Few businesses sell one thing. A bakery sells bread, cakes and pies; a distributor sells fifty lines; a hotel sells rooms, meals and conference space. Each product earns a different contribution on each sale, and the answer to “how many must we sell to break even?” now depends on what we sell.
A cheap line that sells well and a dear line that sells rarely give different answers from the same total of units. To extend CVP analysis to several products we need one more assumption, which looks unrealistic and is the main weakness of the method: the products are sold in a fixed ratio.
This topic explains the two ways of handling the mix (the weighted average unit and the batch), how to find break-even, the margin of safety and target profit for several products, what happens to break-even when the mix changes, how to draw a multi-product chart, and the importance and weaknesses of CVP analysis. It builds on the break-even and profit-volume charts topic.
The next topic covers limiting factor analysis for a single scarce resource.
In 30 seconds
With several products, assume they are sold in a constant sales mix. Weighted average unit method: weighted average contribution per unit = total contribution ÷ total units; break-even units = fixed costs ÷ that figure; split the units by the mix. Batch method: define a batch in the mix ratio; contribution per batch = sum of (mix × contribution per unit); break-even batches = fixed costs ÷ contribution per batch.
The weighted average C/S ratio = total contribution ÷ total sales gives break-even revenue. Margin of safety and target profit follow as for one product. A shift of the mix towards lower-contribution products raises break-even. The multi-product chart plots cumulative revenue against cumulative profit with products ranked by C/S ratio (a bow-shaped line) and a straight line for the budgeted mix.
CVP analysis is useful for sales strategy, cost control and decisions, but rests on weak assumptions. Full marks come from stating the mix assumption and showing the average figures.
Break-even for several products
The method rests on the assumption that products are sold in a set ratio that does not change with volume. With that assumption, one of two approaches can be used. The single weighted average C/S ratio is the same under both.
The weighted average unit approach
- Find the contribution per unit of each product.
- Weight each by its share of the budgeted mix (units sold): weighted average contribution per unit = total budgeted contribution ÷ total budgeted units.
- Break-even units (all products together) = fixed costs ÷ weighted average contribution per unit.
- Share the break-even units between the products in the mix ratio, and multiply by prices for the revenue of each. Or use the weighted average revenue per unit = total budgeted revenue ÷ total units.
The batch approach
- Define a batch as the mix, for example three units of P and one of Q.
- Contribution per batch = sum of (units in the batch × contribution per unit). Revenue per batch likewise.
- Break-even batches = fixed costs ÷ contribution per batch. Break-even revenue = batches × revenue per batch, or fixed costs ÷ the batch C/S ratio.
- Multiply the batches by the units of each product in the batch.
The weighted average C/S ratio = contribution per batch ÷ revenue per batch = total contribution ÷ total sales.
Margin of safety and target profit
The margin of safety compares the budget with break-even, in batches, units or revenue, and is the same percentage in all three. Target profit works as for one product: contribution needed = fixed costs + target profit, divided by the contribution per batch, per average unit, or by the weighted average C/S ratio.
When the mix changes
The weighted average contribution changes whenever the mix changes. If more of the product with the lower contribution is sold, the average falls and the break-even point rises, even though no price or cost has changed. A budget based on a mix that sales staff do not achieve can therefore be wrong although total sales are on target.
The multi-product chart, and the importance and limits of CVP
A break-even chart for several products is usually a P/V chart with two lines. Steps: calculate each product’s C/S ratio; rank the products with the highest first; tabulate revenue, cumulative revenue, contribution and cumulative profit (starting at a loss equal to fixed costs); plot cumulative revenue against cumulative profit.
- The bow-shaped line assumes the most profitable product (highest C/S ratio) is sold first, then the next. Its break-even point is the earliest.
- The straight line assumes sales are in the budgeted mix. Its break-even is where it crosses the horizontal axis.
Importance of CVP analysis
- It gives insight into the relation between cost, volume and profit, and focuses on controllable factors.
- It shows the profit effect of decisions, for example a price cut to win volume.
- It supports sales strategy, cost control and decisions, and shows the minimum sales needed to profit, which helps with new products.
- Costs are often known with accuracy while sales are not, so flexing the model over several volumes is useful.
Limitations and weaknesses
- Cost classification: costs are hard to divide into fixed and variable, and some are semi-variable.
- Constant variable cost per unit: true only over a range; overtime may raise it near capacity, and economies of scale may lower it.
- Constant fixed costs: true only over a range and may step up.
- Constant price: prices change with market conditions.
- Constant sales mix: seldom true, and a single product or constant ratios is not usual in practice.
- The analysis ignores the uncertainty in estimates of costs and revenue.
Layouts
- Contribution per batch = Σ (units of each product in the batch × its contribution per unit). Revenue per batch likewise.
- Break-even batches = fixed costs ÷ contribution per batch. Units of each product = batches × its units per batch.
- Weighted average contribution per unit = contribution per batch ÷ units per batch. Break-even units = fixed costs ÷ it.
- Weighted average C/S ratio = contribution per batch ÷ revenue per batch.
- Target batches = (fixed costs + target profit) ÷ contribution per batch.
- Multi-product chart table: rank by C/S; cumulative revenue; cumulative profit = cumulative contribution − fixed costs.
Money is in naira, units are rounded up, and each answer is rounded once at the end.
Worked examples
Example 1: break-even, margin of safety and target. A company sells products P and Q in the ratio 3 to 1. P sells for ₦90 with a variable cost of ₦60; Q sells for ₦140 with a variable cost of ₦115. Fixed costs are ₦460,000. Budgeted sales are 18,000 units of P and 6,000 of Q. Required: break-even, margin of safety and the sales for a profit of ₦345,000.
- Contribution per unit. P 30; Q 25.
Batch of 3P + 1Q: contribution 3 × 30 + 25 = ₦115
revenue 3 × 90 + 140 = ₦410.
C/S ratio 115 ÷ 410 = 28.0%. - Break-even.
460,000 ÷ 115 = 4,000 batches = 12,000 P and 4,000 Q, 16,000 units.
Revenue 4,000 × 410 = ₦1,640,000.
Weighted average contribution per unit 115 ÷ 4 = ₦28.75
460,000 ÷ 28.75 = 16,000 units, the same. - Budget.
18,000 ÷ 3 = 6,000 batches
revenue 6,000 × 410 = ₦2,460,000. - Margin of safety.
6,000 − 4,000 = 2,000 batches = 8,000 units = ₦820,000, or 2,000 ÷ 6,000 = 33.3%. - Target profit.
Contribution needed 460,000 + 345,000 = ₦805,000
÷ 115 = 7,000 batches = 21,000 P and 7,000 Q, 28,000 units.
Revenue 7,000 × 410 = ₦2,870,000.
Check: 805,000 ÷ 0.2805 = 2,870,000.
Example 2: a change in the mix. The same company sells P and Q in the ratio 1 to 1 instead. Required: the new break-even.
- Weighted average contribution per unit.
(30 + 25) ÷ 2 = ₦27.5. - Break-even.
460,000 ÷ 27.5 = 16,727.3, rounded up to 16,728 units: 8,364 of each. - Comment. A shift towards Q, whose contribution is lower, raises break-even by about 700 units though no price or cost has changed.
Example 3: the multi-product chart. Three products have budgeted revenue and contribution: A ₦1,000,000 and ₦450,000; B ₦1,200,000 and ₦360,000; C ₦800,000 and ₦120,000. Fixed costs are ₦600,000. Required: the chart table and the two break-even points.
- C/S ratios. A 45%; B 30%; C 15%. Rank A, B, C.
- Cumulative profit. Start −600,000. After A (revenue 1,000,000): −150,000. After B (2,200,000): +210,000. After C (3,000,000): +330,000.
- Ranked break-even.
After A the loss is 150,000; B earns 30%, so 150,000 ÷ 0.30 = 500,000 more revenue: ₦1,500,000. - Average mix.
Total contribution 930,000 ÷ revenue 3,000,000 = 31%
break-even 600,000 ÷ 0.31 = ₦1,935,484. - Comment. The ranked line gives an earlier break-even than the budgeted mix, because it sells the best products first. The true figure depends on the mix actually sold.
Expect variants: three products, a mix given as a percentage, a change in price of one product, or a question that asks which method to use.
Common mistakes
- Averaging contributions without weights. Weight them by the mix.
- Dividing batch break-even by the wrong number of units when splitting between products.
- Forgetting the mix assumption when stating the result.
- Using the C/S ratio of one product for the whole business.
- Not ranking products by C/S ratio for the chart.
- Not rounding units up.
Quick revision
- Assume a constant mix. Batch contribution = sum of mix × CPU. Break-even batches = fixed ÷ batch contribution.
- Weighted average CPU = batch contribution ÷ batch units. Weighted average C/S = batch contribution ÷ batch revenue.
- Shift to the lower contribution product raises break-even.
- Chart: ranked bow line and straight budgeted-mix line.
- CVP: useful for strategy and decisions; weak on classification, constancy of price and cost, mix and uncertainty.
How the marks are earned
- Calculate: the batch or average figures, break-even, margin of safety and target.
- State the assumption: constant mix.
- Discuss: each limitation with its effect.
Practice questions
These are original questions, written in the style of the Skills examination. Work each one before you open the answer.
Recall check: multiple choice
Question 1. Multi-product CVP analysis assumes that products are sold:
- In a constant mix
- One at a time
- At falling prices
- At random
Show the answer
A. This is the main weakness of the method.
Question 2. Weighted average contribution per unit is:
- The simple average of unit contributions
- The highest contribution
- Total contribution ÷ total units
- Total sales ÷ units
Show the answer
C. It is weighted by the mix.
Question 3. In the batch approach, break-even batches equal:
- Contribution ÷ fixed costs
- Units ÷ batches
- Revenue ÷ batches
- Fixed costs ÷ contribution per batch
Show the answer
D. A batch is the mix ratio.
Question 4. A shift of the mix towards the lower-contribution product will:
- Lower break-even
- Raise break-even
- Leave it unchanged
- Remove fixed costs
Show the answer
B. The average contribution falls.
Question 5. A multi-product P/V chart plots:
- Cumulative revenue against cumulative profit
- Units against price
- Cost against tax
- Batches against units
Show the answer
A. Products are ranked by C/S ratio.
Question 6. The straight line on a multi-product chart assumes:
- The best product is sold first
- One product only
- Products are sold in the budgeted mix
- No fixed costs
Show the answer
C. It crosses the axis at the break-even point.
Question 7. The bow-shaped line assumes that:
- The worst product is sold first
- The most profitable product is sold first
- Only one product is sold
- Prices fall
Show the answer
B. It reaches break-even earliest.
Question 8. The weighted average C/S ratio is:
- Fixed costs ÷ sales
- Profit ÷ sales
- Price ÷ cost
- Total contribution ÷ total sales
Show the answer
D. It equals batch contribution ÷ batch revenue.
Question 9. A limitation of CVP analysis is that:
- It uses no data
- It needs no assumptions
- Its assumptions of constant price, cost and mix seldom hold
- It cannot be calculated
Show the answer
C. It also ignores uncertainty in estimates.
Question 10. CVP analysis is useful because it:
- Shows the profit effect of decisions at different volumes
- Predicts tax
- Removes risk
- Sets prices
Show the answer
A. It supports sales strategy, cost control and decisions.
Section B style: written questions
Question 11 (10 marks). (a) Explain the weighted average unit and batch approaches to multi-product break-even. (6 marks) (b) State FOUR limitations of CVP analysis. (4 marks)
Question 12 (15 marks). A company sells M and N in the ratio 2 to 1. M sells for ₦100 with a variable cost of ₦70; N sells for ₦160 with a variable cost of ₦118. Fixed costs are ₦816,000. Budgeted sales are 20,000 units of M and 10,000 of N. The target profit is ₦408,000.
(a) Calculate the contribution and revenue per batch and the C/S ratio. (3 marks) (b) Calculate break-even in batches, units of each product and revenue. (5 marks) (c) Calculate the margin of safety in batches, as a percentage and in revenue. (3 marks) (d) Calculate the units of each product and revenue needed for the target profit. (4 marks)
Question 13 (20 marks). A company sells R and S in the ratio 4 to 1. R sells for ₦60 with a variable cost of ₦40. S sells for ₦100 with a variable cost of ₦55. Fixed costs are ₦300,000.
(a) Calculate the weighted average contribution per unit, break-even in units and in revenue, and the units of each product. (8 marks) (b) Calculate break-even if the mix becomes 1 to 1. (4 marks) (c) Explain the effect of the change in mix. (3 marks) (d) Discuss how a multi-product P/V chart is drawn and what it shows. (5 marks)
Model answers
Show the model answer to Question 11
(a) Both assume a constant sales mix (1). Weighted average unit: find the average contribution per unit weighted by the mix, then divide fixed costs by it to get total break-even units, and split by the mix (2.5). Batch: define a batch in the mix ratio, find the contribution per batch, divide fixed costs by it to get batches, then multiply by units per batch (2.5).
(b) Any four:
- costs are hard to classify as fixed or variable
- variable cost per unit may change with activity
- fixed costs may step
- price may change
- the mix is unlikely to stay constant
- estimates are uncertain (1 each)
Show the model answer to Question 12
(a)
- Contribution per unit M 30, N 42.
- Batch of 2M + 1N: contribution 2 × 30 + 42 = ₦102
- revenue 2 × 100 + 160 = ₦360
- C/S ratio 102 ÷ 360 = 28.3% (3).
(b)
- 816,000 ÷ 102 = 8,000 batches = 16,000 M and 8,000 N
- revenue 8,000 × 360 = ₦2,880,000 (5).
(c) Budget 20,000 ÷ 2 = 10,000 batches; margin of safety 2,000 batches; 2,000 ÷ 10,000 = 20.0%; revenue 2,000 × 360 = ₦720,000 (budget revenue ₦3,600,000) (3).
(d)
- Contribution needed 816,000 + 408,000 = ₦1,224,000
- ÷ 102 = 12,000 batches = 24,000 M and 12,000 N
- revenue 12,000 × 360 = ₦4,320,000 (4).
Show the model answer to Question 13
(a)
- Contribution R 20, S 45.
- Batch of 4R + 1S: contribution 4 × 20 + 45 = 125
- weighted average per unit 125 ÷ 5 = ₦25 (2).
- Break-even units 300,000 ÷ 25 = 12,000 (2): R 9,600 and S 2,400 (2).
- Revenue per batch 4 × 60 + 100 = 340, average ₦68 a unit
- break-even revenue 12,000 × 68 = ₦816,000 (2).
(b)
- Mix 1 to 1: average contribution (20 + 45) ÷ 2 = ₦32.5 (2)
- break-even 300,000 ÷ 32.5 = 9,230.8, rounded up to 9,231 units (2).
(c) The 1 to 1 mix sells more of S, which earns the higher contribution, so the average contribution rises and break-even falls by about 2,770 units even though no price or cost changed (3).
(d) Calculate each product’s C/S ratio; rank them with the highest first; tabulate revenue, cumulative revenue, contribution and cumulative profit, starting at a loss equal to fixed costs; plot cumulative revenue against cumulative profit for a bow-shaped line; add a straight line for the budgeted mix.
The chart shows the earliest possible break-even and the break-even at the budgeted mix, and how much profit depends on selling the best products (5).
Pathfinder on Multi-product CVP analysis and its limitations
These are the ICAN Pathfinder questions that test this topic. ICAN owns the questions and the suggested solutions, so we point you to them and do not copy them. Open your Pathfinder, attempt each question under exam conditions, then check your answer against the notes above.
Multi-product break-even in units and value, revised budget, revenue for after-tax return.
Revise first: Layouts
Product contribution/profit, drop a product, extra sales to cover advertising; relevant cost quote for a one-off job; incremental, differential, committed, sunk, opportunity cost.
Revise first: Layouts
Multi-product break-even in units and value; revised mix, target after-tax profit.
Revise first: Layouts
Source: ICAN Pathfinder, Performance Management. Diets covered: March/July Diet, 2020 to May Diet, 2026. All 13 diets in the Pathfinder are mapped.
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