助于在日常生活中做出明智的财务决策。
1. Understanding Comparative Costs | 理解比较成本
Comparative cost is the process of working out which purchase option gives you more quantity per unit of money spent, or which requires less money for the same amount of goods. At its core, you are looking for the lowest cost per standard unit, such as per kilogram, per litre, per metre, or per item. This requires you to convert all the given offers into a common format so they can be compared fairly.
比较成本就是计算出哪一种购买方案能让你用单位金额获得更多数量，或者要得到同等数量的商品哪一种所需花费更少。其核心是寻找每标准单位的最低成本，例如每公斤、每升、每米或每件商品的价格。这需要将所有给出的报价转换成一种共同的形式，才能公平地进行比较。
Often you will encounter situations in supermarkets, mobile phone contracts, travel tickets, or even foreign exchange. The mathematical tools you need include division to find unit rate, multiplication for scaling, and sometimes ratio to express the relationship between price and quantity. Always ask: 'What am I getting for one pound?' or 'How many pence am I paying for one gram?' This mindset will guide you through any comparative cost problem.
你经常会在超市、手机套餐、旅行票务甚至外汇兑换中遇到这种情况。你需要的数学工具包括用于求单价的除法、用于按比例放大的乘法，有时还涉及用比率来表示价格与数量之间的关系。永远要问自己：“我用一英镑能买到多少？”或者“我为每一克支付了多少便士？”这种思维方式能引导你解决任何比较成本的问题。
2. Finding Unit Cost | 计算单位成本
The most fundamental skill in comparative costs is finding the unit cost. The formula is simple: Unit cost = Total price ÷ Number of units. The units can be mass (kg, g), volume (litres, ml), length (m, cm), or even items. Once you have the cost per single unit, you can directly compare offers even if the pack sizes are different.
比较成本中最基本的技能就是计算单位成本。公式很简单：单位成本 = 总价 ÷ 单位数量。单位可以是质量（千克、克）、体积（升、毫升）、长度（米、厘米）甚至是商品个数。一旦得到每个单位的成本，即使包装大小不同，你也能直接进行比较。
For example, a 400 g jar of coffee costs £6.40, while a 250 g jar of the same brand costs £4.25. To compare, we calculate cost per 100 g. For the first jar: £6.40 ÷ 4 = £1.60 per 100 g. For the second: £4.25 ÷ 2.5 = £ r 100 g. Therefore, the larger jar is better value, even though its total price is higher.
例如，一罐400克的咖啡售价6.40英镑，而同一品牌250克的罐装售价4.25英镑。要进行比较，我们计算每100克的成本。第一罐：6.40 ÷ 4 = 每100克1.60英镑。第二罐：4.25 ÷ 2.5 = 每100克1.70英镑。因此，大罐尽管总价更高，但性价比更优。
Be careful with units: if one item is priced in grams and another in kilograms, convert them to the same unit first. It is often easiest to express cost per 100 g or per kg for mass, and per 100 ml or per litre for volume. In the exam, show clear working; writing the division step by step will earn you method marks even if you make a minor arithmetic error.
要注意单位：如果一种商品按克标价，而另一种按千克标价，要先将它们转换成相同的单位。通常，质量用每100克或每千克的成本来表示最为简便，体积则用每100毫升或每升。在考试中，展示清晰的运算步骤；即使出现轻微的算术错误，写出逐步的除法过程也能获得方法分。
3. Comparing Prices with Different Quantities | 比较不同数量的价格
0 g. A common mistake is to compare the total price without considering how much of the product you actually receive.
很多优惠方案包装量各异，你必须判断哪一种更便宜。当数量以不同单位给出时，要先将它们全部转换为相同单位，再计算单位成本。体积方面，1升 = 1000毫升；质量方面，1千克 = 1000克。常见的错误是只比较总价，而不考虑你实际上能得到多少产品。
Consider two bottles of shampoo: Bottle A contains 300 ml for £1.80, and Bottle B contains 500 ml for £ 5 = 55 p. Bottle B is the better buy because 55 p per 100 ml is less than 60 p per 100 ml.
假设有两瓶洗发水：A瓶300毫升售价1.80英镑，B瓶500毫升售价2.75英镑。换算为每100毫升的成本。A瓶：300毫升花费180便士，因此每100毫升成本为180 ÷ 3 = 60便士。B瓶：500毫升花费275便士，因此每100毫升成本为275 ÷ 5 = 55便士。B瓶更划算，因为55便士/100毫升少于60便士/100毫升。
y match. For instance, find the cost of 1500 ml of each: A would need 5 bottles (5 × £1.80 = £9.00); B would need 3 bottles (3 × £2.75 = £8.25). Again, B works out cheaper. Having alternative methods helps verify your answer.
你也可以通过扩大数量使它们相等来比较。例如，计算每种都获得1500毫升的成本：A需要5瓶（5 × 1.80 = 9.00英镑）；B需要3瓶（3 × 2.75 = 8.25英镑）。再次得出B更便宜。掌握替代方法有助于验证你的答案。
4. Using Ratios and Proportions | 使用比例和比率
de a powerful way to express the relationship between cost and quantity. A store may offer '3 for £2' io of items to pounds: 3 : 2. To compare, we can find how much one item costs in the deal: £2 ÷ 3 = £ The single item at 75p is more expensive, so the bundle offer is better if you need three.
比率提供了一种表达成本与数量关系的有力方式。一家商店可能“3个卖2英镑”，而另一家同样的商品每个卖75便士。我们可以将第一个表示为物品数量与英镑的比值：3 : 2。要进行比较，我们可以计算该优惠下每个物品的成本：2 ÷ 3 ≈ 0.667英镑（或66.7便士）。75便士一个更贵，因此如果你需要三个，捆绑销售更划算。
Proportions allow you to solve questions like: If 5 pens cost £2.75, how much would 8 pens cost? This is a direct proportion. First find the cost of 1 pen: £2.75 ÷ 5 = £0.55. Then multiply by 8: £0.55 × 8 = £ stationary sets with proportional reasoning to decide which offers the lower cost per pen.
比例可以帮助你解决这样一类问题：如果5支笔售价2.75英镑，那么8支笔要多少钱？这是一个正比例问题。先求出一支笔的成本：2.75 ÷ 5 = 0.55英镑。再乘以8：0.55 × 8 = 4.40英镑。在比较成本的情境下，你可能需要用比例推理来比较两种不同的文具套装，判断哪一种每支笔的成本更低。
When working with ratios, ensure you keep the order consistent. If you are comparing cost to weight, label clearly 'cost : weight'. You can then calculate the unit rate correctly. Also check for non-linear relationships, but most IGCSE comparative cost questions assume constant price per unit, so proportions are safe to use.
在使用比率时，要确保顺序一致。如果你在比较成本与重量，要清楚地标注“成本 : 重量”。这样你才能正确计算单位比率。同时要检查是否存在非线性关系，但大多数IGCSE比较成本题目都假定单位价格恒定，因此使用比例是可靠的。
5. Converting Currencies | 货币换算
, or other units. To compare them fairly, you must use an exchange rate to convert all prices into the same currency. The exchange rate tells you how much one currency is worth in terms of another, e.g., £1 = $1.25.
许多比较成本问题都会涉及不同的货币，因为面向国际销售的产品可能以英镑、美元、欧元或其他货币标价。要公平地比较它们，你必须使用汇率将所有价格换算成同一种货币。汇率会告诉你一种货币相对于另一种货币的价值，例如 1英镑 = 1.25美元。
Suppose a video game costs £42 in the UK and $55 in the US, with an exchange rate of £1 = $1.30. Convert the US price to pounds: $55 ÷ 1.30 = £ is shows the UK price is slightly cheaper. Alternatively, convert the UK price to dollars: £42 × 1.30 = $54.60, which is less than $55. Always state which deal is better.
假设一款电子游戏在英国售价42英镑，在美国售价55美元，汇率为1英镑 = 1.30美元。将美国价格换算成英镑：55 ÷ 1.30 = 42.31英镑（四舍五入到两位小数）。这表明英国的价格略便宜。或者，将英国价格换算成美元：42 × 1.30 = 54.60美元，低于55美元。始终要说明哪种方案更优。
Be careful with the direction of conversion. If the rate is given as £1 = $ hange rates and need to convert multiple items. Show all workings and write down the converted price clearly before comparing.
要注意换算的方向。如果给出的汇率是1英镑 = 1.20美元，那么将美元兑换成英镑就要除以1.20；将英镑兑换成美元则乘以1.20。在考试中，你可能会看到一张汇率表，并需要转换多个项目。展示所有计算过程，并在比较前清晰地写下转换后的价格。
6. Sales, Discounts and Value for Money | 销售、折扣与性价比
Discounts can significantly affect comparative cost decisions. You must evaluate the final price you pay, not the original one. Common offer types include percentage discounts (e.g., 20% off), fraction discounts (e.g., ⅓ off), and multi-buy offers (e.g., buy one get one free, or 3 for the price of 2). Calculate the effective price per unit after the discount has been applied.
折扣会显著影响比较成本的决策。你必须评估的是实际支付的最终价格，而非原价。常见的优惠类型包括百分比折扣（如降价20%）、分数折扣（如便宜⅓）以及多买优惠（如买一送一，或买三按二收费）。计算应用折扣后的实际单价。
For example, a t-shirt is originally £15 but there is a 30% discount. The saving is 30% of £15 = £4.50, so the final price is £15 − £4.50 = £10.50. Another shop sells a similar t-shirt at £12 with a 'buy one get one half price' offer if you need two. For two shirts, you pay £12 + £6 = £18, so £9 per shirt. Comparatively, the first shop's single shirt is £10.50, but if you buy two from the second, each is £9 – so the second deal is cheaper per shirt when buying two.
例如，一件T恤原价15英镑，但有30%的折扣。折扣金额为15的30%即4.50英镑，最终价格为15 − 4.50 = 10.50英镑。另一家商店类似T恤售价12英镑，并推出“第二件半价”优惠。如果你需要两件，总支付12 + 6 = 18英镑，即每件9英镑。比较而言，第一家店单件10.50英镑，但如果在第二家店购买两件，每件9英镑——因此买两件时第二家每件的成本更低。
Also watch for offers that seem good but require you to buy more than you need, leading to waste. If you only need one item, the multi-buy may not be best. This is where real-life reasoning intersects with mathematics: a comparative cost question might ask whether a larger pack is still 'better value' if half of it goes to waste. You would need to adjust the effective quantity you are getting.
还要注意那些看似划算但却要求你购买超过实际所需的数量，从而导致浪费的优惠。如果你只需要一件商品，多买优惠可能就不是最佳选择。这正是现实生活推理与数学相结合的体现：比较成本题目可能会问，如果大包装有一半会被浪费掉，它是否仍然“更划算”。此时你就需要调整你所得到的有效数量。
7. Multi-step Comparison Problems | 多步骤比较问题
Comparative cost problems in the IGCSE exam often involve more than one step. You might need to convert units, apply a discount, add delivery charges, and then compare total costs. Organising your working logically is key. A table can be a helpful way to present the information for two or more options clearly.
IGCSE考试中的比较成本问题常常涉及多个步骤。你可能需要转换单位、应用折扣、加上运费，然后再比较总成本。有条理地组织你的解题过程至关重要。用表格来清晰地呈现两个或多个选项的信息是一种有效的方法。
Consider a scenario: Online shop X sells a backpack for £25 with free delivery if you spend over £20. Shop Y sells the same backpack for £22, but charges £3.50 for delivery. Which is cheaper? Shop X total = £25. Shop Y total = £22 + £3.50 = £25.50. Shop X saves you 50p. If an additional item was needed, you'd recalculate. Always take into account all compulsory costs.
设想一个情景：网店X销售一款背包，价格25英镑，消费满20英镑免运费。网店Y同款背包售价22英镑，但收取3.50英镑运费。哪家更便宜？X店总价 = 25英镑。Y店总价 = 22 + 3.50 = 25.50英镑。X店让你节省了50便士。如果还需要其他商品，你得重新计算。始终要把所有强制性费用都考虑在内。
olve mixing currencies: a UK student wants to buy a book from a European website priced at €28. The student's bank exchange rate is £1 = €1.15, and there is a 2% gn transaction fee on the sterling amount after conversion. Other UK retailer sells it for £24 with free delivery. To compare, we must convert €28 into pounds: 28 ÷ 1.15 = £24.35 (approx). Then add 2% of £24.35: £0.487, so total £24.84. The UK price of £24 is cheaper. This shows the importance of including all fees.
另一个多步骤问题可能涉及混合货币：一名英国学生想从一家标价28欧元的欧洲网站买书。学生所在银行的汇率是1英镑 = 1.15欧元，并收取转换后英镑金额的2%作为外汇交易手续费。另一家英国零售商以24英镑出售该书并免运费。为了比较，我们需要将28欧元转换成英镑：28 ÷ 1.15 ≈ 24.35英镑。再加上24.35的2%手续费：0.487，总价约24.84英镑。英国24英镑的价格更便宜。这显示了纳入所有费用的重要性。
8. Practical Examples: Groceries and Travel | 实际例子：杂货与旅行
t scenarios. For instance, comparing loose fruit and pre-packaged fruit. Loose apples cost £1.80 per kg, while a pre-packed 1.5 kg bag costs £2.55. The loose apples per kg are £1.80, but the bag gives a unit cost of £2.55 ÷ 1.5 = £.70 per kg. So the bag is better value, provided you can use all 1.5 kg before they spoil.
食品杂货购物提供了无数的比较成本场景。例如，比较散装水果与预先包装水果。散装苹果每千克1.80英镑，而一袋1.5千克的预包装苹果售价2.55英镑。散装苹果每千克1.80英镑，但袋装苹果的单位成本为2.55 ÷ 1.5 = 每千克1.70英镑。因此，只要你能在坏掉之前用完这1.5千克，袋装就更划算。
another rich source. Train tickets may offer different fares: An anytime return ticket is £45, but two single advance tickets cost £ only for specific trains. If flexibility is not needed, total cost for advance: 2 × £19 = £38, which is cheaper than £45. But if there's a chance you need to change plans, the anytime ticket may be worth the extra £7. This integrates non-monetary value with cost comparison.
旅行费用是另一个丰富的来源。火车票可能有不同的票价：一张随时往返票价45英镑，但如果提前预订，两张单程预售票每张19英镑。然而，预售票不可退款且仅限特定车次。如果不需要灵活性，预售票总价：2 × 19 = 38英镑，比45英镑便宜。但如果有可能需要更改计划，那么随时票多付7英镑也许是值得的。这就把非货币价值与成本比较结合起来了。
When comparing travel, also consider group tickets versus individual tickets. A family of four might have the option of a family railcard giving a discount on each ticket, or a group ticket covering all. You would calculate the total for both scenarios to find the lowest cost. Showing this in a table helps:
在比较旅行费用时，也要考虑团体票与个人票。一个四口之家可能有两种选择：使用家庭铁路卡购买每张票享受折扣，或者购买一张涵盖所有人的团体票。你就要计算出两种方案的总费用，找出最低成本。用表格展示会有帮助：
| Option | Cost per person | Total for 4 |
|---|---|---|
| Standard individual | £22 | £88 |
| Railcard (33% off) | £22 × 0.67 = £14.74 | £58.96 |
| Group ticket | £65 total | £65 |
Clearly the railcard offers the best saving, but it has an initial purchase cost for the card itself, so you'd compare over a year.
显然铁路卡最省钱，但购买卡本身有初始成本，因此你需要在更长的时间跨度内进行比较。
9. Common Mistakes to Avoid | 常见错误避免
One frequent error is comparing pack prices without equalising the quantity. Never say 'Pack A is £3 and Pack B is £4, so Pack A is better value' when Pack A contains 200 g and Pack B contains 350 g. The larger pack might have a lower unit cost. Always find the cost per unit.
一个常见的错误是，在未等量化数量的情况下直接比较包装价格。当A包装200克售价3英镑、B包装350克售价4英镑时，绝不能说“A包装3英镑，B包装4英镑，所以A更划算”。大包装的单位成本可能更低。务必求出单位成本。
uestion asks for the cheapest way to buy 3 bottles, and the options are 1 for 90p or 4 for £3, the bundle costs £3 for 4 but you only need 3. You cannot use the bundle price directly because you'd waste one. Here the cheapest is 3 × 90p = £2.70. Using the bundle then discarding one would cost £3, which is worse.
另一类错误是忽略了特殊优惠可能需要购买超过你所需数量的商品。如果题目问买3瓶最便宜的方式，选项是1瓶90便士或4瓶3英镑，那么捆绑装4瓶要花3英镑，但你只需要3瓶。你不能直接使用捆绑价格，因为你将浪费一瓶。此时最便宜的是3 × 90便士 = 2.70英镑。如果购买捆绑装再丢弃一瓶，那将花费3英镑，反而更贵。
Currency mistakes: using the exchange rate the wrong way round. If £1 = $1.30, then to convert $65 to £ two decimal places if they represent currency. Finally, not reading the question carefully – oosing the better deal, not just which one is better. Always answer the specific question.
货币方面的错误：弄错汇率使用方向。如果1英镑 = 1.30美元，那么要将65美元转换为英镑，你应该用65 ÷ 1.30，而不是乘法。还有忘记适当的四舍五入。在货币问题中，如果最终答案表示金额，通常应保留两位小数。最后，不仔细审题——有些题目问的是选择更优方案后你能节省多少钱，而不仅仅是哪个更好。始终要针对具体问题作答。
10. Exam Tips for Comparative Cost Questions | 考试技巧
In the IGCSE Edexcel Mathematics exam, comparative cost questions typically appear in the calculator paper. They are often worth 3 to 5 marks and require clear working. Here are some strategies: First, highlight the key numbers and units in the question. Identify what the question is asking you to find – the better deal, the saving, or a total cost.
在IGCSE爱德思数学考试中，比较成本问题通常出现在可使用计算器的试卷中。这些题目一般占3到5分，并需要清晰的解题步骤。以下是一些策略：首先，标出题目中的关键数字和单位。明确题目要求你求什么——是更优的方案、节省的金额还是总成本。
Set out your work logically, using the same unit for all prices. If you are comparing using cost per 100 g, write that heading. Show the division you perform for each option. If one option requires more than one calculation step, label them 'Option A' and 'Option B'. This makes it easier for the examiner to follow your reasoning and award marks.
逻辑清晰地展示你的答题过程，使用同一单位表示所有价格。如果你是通过每100克的成本来比较，那就写出这个标题。展示你对每种选项所做的除法运算。如果某个选项需要多个计算步骤，就分别标注“选项A”和“选项B”。这能让考官更容易理解你的推理过程，从而给分。
Double-check: Does your answer make sense? A 'better value' product should result in a lower unit price. If you find a higher unit price for the larger pack, re-check your calculations. Make sure your final statement clearly answers the question, e.g., 'So the 1.5 litre bottle is better value, saving 2 pence per 100 ml.' This concluding sentence ties your numbers back to the context and is often required.
仔细复核：你的答案合理吗？“更划算”的产品应该得到更低的单价。如果你发现大包装的单价反而更高，就重新检查计算过程。确保你的最终陈述清晰地回答了问题，例如：“因此1.5升的瓶子更划算，每100毫升能节省2便士。”这种总结性语句将你的数字与情境联系起来，这往往是题目所要求的。
Practice past paper questions regularly. You'll encounter recurring contexts like supermarket deals, phone tariffs, and travel passes. The more familiar you become with the common setups, the quicker you can spot the necessary calculations. Remember, comparative cost is a real-world skill – you're not just learning for an exam, you're learning to manage your own money wisely.
定期练习历年真题。你会遇到反复出现的情境，如超市优惠、电话资费和出行通票。你对这些常见情境越熟悉，就能越快识别出必要的计算。请记住，比较成本是一项现实生活中的技能——你不仅是在为考试学习，更是在学习如何明智地管理自己的钱财。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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