The useful test of a maths answer is whether you can explain what the client should do with it. This practice set asks for both the calculation and the implication.
For each problem, write the equation before solving. Then give a short answer aloud: the result, what it means and the next fact you need. Read the worked answer only after attempting it. All businesses and numbers here are illustrative.
I would rather see you work carefully through two unfamiliar questions and repair the specific mistake than recognise the answers to twenty familiar ones.
1. A delivery business changes its channel mix
Question. A restaurant sells 1,000 identical orders at £20 each. Direct orders have £12 of variable cost; platform orders have £16. The mix moves from 60% direct to 80% direct, with total orders and prices unchanged. A new direct-marketing campaign costs £600. What is the net operating profit effect, assuming all other costs stay fixed?
Setup. Calculate contribution per order in each channel, value the orders that moved and deduct the incremental campaign cost.
Worked answer. Direct contribution is £8 and platform contribution is £4. Two hundred orders move to a channel contributing £4 more, adding £800. After £600 of campaign spending, operating profit improves £200.
You can verify the full totals: earlier contribution is 600 × £8 + 400 × £4 = £6,400; new contribution is 800 × £8 + 200 × £4 = £7,200. The £800 difference matches.
What to say. “The shift improves profit by £200 after marketing, provided sold volume is preserved. The campaign's £3 cost per migrated order is below the £4 contribution benefit.”
If you applied the £8 direct contribution to all 200 migrated orders, calculate the contribution difference between channels instead. Those orders already generated contribution through the old channel.
2. An automation project ramps up slowly
Question. Equipment costs £90,000 upfront. Gross cash savings are £20,000 in year one and £40,000 annually thereafter. Maintenance costs £5,000 every year. When does the investment recover its cash, assuming net benefits accrue evenly within each year?
Setup. Deduct recurring maintenance and accumulate the actual annual net benefits.
Worked answer. Net benefit is £15,000 in year one and £35,000 thereafter. Cumulative benefit reaches £50,000 after two years and £85,000 after three. Recovering the remaining £5,000 takes 5 ÷ 35 of another year. Simple payback is about 3.14 years.
What to say. “The project misses a three-year payback hurdle by £5,000 of unrecovered cash. A £5,000 reduction in upfront cost would make it meet that hurdle at the stated savings.”
Dividing £90,000 by the mature £35,000 would ignore the first-year ramp. Check when staffing or supplier payments can actually fall before treating projected savings as cash.
Quick Case Math Drill
A furniture maker installs a computer-controlled cutting cell. Operators need two years to master it, so savings build gradually before settling at a steady rate.
- Upfront cost
- £110,000
- Maintenance
- £4,000 every year
| Year | Gross saving |
|---|---|
| Year 1 | £14,000 |
| Year 2 | £34,000 |
| Year 3 onwards | £44,000 per year |
If net savings accrue evenly within each year, how many years until payback?
3. A model bill uses prices per million tokens
Question. A service makes 50,000 model calls monthly. Each uses 1,000 input tokens priced at £2 per million and 300 output tokens priced at £10 per million. Ignore caching and other charges. Ten per cent of initial calls need one identical retry. What is the monthly model bill?
Setup. Count total calls including retries, calculate input and output token volumes separately and convert each to millions before applying its price.
Worked answer. Total calls are 55,000. Input is 55 million tokens, costing £110. Output is 16.5 million tokens, costing £165. Total spending is £275 monthly.
What to say. “Retries take the model bill from £250 to £275. Output accounts for £165 of the total despite using fewer tokens, so output length is a useful cost lever to test against quality.”
The answer describes model spending, not the full delivered-service cost. Add caching, a spending limit and cost per accepted task before choosing a provider.
4. Extra hours do not all become completed appointments
Question. Three clinicians each add two scheduled hours weekly. Seventy-five per cent of that time is available for 20-minute appointments. All slots are booked and attendance is 80%. The clinic operates for 50 weeks. How many extra completed visits should it expect annually?
Setup. Convert scheduled hours to productive hours, then slots, then completions. Keep weekly and annual units separate.
Worked answer. Scheduled time is 3 × 2 = six hours weekly. Productive time is 4.5 hours, or 270 minutes. That creates 13.5 average slots at 20 minutes each. At 80% attendance, expected completions are 10.8 weekly, or 540 annually.
What to say. “The schedule adds about 540 completed visits annually under the booking and attendance assumptions. I would compare that with new demand and the existing backlog before claiming it solves the waiting-list problem.”
The unadjusted upper bound is 18 visits weekly; 10.8 is below it. Compare extra completed visits with new arrivals to calculate whether the queue shrinks.
5. A higher subscription price loses customers
Question. A cohort of 5,000 customers pays £20 monthly and costs £8 per active customer to serve. All would remain at the old price for the compared month. The proposed price is £22 with the same variable cost. What minimum retention preserves contribution? What happens if 4,500 customers stay?
Setup. Preserve existing total contribution, then divide by contribution per retained customer at the new price.
Worked answer. Existing contribution is 5,000 × £12 = £60,000. New unit contribution is £14, so required customers are £60,000 ÷ £14 = 4,285.71: at least 4,286 whole customers, roughly 85.7% retention.
At 4,500 customers, contribution is £63,000, up £3,000. Revenue is £99,000, down £1,000 from the original £100,000.
What to say. “At 90% retention, the increase improves contribution by £3,000 despite slightly lower revenue. I would test paid renewal and plan mix against the roughly 85.7% contribution-preservation threshold.”
Check downgrades and renewal timing before treating the retained revenue as a full-year gain.
Quick Answer Repair Drill
A vinyl record club plans to raise its monthly subscription. All 4,000 members would stay at today's price, and fixed costs will not change.
- Analyst's threshold
- At least 3,334 of 4,000 members must stay
| Plan | Monthly fee | Records and postage per member |
|---|---|---|
| Current | £25 | £15 |
| Proposed | £30 | £15 |
What is the main flaw in the analyst's retention threshold?
6. Two jobs compete for paid-for capacity
Question. One committed compute block can either earn £24,000 of external contribution or avoid £18,000 of internal cash spending. Each job needs the whole block. A second equivalent block is available for £12,000. All figures include other incremental costs. What should the company do?
Setup. Identify the best use of existing capacity, then compare the benefit unlocked by another block with its price.
Worked answer. External work is the best use of the existing block, beating internal savings by £6,000. Adding the second block enables both jobs: £24,000 + £18,000 − £12,000 = £30,000 of incremental benefit before the original common lease cost. That is £6,000 better than external work alone.
What to say. “Use the existing block externally and buy the second for internal work. The second block is worthwhile below £18,000, which is the internal spending it avoids.”
A common committed cost does not change the ranking. Allocate capacity to the higher contribution first, up to the demand available for that work.
Choose your next exercise from the mistake
If your equation was wrong, practise translating a fresh prompt into the requested quantity. If the equation was right but arithmetic failed, isolate that operation. If the number was right but your explanation stopped there, practise giving the business implication first.
Review setup, calculation and interpretation separately. “Bad at maths” does not tell you which exercise would help.
Record one correction in your practice log, then retest on a new problem. Repeating the same solution immediately can demonstrate memory without showing that the correction transfers.
Frequently asked questions
Channel mix, payback and token pricing
What is the net profit effect of shifting the delivery mix to 80% direct?
It is a gain of £200 on these figures. Direct orders contribute £8 and platform orders £4, so moving 200 orders to direct adds £800. Subtracting the £600 campaign leaves £200. Prices and order numbers are fixed, so revenue does not change and the whole effect comes from lower variable cost.
When does the automation equipment pay for itself?
Roughly 3.1 years after purchase. Net benefit is £15,000 in year one and £35,000 afterwards, so cumulative benefit is £50,000 after two years and £85,000 after three. The remaining £5,000 is a seventh of year four's benefit, assuming savings accrue evenly within each year.
Can I estimate payback by dividing the cost by the annual saving?
Not here. £90,000 divided by £40,000 gives 2.25 years, which ignores both the slower first year and the £5,000 annual maintenance. That shortcut understates payback by nearly a year. Simple division only works when net benefits are level from the start, so check for a ramp-up before choosing a method.
How do I work out a monthly model bill priced per million tokens?
Count the calls first, then price input and output separately. With retries there are 55,000 calls. Input is 55 million tokens at £2 per million, or £110. Output is 16.5 million tokens at £10 per million, or £165. The monthly bill is £275, before the caching and other charges the question tells you to ignore.
Do the retries really make a difference to the token bill?
Yes. Each retry is identical to the original call, so retries add a tenth to the bill. Without them, 50,000 calls cost £250. With 5,000 retries the total is £275. The business implication is that reducing failed calls directly reduces cost, so a sensible next fact to ask for is why calls need retrying.
Are the token prices in the model bill problem real market rates?
No. Every business and number in this set is illustrative, including the per-million-token prices. Use the problem to practise the method of separating input and output volumes and converting units. Before applying the calculation to a real decision, check the current prices with the actual provider.
Capacity, pricing and opportunity cost
How many extra clinic visits do the additional hours actually produce?
About 540 completed visits a year. Six scheduled hours weekly become 4.5 productive hours at 75%, which hold 13.5 twenty-minute slots. At 80% attendance that is 10.8 completed visits a week, and 50 operating weeks gives 540. Reporting scheduled hours or booked slots instead would overstate what the clinic gains.
What retention keeps contribution unchanged after the price rise?
The company needs at least 4,286 of its 5,000 customers, roughly 86%. Current contribution is 5,000 × £12, or £60,000. At £22, each retained customer contributes £14, and £60,000 divided by £14 is just over 4,285. You round up because you cannot retain part of a customer.
Is the £22 price worth it if 4,500 customers stay?
Yes, on contribution alone. 4,500 customers at £14 each produce £63,000, which is £3,000 more than today despite losing 500 customers. Because 4,500 sits comfortably above the break-even of about 4,286, the decision depends mainly on how reliable that retention estimate is. That is the next fact to test.
Should the company buy the second compute block?
Yes, on the stated figures. The existing block should take the external job, worth £24,000. A second block lets the internal job run as well, avoiding £18,000 of spending for a £12,000 price, which adds a further £6,000. Total benefit rises from £24,000 to £30,000, and all figures already include other incremental costs.
Why not run the internal job on the existing block?
Because it would give up more than it saves. Using the block internally avoids £18,000 of spending but forgoes £24,000 of external contribution, which leaves you £6,000 worse off than the best use. Opportunity cost means judging each use against the best alternative for the same capacity, not against zero.
Reviewing your attempts
Can I treat a correct number as a finished answer?
No. The set expects three things: the result, what it means and the next fact you need. If your number was right but your explanation stopped there, practise giving the business implication first. For example, say that the channel shift earns only £200 after the campaign, then show the working.
I keep getting problems wrong. Which exercise should I do next?
Choose it from where the attempt first failed. A wrong equation calls for practice translating fresh prompts into the requested quantity. A correct setup with faulty arithmetic calls for isolating that operation. Review setup, calculation and interpretation separately, and record one correction in your practice log rather than labelling yourself bad at maths.