A formula sheet is most useful when you can find an equation by the question it answers. In a case, nobody announces that it is time to use weighted averages. You hear “our margins fell even though every product became more profitable” and have to decide what to calculate.
Use this page as a reference while practising. Start with the business question, choose the relationship, and write the units beside the inputs. All numerical examples below are illustrative.
How much does the business earn?
| Question | Formula | Quick example |
|---|---|---|
| What is revenue? | Units sold × price per unit | 4,000 orders × £25 = £100,000 |
| What does one sale contribute? | Price − all variable costs per unit | £25 − £15 = £10 per order |
| What is total contribution? | Revenue − total variable costs | £100,000 − £60,000 = £40,000 |
| What is operating profit? | Contribution − operating fixed costs | £40,000 − £30,000 = £10,000 |
| What is operating margin? | Operating profit ÷ revenue | £10,000 ÷ £100,000 = 10% |
Use the same period throughout. In this example, contribution margin is 40%, while operating margin is 10%. The different names identify which costs have been deducted.
For multiple products, add each product's price × quantity to get total revenue. Using a simple average price assumes equal volumes unless you explicitly weight it.
Gross profit uses the cost-of-goods-sold definition supplied in the case. It is not automatically contribution: some production costs can be fixed, while some selling costs can vary with each order. Use the case’s cost definitions and state which costs change when one more unit is sold.
How much did a number change?
| Question | Formula | Quick example |
|---|---|---|
| Absolute change | New − old | £120k − £100k = £20k |
| Relative change | (New − old) ÷ old | £20k ÷ £100k = 20% |
| Percentage-point change | New rate − old rate | 25% − 20% = 5 points |
| New amount after growth | Old × (1 + growth rate) | £100k × 1.20 = £120k |
| Original amount before growth | New ÷ (1 + growth rate) | £120k ÷ 1.20 = £100k |
Treat percentages as fractions when multiplying: 20% is 0.20. If a margin rises from 20% to 25%, that is five percentage points and a 25% relative increase in the margin.
When both price and volume change, multiply the factors. Price up 10% and volume down 10% gives 1.10 × 0.90 = 0.99, so revenue falls 1%. Write the starting base beside each percentage before multiplying the factors.
Quick Answer Repair Drill
A holiday-rental website redesigned its booking page. Presenting the results to the client, an analyst says: 'Conversion improved by 1%, a modest result.'
- Conversion before redesign
- 4% of visitors book
- Conversion after redesign
- 5% of visitors book
- Visitor traffic
- Unchanged
Which statement correctly restates the result for the client?
How do you combine segments or periods?
Weighted average = sum of each value × its weight ÷ total weights.
If 80 orders contribute £10 each and 20 contribute £30 each, average contribution is (80 × £10 + 20 × £30) ÷ 100 = £14 per order.
Choose weights from the measure's denominator. Weight margin percentages by revenue, prices by units and conversion rates by eligible prospect counts. When possible, calculate the total numerator and denominator directly; that makes the logic visible.
Compound growth: ending value = starting value × (1 + growth rate)^number of periods.
At 10% annual growth, £100 becomes £121 after two years. For differing annual growth rates, multiply each year's factor in sequence.
CAGR = (ending value ÷ starting value)^(1 ÷ years) − 1.
Growth from £100 to £121 over two years has a 10% compound annual growth rate. Count intervals: from the end of 2023 to the end of 2025 is two years. CAGR is a smoothed rate between endpoints; actual yearly growth may differ. The usual formula requires positive values.
How much volume covers the costs?
Break-even units = fixed costs ÷ contribution per unit.
At £10 contribution and £30,000 monthly fixed costs, break-even is 3,000 orders per month.
Units for target profit = (fixed costs + target operating profit) ÷ contribution per unit.
To earn £20,000 monthly operating profit with those costs, sell 5,000 orders. Check whether reaching that volume adds another shift, site or fixed charge.
Break-even revenue = fixed costs ÷ contribution margin ratio.
At a 40% contribution margin, £30,000 of fixed costs requires £75,000 of revenue. These relationships assume the relevant price and cost structure remains valid over the volume range.
Quick Case Math Drill
An online plant nursery ships boxes from one packing site. Management wants the monthly volume needed to reach its operating profit target.
- Contribution per box
- £12
- Profit target
- £30,000 operating profit per month
| Monthly boxes packed | Packing shifts | Monthly fixed costs |
|---|---|---|
| Up to 5,000 | One | £36,000 |
| 5,001 to 8,000 | Two | £45,000 |
How many boxes a month must the nursery sell to hit its target?
When does the investment recover its cash?
Simple payback = upfront investment ÷ annual net cash benefit, when annual cash benefits are level.
A £120,000 investment saving £50,000 annually after recurring costs has 2.4-year payback if savings accrue evenly from the start. With a ramp, write each year's net cash flow and find when cumulative benefit covers the investment.
Net present value = sum of each future net cash flow ÷ (1 + discount rate)^its period − initial investment.
For a single £110 receipt one year from now, a 10% annual discount rate gives present value £100. Paying £100 now for that receipt has zero NPV under those assumptions.
Use NPV when the case calls for discounted value and provides the required inputs. Simple payback answers recovery speed. It does not capture all later value. Keep tax, working capital, asset replacement and terminal value consistent with the stated case rather than quietly assuming they disappear.
How much can an operation deliver?
| Question | Relationship | What to check |
|---|---|---|
| Appointment slots | Productive hours ÷ hours per appointment | Convert minutes to hours first |
| Completed appointments | Available slots × booking fill × attendance | Do not apply a rate twice if the supplied figure already includes it |
| Process capacity | Capacity of the limiting required stage | Use comparable good-output units and time periods |
| Utilisation | Actual output ÷ relevant capacity | Identify which capacity definition is used |
| Average reusable fleet in circulation | Uses per day × average full cycle days | Add the stated reserve separately; use a stable-flow assumption |
For example, 10 productive hours create 20 half-hour slots. At 80% booking fill and 90% attendance, expect 14.4 completed visits across comparable periods. The schedule has 20 slots; expected completed demand is lower.
How large is the opportunity?
A useful demand model is eligible customers × adoption × purchases per customer per period. Multiply by price to convert units into revenue.
For example, 50,000 eligible businesses × 20% adoption × one £1,000 annual subscription gives £10 million of annual market revenue. That is a market estimate, not automatically the client's revenue. Apply its attainable share separately.
For replacement markets, a starting relationship is installed stock ÷ average replacement life, with growth, eligibility and adoption handled explicitly. A stock of one million devices does not mean one million new purchases every year.
Check the equation before doing the arithmetic
Ask yourself three things: will the units cancel into the output I need, does the cost or population definition match, and does the result answer the client's question?
Then test it with small numbers if you are uncertain. Five orders at £20 each must create £100 revenue. A 10% margin on that revenue must mean £10 profit. Small examples expose a reversed ratio quickly.
Use the monthly store break-even drill to practise selecting and explaining a formula without keeping this sheet open.
Case math drill
Can you talk an MBB interviewer through the numbers?
Monthly Breakeven Units for New Store
The next problem asks for monthly breakeven. Explain the contribution per unit and the cost period to Casey, Case Room’s AI interviewer, before calculating.
Frequently asked questions
Revenue, contribution and change
What's the difference between revenue and contribution?
Revenue is units sold × price, while contribution per unit is price minus all variable costs per unit. With 4,000 orders at £25, revenue is £100,000; if variable costs are £15 per order, each order contributes £10 towards fixed costs and profit.
Is a rise from 20% to 25% a 5% increase?
No, it is a 5 percentage-point increase. Percentage-point change is new rate minus old rate. Relative change, (new − old) ÷ old, would describe the same move as 25% higher. Confusing the two is an easy way to misstate margins or conversion rates.
How do I calculate a percentage change correctly?
Divide the change by the old value. Going from £100k to £120k is an absolute change of £20k and a relative change of £20k ÷ £100k, or 20%. Dividing by the new figure instead would understate the growth.
Why can average margin fall when every product became more profitable?
Because the mix shifted. An overall figure is a weighted average, so more sales of lower-margin products can pull it down even when each improves. Calculate value × weight summed, divided by total weights: 80 orders at £10 and 20 at £30 average £14, not £20.
Which weights should I use in a weighted average?
Take them from the measure's denominator. Weight margin percentages by revenue, prices by units and conversion rates by eligible prospect counts. Using the wrong base, such as averaging margins equally across products, gives a figure that will not match the business total.
Break-even, payback and capacity
How do I calculate break-even volume, and volume for a profit target?
Divide fixed costs by contribution per unit. With £30,000 monthly fixed costs and £10 contribution, break-even is 3,000 orders a month. For a target, add it to fixed costs first: £20,000 monthly operating profit needs 5,000 orders.
Does the break-even formula still hold at much higher volumes?
Only if fixed costs genuinely stay fixed. Reaching a higher volume may require another shift, site or fixed charge, which raises the numerator and moves the break-even point. Check for such steps before relying on one calculation across a wide volume range.
How do I work out payback when savings ramp up?
Simple payback, upfront investment ÷ annual net cash benefit, only works when benefits are level. A £120,000 investment saving £50,000 a year after recurring costs pays back in 2.4 years if savings accrue evenly. With a ramp, list each year's net cash flow and find when the cumulative total covers the investment.
How do I calculate how many appointments an operation can complete?
Divide productive hours by hours per appointment, converting minutes to hours first, to get available slots. Then multiply by booking fill and attendance. Check whether a supplied figure already includes one of those rates, so you do not apply it twice.
Market size and checking your work
Is a market revenue estimate the same as the client's revenue?
No. Eligible customers × adoption × purchases per period × price gives market revenue: 50,000 businesses × 20% × one £1,000 annual subscription is £10 million. To reach the client's revenue, apply its attainable share separately on top of that figure.
How can I catch a reversed ratio quickly?
Test the formula with small, obvious numbers. Five orders at £20 each must create £100 revenue, and a 10% margin on that revenue must mean £10 profit. If your equation gives anything else with simple inputs, fix it before using the case data.
What should I check before starting the arithmetic?
Ask three things: will the units cancel into the output you need, does the cost or population definition match, and does the result answer the client's question? Flawless arithmetic on a mismatched population or cost base still produces the wrong answer.
How should I use a formula sheet while practising cases?
Start from the business question, choose the relationship, then write units beside each input. Cases rarely announce which formula applies; you might hear that margins fell despite every product improving and need to recognise a weighted-average problem. Finding equations by question builds that habit.