Interest, break-even, depreciation, margins, growth (CAGR), yield, index numbers, real-terms.
Compute the margin for each period (profit ÷ revenue), then the percentage change between the margins — not the percentage-point gap.
A firm's revenue and costs were £800m and £600m in 2023, and £900m and £630m in 2024. By what percentage did its profit margin change?
Answer: 20.0%
Margins: 200/800 = 25% and 270/900 = 30%. Change = (30−25)/25 = 20%. The 5.0% foil gives the percentage-POINT gap; 35% the change in profit; 12.5% the revenue change.
Simple interest = principal × rate × years — the interest doesn't compound.
£2,000 is invested at 4% simple interest per year for 3 years. How much interest is earned?
Answer: £240
Simple interest = principal × rate × time = 2,000 × 0.04 × 3 = £240. The £249.73 foil compounds; £80 forgets the three years; £2,240 gives the total balance; £480 doubles the period.
Compound interest: final value = principal × (1 + rate)ⁿ for n years.
£5,000 is invested at 5% compound interest per year for 3 years. What is it worth at the end? (to the nearest £)
Answer: £5,788
Compound: 5,000 × 1.05³ = £5,788. The £5,750 foil uses simple interest; £5,513 and £6,078 use the wrong number of years; £788 gives the interest, not the final value.
Contribution per unit = price − variable cost. Break-even units = fixed costs ÷ contribution.
A product sells for £25, has a variable cost of £15 per unit, and the company has fixed costs of £8,000. How many units must be sold to break even?
Answer: 800
Contribution per unit = 25 − 15 = £10. Break-even = fixed costs ÷ contribution = 8,000 ÷ 10 = 800 units. The 320 foil divides by price; 533 by variable cost; 200 by price+variable; 400 uses a wrong contribution.
CAGR = (ending ÷ beginning)^(1/years) − 1 — annualised, not the simple average of total growth.
Carraway plc's revenue grew from £320m in 2021 to £625m in 2024. What was the compound annual growth rate (CAGR) over this period?
Answer: 25.0%
CAGR = (ending/beginning)^(1/years) − 1 = (625/320)^(1/3) − 1 = 1.953125^(1/3) − 1 = 1.25 − 1 = 25.0%. The 31.8% foil takes the simple average (total growth ÷ 3); 95.3% gives total growth un-annualised; 18.0% and 39.8% use the wrong number of periods (4 and 2 instead of 3).
Derive the budget first, then variance % = (actual − budget) ÷ budget × 100.
A department's budget was set at 1,500 units at a standard cost of £30 per unit. Actual spend was £52,200. By what percentage did actual spend exceed the budget?
Answer: 16.0%
First derive the budget: 1,500 × £30 = £45,000. Variance = (52,200 − 45,000)/45,000 = 7,200/45,000 = 16.0%. The 13.8% foil divides by actual instead of budget; 116.0% reports actual as a % of budget (the level, not the variance); 86.2% inverts it; 4.8% divides the £7,200 overspend by the 1,500 units.
Income = shares held × price per share × yield. Watch pence vs pounds.
You hold 297 shares of Optimum, bought at 368p each, with an annual dividend yield of 3.68%. What is your total dividend income over one year (dividends not reinvested)?
Answer: £40.22
Dividend per share = 368p × 3.68% = 13.54p. Total = 297 × 13.54p = 4,022p = £40.22. The £1,092.96 foil gives the value of the holding, not the dividend; £4,022.09 leaves the answer in pence.
Deflate to base-year prices first (÷ the price index), then compute the percentage change on the real values.
Using the price index to adjust for inflation, by what percentage did Delphine Foods' revenue change in REAL terms from 2020 to 2024?
Answer: 4.0%
Deflate 2024 revenue to 2020 prices: 260 × (100/125) = £208m. Real change = (208 − 200)/200 = 4.0%. The 30.0% foil ignores inflation (nominal); 5.0% wrongly subtracts the 25% inflation from the 30% nominal growth (additive); 24.0% deflates the growth instead of the level; 62.5% multiplies by the index instead of dividing.
An index of i means i% of the base: value = base value × index ÷ 100.
With 2020 as the base year (index = 100), a materials cost index reached 128 by 2024. A material cost £45 per tonne in 2020. Tracking the index, what would it cost in 2024?
Answer: 57.60
Indexed value = base value × index/100 = 45 × 128/100 = £57.60. The £73 foil adds the 28 index points to the price; £5,760 forgets to divide by 100; £35.20 inverts the index; £12.60 gives only the 28% increase.
Compound growth: value × (1 + rate)ⁿ for n years ahead.
A company's revenue is £400m and grows 6% per year. To the nearest £m, what will it be in 3 years?
Answer: 476
Compound: 400 × 1.06³ = 400 × 1.19102 = £476m. The 472 foil applies a simple 18% (6%×3); 449 and 505 use the wrong number of years; 424 grows for one year only.
Linear growth: add the constant yearly amount × the number of years.
A company's revenue rose by a constant £45m each year, reaching £520m in 2024. At this linear rate, what will revenue be in 2027?
Answer: 655
Linear: add £45m for each of the three years to 2027: 520 + 45×3 = £655m. The 610 and 700 foils use the wrong number of years; 565 adds once; 385 subtracts.
Reducing balance: value × (1 − rate)ⁿ — the depreciation amount shrinks each year.
A machine worth £24,000 depreciates 15% per year on a reducing-balance basis. What is its value after 2 years? (to the nearest £)
Answer: £17,340
Reducing balance: 24,000 × 0.85² = £17,340. The £16,800 foil uses straight-line (30% flat); £20,400 depreciates one year; £14,739 uses three years; £31,734 grows instead of shrinking.
Margin = profit ÷ selling price. Dividing by cost gives markup — a different number.
A product costs £45 and sells for £60. What is the profit margin?
Answer: 25.0%
Margin = profit ÷ selling price = (60−45)/60 = 25.0%. The 33.3% foil divides by cost (that's markup, not margin); 75.0% is cost as a share of price; 15.0% quotes the £15 profit; 20.0% uses a wrong base.
Tax each band separately at its own rate and add the parts — the top rate applies only to income inside the top band.
Income tax is charged at 0% on the first £12,000, 20% on income between £12,000 and £50,000, and 40% above £50,000. How much tax is due on an income of £60,000?
Answer: £11,600
Tax the bands separately: 20% on (50,000−12,000)=£7,600, plus 40% on (60,000−50,000)=£4,000, total £11,600. The £24,000 foil applies the top rate to everything; £19,200 applies 40% above the allowance; the others apply a single flat rate.
Derive the full variable cost per unit first; contribution = price − variable cost; break-even = fixed ÷ contribution.
A product sells for £40. Materials cost £18 per unit and labour £6 per unit. Fixed costs are £8,000. How many units must be sold to break even?
Answer: 500
Contribution = 40 − (18 + 6) = £16 per unit. Break-even = 8,000 ÷ 16 = 500 units. The 333 foil forgets the labour cost; 200 divides by price; 275 by total variable cost as the divisor mishandled.
Track the money in stages: total cost, then revenue from each selling leg, then profit = revenue − cost.
A retailer buys 200 lamps at £18 each. It sells 60% of them at a 50% markup on cost, and the remaining lamps at a 25% discount on cost. What was the retailer's total profit?
Answer: £720
Cost = 200 × £18 = £3,600. Markup leg: 120 lamps × (18×1.5 = £27) = £3,240. Discount leg: 80 lamps × (18×0.75 = £13.50) = £1,080. Total revenue = £4,320, so profit = 4,320 − 3,600 = £720. (Equivalently: +£1,080 profit on the markup leg, −£360 loss on the discount leg.) The £4,320 foil reports revenue; £1,800 assumes all at markup; £1,080 counts only the markup-leg profit; £1,440 treats the discount leg as a gain.
Revenue = quantity × yield × price; cost = quantity × unit cost; profit is the difference.
One tonne of maize ferments into 380 litres of fuel ethanol, which sells for £1.21 per litre. Maize costs £200.23 per tonne. What profit is generated by buying 47 tonnes, fermenting it, and selling the ethanol (no other costs)?
Answer: £12,199.79
Revenue = 47 × 380 × £1.21 = £21,610.60. Cost = 47 × £200.23 = £9,410.81. Profit = £12,199.79. The £21,610.60 foil forgets the maize cost; £259.57 forgets to scale by 47 tonnes.
Derive the hidden rate from the pair given (part ÷ whole), then apply or reverse it as asked.
Commission is a flat percentage of sales, the same for every consultant. One consultant earned £28,057 in commission on £140,287 of sales. What sales total is required to earn £60,000 in commission?
Answer: £300,000
First derive the rate: 28,057 ÷ 140,287 = 20%. Then reverse: £60,000 ÷ 0.20 = £300,000. The £12,000 foil multiplies instead of reversing; £60,000 just echoes the target.
Index − 100 = % growth from the base. Compare the growth figures, and mind points vs percent.
Sales are indexed to 2020 = 100. By 2024, Region A's index is 140 and Region B's is 125. Which region grew faster, and by how much more?
Answer: Region A, by 15 percentage points
An index of 140 means 40% growth from the 2020 base; 125 means 25%. Region A grew faster, by 40 − 25 = 15 percentage points. The relative-framing and index-ratio foils mis-express the gap.
Compute each margin as profit ÷ selling price, then compare the margins — not the raw profits.
Product A sells for £500 at a cost of £350. Product B sells for £800 at a cost of £600. What is the higher of the two profit margins?
Answer: 30.0%
Margin = profit/price. A: 150/500 = 30%. B: 200/800 = 25%. The higher is A at 30%. The 42.9%/33.3% foils divide by cost (markup); B has more profit in £ but a lower margin.
Compute the per-unit rate for each period first, then the percentage change between the rates.
A retailer's sales were £4.8m across 8 stores in 2022, and £7.5m across 12 stores in 2024. By what percentage did sales PER STORE change?
Answer: 4.2%
Sales per store: 4.8m/8 = £600k and 7.5m/12 = £625k. Change = (625−600)/600 = 4.2%. The 56.3% foil uses total sales (ignoring that store count grew); 50% is the change in store count.
Real growth = (1 + nominal) ÷ (1 + inflation) − 1 — divide the factors, don't subtract the rates.
An investment grew at a nominal CAGR of 12% per year while inflation ran at 5% per year. What was the real (inflation-adjusted) CAGR?
Answer: 6.7%
Real CAGR = (1 + nominal)/(1 + inflation) − 1 = 1.12/1.05 − 1 = 6.7%. The 7.0% foil simply subtracts inflation (the classic shortcut); 12% ignores it.
Cost = quantity × buy price × (1 + commission); proceeds = quantity × sell price × (1 − commission); profit is the difference.
You buy 3.7 tonnes of aluminium at £884.37/tonne and later sell it at £1,525.59/tonne, through a broker charging 2% commission on each transaction. What profit is made?
Answer: £2,194.18
Buy cost incl. 2% = 3.7×884.37×1.02 = £3,337.61. Sale proceeds after 2% = 3.7×1525.59×0.98 = £5,531.79. Profit = 5,531.79 − 3,337.61 = £2,194.18. The £2,372.51 foil ignores commission; £641.22 forgets the 3.7 tonnes.