Worksheet Chapter 27: Costs and Break-even Analysis | 第27章 练习题：成本与盈亏平衡分析

📚 Worksheet Chapter 27: Costs and Break-even Analysis | 第27章 练习题：成本与盈亏平衡分析

This worksheet covers the essential concepts of costs and break-even analysis for Cambridge A-Level Business. You will learn how to classify costs, calculate total and average costs, apply contribution analysis, construct break-even charts, and evaluate the usefulness of break-even analysis for business decisions.

本练习册涵盖剑桥A-Level商务课程中成本与盈亏平衡分析的核心概念。你将学习成本分类、总成本与平均成本的计算、边际贡献分析的应用、盈亏平衡图的绘制，以及评估盈亏平衡分析对企业决策的有效性。

1. Fixed and Variable Costs | 固定成本与可变成本

Fixed costs are costs that do not change with the level of output. They must be paid even if the business produces nothing. Examples include rent, insurance, and salaries of permanent staff.

固定成本是指不随产量水平变化而变化的成本。即使企业没有任何产出，也必须支付这些费用。例如租金、保险和长期职员的工资。

Variable costs are costs that change directly in proportion to the level of output. If output increases, variable costs rise; if output falls, variable costs fall. Typical variable costs include raw materials and direct labour paid per unit produced.

可变成本是随产量水平按比例直接变化的成本。如果产量增加，可变成本上升；如果产量下降，可变成本下降。典型的可变成本包括原材料和按件计酬的直接人工。

• Semi-variable costs have both fixed and variable elements, such as a telephone bill with a fixed monthly rental plus charges for extra calls.

  半可变成本同时包含固定与可变因素，例如电话账单包含固定月租费以及额外通话的费用。

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2. Total Costs and Other Cost Concepts | 总成本及其他成本概念

Total cost is the sum of fixed costs and variable costs at a given level of output.

总成本是在给定产量水平下固定成本与可变成本之和。

Total Cost = Fixed Costs + Variable Costs

总成本 = 固定成本 + 可变成本

Average cost (unit cost) is the cost of producing one unit. It is calculated by dividing total cost by output. As output increases, average cost may fall due to economies of scale, but it can rise if fixed costs are spread too thinly or variable costs increase.

平均成本（单位成本）是生产一单位产品的成本。计算方法是用总成本除以产量。随着产量增加，平均成本可能因规模经济而下降，但如果固定成本分摊过度或可变成本上升，也可能上升。

Average Cost = Total Cost ÷ Output

平均成本 = 总成本 ÷ 产量

Sales revenue is the income from selling goods or services. It is calculated by multiplying price per unit by the quantity sold.

销售收入是销售商品或服务所获得的收入。计算方法为单价乘以销售量。

Sales Revenue = Price × Quantity Sold

销售收入 = 单价 × 销售量

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3. Contribution and Break-even Point | 边际贡献与盈亏平衡点

Contribution per unit is the amount each unit contributes towards covering fixed costs and then generating profit. It is calculated as selling price minus variable cost per unit.

单位边际贡献是每单位产品用于弥补固定成本并产生利润的金额。其计算方法为销售单价减去单位可变成本。

Contribution per unit = Price – Variable Cost per Unit

单位边际贡献 = 单价 – 单位可变成本

Total contribution is the contribution per unit multiplied by the quantity sold. It shows the total amount left after variable costs to cover fixed costs.

总边际贡献是单位边际贡献乘以销售量。它表示在弥补可变成本后，可用于覆盖固定成本的总额。

Total Contribution = Contribution per Unit × Quantity Sold

总边际贡献 = 单位边际贡献 × 销售量

The break-even point is the level of output at which total revenue equals total cost, meaning neither profit nor loss is made. It can be expressed in units or in sales value.

盈亏平衡点是总收入等于总成本时的产量水平，即既不盈利也不亏损。它可以用单位数量或销售额来表示。

Break-even Point (units) = Fixed Costs ÷ Contribution per Unit

盈亏平衡点（单位）= 固定成本 ÷ 单位边际贡献

For break-even revenue, divide fixed costs by the contribution ratio (contribution per unit ÷ price).

盈亏平衡销售额的计算方法为固定成本除以边际贡献率（单位边际贡献 ÷ 单价）。

Break-even Revenue = Fixed Costs ÷ (Contribution per Unit ÷ Price)

盈亏平衡销售额 = 固定成本 ÷（单位边际贡献 ÷ 单价）

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4. Constructing a Break-even Chart | 绘制盈亏平衡图

A break-even chart is a graphical representation showing fixed costs, variable costs, total costs, and sales revenue on the same axes. The x-axis shows output, and the y-axis shows costs and revenue.

盈亏平衡图是一种在同一坐标轴上展示固定成本、可变成本、总成本和销售收入的图形。横轴表示产量，纵轴表示成本和收入。

To construct the chart:

绘制该图的步骤：

• Draw the fixed cost line as a horizontal line starting from the origin level of fixed costs.

  绘制一条从固定成本金额处开始的水平固定成本线。

• Draw the variable cost line starting from zero, but for the total cost line, start from the fixed cost line and add variable costs at each level of output.

  绘制从零点开始的可变成本线；总成本线则从固定成本线开始，在每个产量水平上加上可变成本。

• Draw the revenue line starting from the origin with a slope equal to the selling price per unit.

  绘制从原点开始、斜率等于单价的收入线。

• The break-even point is where the revenue line intersects the total cost line.

  盈亏平衡点是收入线与总成本线的交点。

Below the break-even point, the business makes a loss; above it, a profit is made. The vertical distance between revenue and total cost lines at any output shows profit or loss.

在盈亏平衡点以下，企业发生亏损；在盈亏平衡点以上，企业获得盈利。任何产量水平下收入线与总成本线之间的垂直距离表示利润或亏损。

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5. Margin of Safety | 安全边际

The margin of safety is the amount by which actual or budgeted sales exceed the break-even level of sales. It shows how much sales can fall before the business starts to make a loss.

安全边际是实际或预算销售量超过盈亏平衡销售量的部分。它表示销售在开始亏损之前可以下降多少。

Margin of Safety (units) = Expected Sales – Break-even Sales

安全边际（单位）= 预计销售量 – 盈亏平衡销售量

The margin of safety can also be expressed as a percentage of expected sales:

安全边际也可以用预计销售量的百分比表示：

Margin of Safety (%) = Margin of Safety ÷ Expected Sales × 100%

安全边际（%）= 安全边际 ÷ 预计销售量 × 100%

With a margin of safety of 4,000 units on expected sales of 10,000 units, the margin is 40%. This means sales could drop by 40% before losses occur.

例如，预计销售量为10,000件，安全边际为4,000件，则安全边际率为40%。这意味着销售量下降40%才会开始亏损。

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6. Advantages and Limitations of Break-even Analysis | 盈亏平衡分析的优缺点

Break-even analysis is widely used because it is simple, quick, and helps managers understand the relationship between costs, volume, and profit.

盈亏平衡分析被广泛使用，因为它简单、快捷，并帮助管理者理解成本、产量和利润之间的关系。

• Advantages: It provides a clear target for sales, helps evaluate pricing strategies, supports decision-making about production levels, and allows sensitivity analysis by changing one variable.

  优点：它为销售提供明确目标，帮助评估定价策略，支持产量决策，并允许通过改变一个变量进行敏感性分析。

• Limitations: It assumes that all costs are either fixed or variable, but in reality many costs are semi-variable. It also assumes that selling price and variable cost per unit remain constant, which may not be true if discounts or bulk buying are involved. Furthermore, it is a linear model that may not reflect real-world complexity.

  缺点：它假设所有成本要么固定要么可变，但现实中许多成本是半可变的。它还假设销售单价和单位可变成本保持不变，但在打折或批量购买时可能不成立。此外，它是线性模型，可能无法反映现实世界的复杂性。

Therefore, break-even analysis should be used as a planning tool, not as a complete prediction of actual results.

因此，盈亏平衡分析应作为规划工具使用，而不是对实际结果的完整预测。

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7. Worked Example | 例题示范

Let us work through a typical exam-style question. A business sells hand-made chairs. The selling price is $40 per unit. The variable cost per unit is $24. Fixed costs are $32,000 per year. The business expects to sell 2,500 chairs next year.

我们来看一道典型的考试题。一家企业销售手工椅子。每单位售价为40美元。单位可变成本为24美元。年固定成本为32,000美元。该企业预计明年销售2,500把椅子。

First, calculate the contribution per unit:

首先计算单位边际贡献：

$40 – $24 = $16

Next, calculate the break-even point in units:

接下来计算盈亏平衡点（单位）：

$32,000 ÷ $16 = 2,000 units

Now calculate the margin of safety in units and as a percentage:

然后计算安全边际（单位及百分比）：

2,500 – 2,000 = 500 units
500 ÷ 2,500 × 100% = 20%

This means sales can fall by 20% before the business breaks even.

这意味着销售额在盈亏平衡之前可以下降20%。

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8. Worksheet Practice Questions | 练习题目

Attempt these questions on your own before checking the answers below. Show your working and state any assumptions you make.

请先独立完成以下题目，再对照下面的答案。写出计算过程并说明你所做的假设。

1. Define the terms ‘fixed cost’ and ‘variable cost’. Give one example of each.

   解释“固定成本”和“可变成本”的含义，并各举一例。

2. A company has fixed costs of $50,000. The selling price per unit is $12, and the variable cost per unit is $7. Calculate the break-even point in units.

   某公司固定成本为50,000美元。销售单价为12美元，单位可变成本为7美元。计算盈亏平衡点（单位）。

3. Using the data above, what is the contribution per unit? What is the total contribution if the company sells 12,000 units? Would the company make a profit or loss? Show your profit or loss.

   根据以上数据，单位边际贡献是多少？如果公司销售12,000件，总边际贡献是多少？公司会盈利还是亏损？请计算利润或亏损。

4. Explain the meaning of ‘margin of safety’. Calculate the margin of safety if budgeted sales are 12,000 units and break-even output is 10,000 units.

   解释“安全边际”的含义。如果预算销售量为12,000件，盈亏平衡产量为10,000件，计算安全边际。

5. Discuss two limitations of break-even analysis.

   讨论盈亏平衡分析的两个局限性。

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9. Answers and Key Formulas | 答案与关键公式

Check your answers to the practice questions below. Keep this summary of formulas for your revision.

请对照下面的练习答案。保留这份公式摘要以备复习。

Answers | 答案

Question 1: Fixed cost is a cost that does not change with output, e.g. rent. Variable cost is a cost that changes directly with output, e.g. raw materials.

问题1：固定成本是不随产量变化而变化的成本，例如租金。可变成本是随产量直接变化的成本，例如原材料。

Question 2: Contribution per unit = $12 – $7 = $5. Break-even point = $50,000 ÷ $5 = 10,000 units.

问题2：单位边际贡献 = 12美元 – 7美元 = 5美元。盈亏平衡点 = 50,000美元 ÷ 5美元 = 10,000件。

Question 3: Contribution per unit = $5. Total contribution at 12,000 units = 12,000 × $5 = $60,000. After covering fixed costs of $50,000, profit = $10,000.

问题3：单位边际贡献 = 5美元。12,000件时的总边际贡献 = 12,000 × 5美元 = 60,000美元。扣除50,000美元固定成本后，利润 = 10,000美元。

Question 4: Margin of safety = 12,000 – 10,000 = 2,000 units. This is the amount by which sales can fall before loss is made.

问题4：安全边际 = 12,000 – 10,000 = 2,000件。这是销售在发生亏损前可以下降的数量。

Question 5: Two limitations: (1) Break-even analysis assumes costs and revenues are linear. (2) It ignores the impact of changes in stock levels and assumes all output is sold.

问题5：两个局限性：（1）盈亏平衡分析假设成本和收入是线性的。（2）它忽略了存货水平变动的影响，并假设所有产出都已售出。

Key Formulas | 关键公式

Formula | 公式	Calculation | 计算
Total Cost | 总成本	Fixed Costs + Variable Costs | 固定成本 + 可变成本
Average Cost | 平均成本	Total Cost ÷ Output | 总成本 ÷ 产量
Contribution per Unit | 单位边际贡献	Price – Variable Cost per Unit | 单价 – 单位可变成本
Break-even (units) | 盈亏平衡点（件）	Fixed Costs ÷ Contribution per Unit | 固定成本 ÷ 单位边际贡献
Margin of Safety (units) | 安全边际（件）	Expected Sales – Break-even Sales | 预计销售量 – 盈亏平衡销售量

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Use this worksheet as a structured revision tool. Practice drawing break-even charts and calculating missing values using the formulas. This is a key topic for Cambridge A-Level Business exams, especially in Paper 2 and Paper 3.

将本练习册作为系统的复习工具。练习绘制盈亏平衡图并使用公式计算缺失值。这是剑桥A-Level商务考试中的关键专题，尤其是在Paper 2和Paper 3中。

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