Comparative Costs | 比较成本

📚 Comparative Costs | 比较成本

In IGCSE Mathematics, comparative costs problems ask us to make sensible financial decisions by comparing the value of different options. Whether we are choosing between two products, tariffs, or travel plans, the key skill is to express each option in a common unit and then compare them fairly.

在IGCSE数学中，比较成本问题要求我们通过比较不同选项的价值来做出合理的财务决策。无论是选择两种商品、资费方案还是旅行计划，关键技能是先将每个选项表示为统一的单位，然后公平地进行比较。

1. Understanding Unit Cost | 理解单位成本

The unit cost is the price of one unit of an item or service. For example, if a 500 g box of cereal costs $2.50, the unit cost is $2.50 ÷ 500 = $0.005 per gram, or $5 per kilogram.

单位成本是指一件物品或一项服务中每一个单位的价格。例如，如果一盒500克的麦片售价2.50美元，那么单位成本为2.50 ÷ 500 = 每克0.005美元，即每千克5美元。

To find the unit cost, divide the total price by the quantity. The quantity may be in grams, litres, metres, hours, or any other measurable unit. Always write down the units so the result makes sense.

要计算单位成本，用总价除以数量。数量可以用克、升、米、小时或其他任何可测量的单位表示。始终写下单位，这样结果才有意义。

Unit cost = Total cost ÷ Quantity

单位成本 = 总成本 ÷ 数量

2. Comparing Two Products | 比较两种产品

Suppose a shop sells the same brand of orange juice in two bottle sizes: a 1 litre bottle for $1.80 and a 1.5 litre bottle for $2.60. Which is better value?

假设一家商店以两种瓶装规格销售同一品牌的橙汁：1升装售价1.80美元，1.5升装售价2.60美元。哪一种更划算？

1 L bottle: $1.80 ÷ 1 = $1.80 per litre

1升装：1.80 ÷ 1 = 每升1.80美元
1.5 L bottle: $2.60 ÷ 1.5 = $1.73 per litre (to the nearest cent)

1.5升装：2.60 ÷ 1.5 = 每升1.73美元（精确到分）

The 1.5 litre bottle is cheaper per litre, so it gives better value. Even though the total price is higher, the unit cost is lower.

1.5升装每升更便宜，因此更划算。尽管总价更高，但单位成本更低。

3. Using the Unitary Method | 使用归一法

The unitary method is a formal way of solving comparative cost problems. First find the cost of one unit, then multiply to find the cost of the required number of units.

归一法是解决比较成本问题的一种规范方法。首先求出一个单位的成本，然后乘以所需单位的数量，得到总成本。

For example, if 8 pens cost $4.80, find the cost of 11 pens. One pen costs $4.80 ÷ 8 = $0.60. Then 11 pens cost 11 × $0.60 = $6.60.

例如，如果8支笔售价4.80美元，求11支笔的成本。一支笔的成本为4.80 ÷ 8 = 0.60美元。那么11支笔的成本为11 × 0.60 = 6.60美元。

Cost of n items = (cost of one item) × n

n件物品的成本 = (一件物品的成本) × n

4. Best Buy Problems | 最佳购买问题

Best buy problems ask you to choose between different pack sizes or brands. The standard technique is to calculate the unit price of each option and select the smallest unit price.

最佳购买问题要求你在不同包装规格或品牌之间做出选择。标准技巧是计算每个选项的单位价格，然后选择单位价格最小的选项。

Option	Price	Quantity	Unit price
A	$3.20	400 g	$0.008 per g
B	$5.70	750 g	$0.0076 per g

Option B has the lower unit price, so it is the best buy. However, always check whether the larger size is truly needed, because buying more than you need may still be wasteful.

选项B的单位价格更低，因此是最佳购买。然而，始终要检查是否真的需要更大的包装，因为买得过多仍然可能造成浪费。

5. Currency Conversion | 货币兑换

When comparing costs across countries, we must convert currencies using exchange rates. The exchange rate tells us how much one currency is worth in another currency.

当比较不同国家的成本时，我们必须使用汇率进行货币兑换。汇率告诉我们一种货币相对于另一种货币的价值。

For example, if £1 = $1.25, then an item costing $50 is equivalent to £50 ÷ 1.25 = £40. To convert from dollars to pounds, divide by the exchange rate.

例如，如果£1 = $1.25，那么一件售价50美元的商品相当于50 ÷ 1.25 = £40。要将美元转换为英镑，需要用汇率相除。

Amount in foreign currency = Amount in home currency × Exchange rate

外币金额 = 本币金额 × 汇率

6. Comparing Tariffs and Plans | 比较资费套餐

Mobile phone plans and energy tariffs often have a fixed charge plus a variable charge per unit used. To compare such plans, we need to evaluate the total cost for a realistic usage level.

手机套餐和能源费率通常包含固定费用加上按使用单位计算的变动费用。要比较这类套餐，我们需要针对一个实际的使用水平评估总成本。

Plan X: $10 per month plus $0.05 per minute. Plan Y: $20 per month plus $0.02 per minute. For 300 minutes of calls:

套餐X：每月10美元，另加每分钟0.05美元。套餐Y：每月20美元，另加每分钟0.02美元。对于300分钟通话：

Plan X: $10 + 300 × $0.05 = $10 + $15 = $25

套餐X：10 + 300 × 0.05 = 10 + 15 = 25美元
Plan Y: $20 + 300 × $0.02 = $20 + $6 = $26

套餐Y：20 + 300 × 0.02 = 20 + 6 = 26美元

For 300 minutes, Plan X is cheaper. But for 1000 minutes, Plan X costs $10 + $50 = $60, while Plan Y costs $20 + $20 = $40, so Plan Y is cheaper. The best plan depends on usage.

对于300分钟，套餐X更便宜。但对于1000分钟，套餐X花费10 + 50 = 60美元，而套餐Y花费20 + 20 = 40美元，因此套餐Y更便宜。最佳套餐取决于使用量。

7. Comparing Fuel Efficiency | 比较燃油效率

Fuel costs are often compared using miles per gallon (mpg) or litres per 100 km. To compare the cost of driving two different cars, we can calculate the cost per mile or per kilometre.

燃油成本通常用每加仑英里数（mpg）或每100公里升数来比较。为了比较驾驶两辆不同汽车的成本，我们可以计算每英里或每公里的成本。

A car uses 6 litres per 100 km, and petrol costs $1.50 per litre. The fuel cost per 100 km = 6 × $1.50 = $9.00, so the cost per km is $9.00 ÷ 100 = $0.09 per km.

一辆汽车每100公里消耗6升汽油，汽油价格为每升1.50美元。每100公里的燃油成本 = 6 × 1.50 = 9.00美元，因此每公里成本为9.00 ÷ 100 = 每公里0.09美元。

If another car uses 4.5 litres per 100 km, its fuel cost per km = 4.5 × $1.50 ÷ 100 = $0.0675 per km. The second car is cheaper to run in terms of fuel.

如果另一辆汽车每100公里消耗4.5升，其每公里燃油成本 = 4.5 × 1.50 ÷ 100 = 每公里0.0675美元。就燃油而言，第二辆车运行成本更低。

8. Comparing Travel Options | 比较出行方式

Travel costs may involve a fixed ticket price plus per-kilometre costs, or a daily pass versus single tickets. We can use comparative cost analysis to decide which is better.

出行成本可能涉及固定票价加上每公里成本，或者日票与单程票的比较。我们可以使用比较成本分析来决定哪种更划算。

Daily pass costs $12 for unlimited travel. A single ticket costs $2.50. If you make n trips in a day, the cost of single tickets is $2.50 × n. The daily pass is better when $2.50n > $12, i.e. n > 4.8. So from 5 trips onwards, the daily pass is better.

日票价格为12美元，不限次数。单程票价格为2.50美元。如果你一天乘坐n次，单程票的成本为2.50 × n美元。当日票更划算的条件是2.50n > 12，即n > 4.8。因此从第5次乘坐开始，日票更划算。

Break-even point: number of trips where both costs are equal

盈亏平衡点：两种成本相等时的乘坐次数

9. Discounts and Special Offers | 折扣与特价

When comparing costs, we must also consider percentage discounts, “buy one get one free” offers, and multiple-purchase savings. Always calculate the actual amount paid per unit after the discount.

在比较成本时，我们还必须考虑百分比折扣、“买一赠一”优惠以及多件购买节省。始终计算折扣后实际支付的每单位金额。

Example: A packet of 6 bottles costs $7.50, but there is a 20% discount. After discount the price is $7.50 × 0.8 = $6.00. The cost per bottle is $6.00 ÷ 6 = $1.00.

示例：一包6瓶饮料售价7.50美元，但享有20%折扣。折扣后价格为7.50 × 0.8 = 6.00美元。每瓶成本为6.00 ÷ 6 = 1.00美元。

Another offer: “Buy 2 get 1 free” at $2.50 each means 3 bottles cost $5.00, so the average cost per bottle is $5.00 ÷ 3 ≈ $1.67. This is more expensive than the discounted pack.

另一个优惠：“买2送1”，每瓶2.50美元，意味着3瓶花费5.00美元，因此每瓶平均成本为5.00 ÷ 3 ≈ 1.67美元。这比折扣包装更贵。

10. Comparing Loans and Simple Interest | 比较贷款与单利

In financial mathematics, comparative costs may involve the total interest paid on different loans. Simple interest is calculated using the formula I = PRT, where P is principal, R is annual rate, and T is time in years.

在金融数学中，比较成本可能涉及不同贷款所支付的总利息。单利使用公式I = PRT计算，其中P为本金，R为年利率，T为以年为单位的时间。

Loan A: borrow $1000 at 8% per year for 2 years. Interest = 1000 × 0.08 × 2 = $160. Total repayment = $1000 + $160 = $1160.

贷款A：以每年8%的利率借款1000美元，期限2年。利息 = 1000 × 0.08 × 2 = 160美元。总还款额 = 1000 + 160 = 1160美元。

Loan B: borrow $1000 at 6% per year for 2 years. Interest = 1000 × 0.06 × 2 = $120. Total repayment = $1120. Loan B has a lower comparative cost.

贷款B：以每年6%的利率借款1000美元，期限2年。利息 = 1000 × 0.06 × 2 = 120美元。总还款额 = 1120美元。贷款B的比较成本更低。

11. Using Tables to Compare Costs | 使用表格比较成本

A table is a clear way to organise comparisons. Let us compare the total cost of hiring a car for 3 days from two companies.

表格是组织比较的清晰方式。让我们比较从两家公司租车3天的总成本。

Company	Daily rate	Insurance (3 days)	Total for 3 days
A	$45	$30	3 × 45 + 30 = $165
B	$50	$15	3 × 50 + 15 = $165

Both companies cost the same for 3 days. This is an equal-cost situation. For longer hires, calculate again: for 5 days, A costs 5 × 45 + 30 = $255, B costs 5 × 50 + 15 = $265, so A becomes cheaper.

两家公司在3天内的成本相同。这是一种成本相等的情况。对于更长的租期，再次计算：对于5天，A花费5 × 45 + 30 = 255美元，B花费5 × 50 + 15 = 265美元，因此A变得更便宜。

12. Putting It All Together | 综合应用

To solve any comparative cost problem, follow these steps: identify the quantities, choose a common unit, calculate the relevant total or unit cost, compare the values, and give an answer with units. Do not forget to round only at the final step to avoid rounding errors.

要解决任何比较成本问题，请遵循以下步骤：确定数量，选择统一单位，计算相关的总成本或单位成本，比较数值，并给出带单位的答案。不要忘记只在最后一步进行四舍五入，以避免舍入误差。

Always check whether the question asks for the better value, the cheaper total, or the difference in cost. Understanding the context is just as important as performing the arithmetic correctly.

始终检查问题是要求更划算、更便宜的总价还是成本差异。理解问题背景与正确执行算术同样重要。

With practice, comparative cost problems become straightforward. Remember to show all working clearly, because method marks are often awarded in IGCSE exams, even if the final numerical answer is correct.

通过练习，比较成本问题会变得简单明了。记住要清晰地展示所有步骤，因为在IGCSE考试中，即使最终数字答案正确，方法分也经常被给予。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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Comparative Costs | 比较成本