Cost Comparison: Calculations and Applications | 成本比较的计算与应用

📚 Cost Comparison: Calculations and Applications | 成本比较的计算与应用

Cost comparison is one of the most practical topics in IGCSE Mathematics. It appears in everyday decisions such as choosing the best value product, selecting a mobile phone plan, or comparing energy tariffs. This article will guide you through the key calculation methods and real-life applications you need to master for your Edexcel IGCSE exam.

成本比较是IGCSE数学中最实用的课题之一。它出现在日常决策中，例如选择最划算的商品、挑选手机套餐或比较能源收费标准。本文将引导你掌握Edexcel IGCSE考试所需的关键计算方法和实际应用。

1. Unit Price | 单位价格

The unit price is the cost per single unit of measure, such as per kilogram, per litre, or per item. It allows us to compare products of different sizes fairly. To calculate the unit price, divide the total cost by the number of units.

单位价格是每单位度量的成本，例如每公斤、每升或每件。它使我们能够公平地比较不同规格的产品。计算单位价格时，用总成本除以单位数量。

Unit Price = Total Cost ÷ Number of Units

For example, a 500 g box of cereal costs £2.40 and a 750 g box costs £3.45. The unit price of the first is £2.40 ÷ 0.5 kg = £4.80 per kg. The unit price of the second is £3.45 ÷ 0.75 kg = £4.60 per kg. Therefore, the 750 g box is better value per kilogram.

例如，一盒500克麦片售价2.40英镑，一盒750克麦片售价3.45英镑。第一盒的单位价格为2.40英镑 ÷ 0.5千克 = 每千克4.80英镑。第二盒的单位价格为3.45英镑 ÷ 0.75千克 = 每千克4.60英镑。因此，750克装每公斤更划算。

Always check the units before comparing (kg, g, L, mL, etc.).
Convert all quantities to the same unit first.
Round the unit price to a sensible degree of accuracy, such as 2 decimal places.

比较前务必检查单位（kg、g、L、mL等）。
先将所有数量转换为相同单位。
将单位价格四舍五入到合理的精确度，例如两位小数。

2. Percentage Difference | 百分比差异

A percentage difference expresses how much one value is greater or smaller than another, relative to the reference value. This is useful when comparing discounts, price increases, or efficiency between two options.

百分比差异表示一个值相对于参考值比另一个值大多少或小多少。这在比较折扣、价格上涨或两个选项的效率时非常有用。

Percentage Difference = (Difference ÷ Original Value) × 100%

Suppose a shop sells a jacket for £60 and another shop sells the same jacket for £48. The difference is £12. The percentage saving compared to the original price is (£12 ÷ £60) × 100% = 20%. This tells the consumer exactly how much they save relative to the higher price.

假设一家商店以60英镑出售夹克，另一家商店以48英镑出售同一件夹克。差价为12英镑。与原价相比的百分比节省为（12英镑 ÷ 60英镑）× 100% = 20%。这告诉消费者相对于较高价格，他们究竟节省了多少。

New Value = Original Value × (1 ± Percentage Change)

This formula is crucial for reverse percentage problems, where you are given the final price after a percentage change and must find the original price. For example, if a price after a 15% discount is £85, then the original price is £85 ÷ 0.85 = £100.

这个公式对于逆百分比问题至关重要，即已知百分比变化后的最终价格，必须求出原价。例如，如果15%折扣后的价格为85英镑，那么原价为85英镑 ÷ 0.85 = 100英镑。

3. Fixed and Variable Costs | 固定成本与变动成本

In cost comparison problems, we often separate costs into two categories. Fixed costs remain constant regardless of the quantity produced or consumed. Variable costs change in proportion to the quantity. Understanding this distinction is essential when comparing two business options or utility tariffs.

在成本比较问题中，我们通常将成本分为两类。固定成本不随生产或消费数量变化而保持不变。变动成本随数量按比例变化。理解这一区别对于比较两种商业方案或公用事业费率至关重要。

Fixed costs include rent, insurance, and monthly subscription fees.
Variable costs include raw materials, electricity per unit, and fuel per kilometre.
Total cost = Fixed Cost + (Variable Cost per Unit × Number of Units)

固定成本包括租金、保险费和月订阅费。
变动成本包括原材料、每单位电费和每公里燃油费。
总成本 = 固定成本 +（每单位变动成本 × 单位数量）

For example, a gym membership costs £30 per month plus £2 per visit. If you visit 10 times in a month, the total cost is £30 + (£2 × 10) = £50. Without the distinction between fixed and variable costs, you cannot accurately compare this with a pay-as-you-go option that charges £6 per visit.

例如，健身房会员每月30英镑外加每次访问2英镑。如果你一个月访问10次，总成本为30英镑 +（2英镑 × 10）= 50英镑。如果不区分固定成本和变动成本，就无法准确地将此与每次访问收费6英镑的按次付费选项进行比较。

4. Break-Even Point | 盈亏平衡点

The break-even point is the level of output at which total revenue equals total cost. Below this point, the business makes a loss; above it, the business makes a profit. This is one of the most tested applications of cost comparison in the IGCSE syllabus.

盈亏平衡点是总收入等于总成本时的产出水平。低于此点，企业亏损；高于此点，企业盈利。这是IGCSE大纲中成本比较最常考查的应用之一。

Break-Even Quantity = Fixed Cost ÷ (Selling Price per Unit − Variable Cost per Unit)

The denominator is called the contribution margin per unit. For instance, a company sells pens for £1.50 each. The variable cost is £0.70 per pen and the fixed cost is £400 per week. The contribution margin is £1.50 − £0.70 = £0.80. The break-even quantity is £400 ÷ £0.80 = 500 pens per week.

分母称为单位边际贡献。例如，一家公司以每支1.50英镑的价格出售钢笔。每支笔的变动成本为0.70英镑，每周固定成本为400英镑。边际贡献为1.50英镑 − 0.70英镑 = 0.80英镑。盈亏平衡数量为400英镑 ÷ 0.80英镑 = 每周500支。

If the company sells 600 pens, the profit is (600 − 500) × £0.80 = £80. This calculation shows how break-even analysis links directly to profit prediction.

如果该公司销售600支笔，利润为（600 − 500）× 0.80英镑 = 80英镑。这一计算展示了盈亏平衡分析如何直接与利润预测相关联。

5. Best Buy Problems | 最佳购买问题

Best buy problems require comparing the value of different package sizes or promotional offers. The strategy is to calculate the cost per unit for each option and select the lowest cost per unit, or to calculate the quantity obtained per pound.

最佳购买问题需要比较不同包装规格或促销优惠的价值。策略是计算每个选项的单位成本，并选择单位成本最低的选项，或者计算每英镑可获得的数量。

Consider a supermarket selling orange juice: a 1-litre bottle costs £1.80, while a 250-mL carton costs £0.55. The unit price of the bottle is £1.80 per litre. The unit price of the carton is £0.55 ÷ 0.25 L = £2.20 per litre. Therefore, the 1-litre bottle is much better value.

考虑一家超市销售的橙汁：1升装售价1.80英镑，而250毫升装售价0.55英镑。瓶装的单位价格为每升1.80英镑。盒装的单位价格为0.55英镑 ÷ 0.25升 = 每升2.20英镑。因此，1升装划算得多。

Sometimes promotions complicate comparisons. A “buy one get one free” offer on a £2.00 item effectively gives you 2 units for £2.00, so the unit price is £1.00. A “3 for £5” offer gives a unit price of £5 ÷ 3 ≈ £1.67. Always compute the effective unit price before deciding.

有时促销会使比较变得复杂。一项价值2.00英镑商品的”买一送一”优惠实际上让你以2.00英镑获得2件商品，因此单位价格为1.00英镑。”3件5英镑”优惠的单位价格为5英镑 ÷ 3 ≈ 1.67英镑。在决定之前务必计算有效单位价格。

6. Comparing Energy Tariffs | 比较能源费率

Energy companies charge customers using a standing charge (fixed cost per day) plus a unit rate (variable cost per kWh). Comparing two tariffs requires setting up a total cost equation for each and then evaluating for a given consumption level.

能源公司通过日固定费（固定成本）加单位费率（每千瓦时变动成本）向客户收费。比较两种费率需要为每种方案建立总成本方程，然后针对给定的消费水平进行评估。

Tariff A: standing charge £0.30 per day, unit rate £0.15 per kWh.

Tariff B: standing charge £0.50 per day, unit rate £0.12 per kWh.

Cost A = 0.30d + 0.15k

Cost B = 0.50d + 0.12k

For a month with 30 days and 300 kWh used, Tariff A costs £9.00 + £45.00 = £54.00, while Tariff B costs £15.00 + £36.00 = £51.00. Tariff B is cheaper for this consumption level. However, if consumption is only 100 kWh, Tariff A costs £9.00 + £15.00 = £24.00 and Tariff B costs £15.00 + £12.00 = £27.00, so Tariff A is cheaper.

对于一个月使用30天和300千瓦时的情况，方案A成本为9.00英镑 + 45.00英镑 = 54.00英镑，而方案B成本为15.00英镑 + 36.00英镑 = 51.00英镑。在此消费水平下，方案B更便宜。然而，如果消费仅为100千瓦时，方案A成本为9.00英镑 + 15.00英镑 = 24.00英镑，方案B成本为15.00英镑 + 12.00英镑 = 27.00英镑，因此方案A更便宜。

The crossover point occurs when 0.30d + 0.15k = 0.50d + 0.12k. Solving gives 0.03k = 0.20d, so k = (0.20 ÷ 0.03)d ≈ 6.67d. For d = 30, k = 200 kWh. Above 200 kWh, Tariff B is cheaper; below it, Tariff A is cheaper.

交叉点出现在0.30d + 0.15k = 0.50d + 0.12k时。求解得0.03k = 0.20d，因此k =（0.20 ÷ 0.03）d ≈ 6.67d。当d = 30时，k = 200千瓦时。高于200千瓦时，方案B更便宜；低于此值，方案A更便宜。

7. Comparing Phone Plans | 比较手机套餐

Mobile phone plans typically combine a fixed monthly charge with a variable cost per minute or per gigabyte. You should know how to model each plan as a linear equation and determine which plan is more economical for a given usage level.

手机套餐通常将固定月费与每分钟或每千兆字节的变动成本相结合。你应该知道如何将每种套餐建模为线性方程，并确定在给定使用水平下哪种套餐更经济。

Plan	Monthly Fee	Cost per Extra Minute
X	£20	£0.10
Y	£30	£0.05

Plan X total cost = 20 + 0.10m, where m is the number of extra minutes beyond the included allowance. Plan Y total cost = 30 + 0.05m. To find when they are equal: 20 + 0.10m = 30 + 0.05m, so 0.05m = 10, giving m = 200 minutes.

方案X总成本 = 20 + 0.10m，其中m为超出含括额度的额外分钟数。方案Y总成本 = 30 + 0.05m。求两者相等时：20 + 0.10m = 30 + 0.05m，因此0.05m = 10，得m = 200分钟。

If you expect to use fewer than 200 extra minutes, Plan X is cheaper. If more than 200 minutes, Plan Y becomes the better choice. Drawing both lines on a graph can visualise this comparison clearly.

如果你预计使用少于200分钟的额外时间，方案X更便宜。如果超过200分钟，方案Y成为更优选择。在图上绘制两条直线可以清晰地可视化这一比较。

8. Comparing Hiring vs. Buying | 租赁与购买比较

Another common application involves deciding whether to hire equipment or buy it. Hiring has lower upfront cost but ongoing charges, while buying has a large initial cost but no regular payments. Comparing them requires considering the period of use.

另一个常见应用涉及决定是租用设备还是购买设备。租用前期成本较低，但有持续费用，而购买初始成本较大，但没有定期付款。比较它们需要考虑使用期限。

Suppose a professional camera can be hired for £25 per day or purchased for £400. The comparison is straightforward: the hire cost for d days is 25d. Buying is cheaper when 400 < 25d, which means d > 16 days. If you need the camera for more than 16 days, buying is economically sensible.

假设一台专业相机可以按每天25英镑租用或以400英镑购买。比较很简单：d天的租赁成本为25d。当400 < 25d时，购买更便宜，这意味着d > 16天。如果你需要相机超过16天，购买在经济上是合理的。

This type of problem may also include a residual value for the purchased item, such as reselling the camera after use for £150. Then the net cost of buying is £400 − £150 = £250, and the break-even point becomes d = £250 ÷ £25 = 10 days.

此类问题还可能包括购买物品的残值，例如使用后以150英镑转卖相机。那么购买的净成本为400英镑 − 150英镑 = 250英镑，盈亏平衡点变为d = 250英镑 ÷ 25英镑 = 10天。

9. Cost Comparison with Percentages | 含百分比的成本比较

Some cost comparison problems incorporate percentage increases or decreases, such as VAT, discount, or service charge. You must be able to apply a percentage change to a cost and then compare the results.

一些成本比较问题包含百分比增减，例如增值税、折扣或服务费。你必须能够对成本应用百分比变化，然后比较结果。

For example, a laptop is priced at £800 including 20% VAT. The price excluding VAT is £800 ÷ 1.20 = £666.67. If a competitor offers the same laptop at £720 including VAT but with a 5% store discount, the final price is £720 × 0.95 = £684. Comparing the two, the first deal is cheaper even though the sticker price is higher.

例如，一台笔记本电脑含20%增值税的定价为800英镑。不含增值税的价格为800英镑 ÷ 1.20 = 666.67英镑。如果竞争对手以含税720英镑的价格提供同一台笔记本电脑，但提供5%的商店折扣，最终价格为720英镑 × 0.95 = 684英镑。比较两者，第一个交易更便宜，即使标价更高。

Final Price = Original Price × (1 + r₁) × (1 − r₂)

Here r₁ represents a tax rate, and r₂ represents a discount rate. Sequential percentage changes cannot be added or subtracted; they must be multiplied. This is a common trap in exams.

这里r₁代表税率，r₂代表折扣率。连续的百分比变化不能相加或相减；必须相乘。这是考试中常见的陷阱。

10. Exam Tips and Worked Strategy | 考试技巧与解题策略

Cost comparison questions in the Edexcel IGCSE exam often appear in Paper 1 (non-calculator) and Paper 2 (calculator). You should always set up clear expressions for each option before performing calculations.

Edexcel IGCSE考试中的成本比较问题通常出现在Paper 1（不可用计算器）和Paper 2（可用计算器）中。在执行计算之前，你应该始终为每个选项建立清晰的表达式。

Read the question carefully to identify fixed costs, variable costs, and units.
Express every quantity in consistent units before comparing.
Write down the equation for each option, then solve or substitute as needed.
Check whether the question asks for unit price, total cost, savings, or break-even quantity.
Round only at the final step unless instructed otherwise.

仔细阅读问题，找出固定成本、变动成本和单位。
在比较之前，将所有数量表示为一致的单位。
写出每个选项的方程，然后根据需要求解或代入。
检查问题要求的是单位价格、总成本、节省额还是盈亏平衡数量。
除非另有说明，否则仅在最后一步四舍五入。

A useful strategy is the “unit comparison table”. Create a table with columns for option, formula, and result. This keeps your work organised and prevents arithmetic errors, especially in multi-part questions worth 4 to 6 marks.

一个有用的策略是”单位比较表”。创建一张包含选项、公式和结果列的表格。这能使你的解答条理清晰，防止算术错误，特别是在价值4到6分的多小题问题中。

11. Worked Exam-Style Example | 考试风格例题精讲

A company prints custom T-shirts. The setup fee is £50. Printing costs £4 per shirt from Supplier A, or £2.50 per shirt from Supplier B with a setup fee of £90. For what number of T-shirts does Supplier B become cheaper?

一家公司印制定制T恤。制版费为50英镑。供应商A每件衬衫印刷费为4英镑，或者供应商B每件印刷费为2.50英镑，制版费为90英镑。当T恤数量为多少时，供应商B变得更便宜？

Cost A = 50 + 4n

Cost B = 90 + 2.5n

Set the two costs equal: 50 + 4n = 90 + 2.5n. Subtract 2.5n from both sides: 1.5n = 40. Therefore n = 40 ÷ 1.5 ≈ 26.67. Since the number of T-shirts must be a whole number, Supplier B becomes cheaper from 27 shirts onwards.

令两个成本相等：50 + 4n = 90 + 2.5n。两边减去2.5n：1.5n = 40。因此n = 40 ÷ 1.5 ≈ 26.67。由于T恤数量必须是整数，供应商B从第27件起变得更便宜。

For 26 shirts, Cost A = 50 + 104 = £154, and Cost B = 90 + 65 = £155, so A is cheaper. For 27 shirts, Cost A = 50 + 108 = £158, and Cost B = 90 + 67.5 = £157.50, so B is cheaper. This confirms the inequality boundary.

对于26件衬衫，成本A = 50 + 104 = 154英镑，成本B = 90 + 65 = 155英镑，因此A更便宜。对于27件衬衫，成本A = 50 + 108 = 158英镑，成本B = 90 + 67.5 = 157.50英镑，因此B更便宜。这确认了不等式边界。

12. Common Mistakes to Avoid | 常见错误避坑指南

Students frequently make errors in cost comparison problems due to unit confusion, incorrect formula setup, or premature rounding. Awareness of these pitfalls is the first step to avoiding them.

学生在成本比较问题中经常因单位混淆、公式设置错误或过早四舍五入而出错。意识到这些陷阱是避免它们的第一步。

Adding percentage changes instead of multiplying them.
Comparing prices without converting units (e.g., comparing 500 g with 1 kg directly).
Using the wrong reference value when calculating percentage saving.
Forgetting to include the fixed cost in total cost calculations.
Solving break-even equations but not interpreting the result in context.

将百分比变化相加而非相乘。
不转换单位就比较价格（例如，直接比较500克和1千克）。
计算百分比节省时使用错误的参考值。
在总成本计算中忘记包含固定成本。
求解盈亏平衡方程但未结合情境解释结果。

Always check the reasonableness of your answer. If the break-even quantity is negative or absurdly large, revisit your equation. Also verify which option is cheaper on each side of the boundary using a simple test value.

始终检查你答案的合理性。如果盈亏平衡数量为负数或大得离谱，重新检查你的方程。同时使用一个简单的测试值来验证边界两侧哪个选项更便宜。

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Cost Comparison: Calculations and Applications | 成本比较的计算与应用