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Case maths cheat sheet
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1Fractions to percentages
| Halves: 1/250% | |||
| Thirds: 1/333.3% | 2/366.7% | ||
| Quarters: 1/425% | 3/475% | ||
| Fifths: 1/520% | 2/540% | 3/560% | 4/580% |
| Sixths: 1/616.7% | 5/683.3% | ||
| Sevenths: 1/714.3% | 2/728.6% | 3/742.9% | 4/757.1% |
| Eighths: 1/812.5% | 3/837.5% | 5/862.5% | 7/887.5% |
| Ninths: 1/911.1% | 2/922.2% | 4/944.4% | 5/955.6% |
| Tenths: 1/1010% | 3/1030% | 7/1070% | 9/1090% |
| Elevenths: 1/119.1% | 2/1118.2% | 3/1127.3% | |
| Twelfths: 1/128.3% | 5/1241.7% | 7/1258.3% | 11/1291.7% |
| Others: 1/156.7% | 1/166.25% | 1/205% | 1/254% |
- Dividing by a percentage is multiplying by its fraction flipped: ÷ 12.5% = × 8, ÷ 20% = × 5, ÷ 25% = × 4.
- x% of y = y% of x: 16% of 25 is 25% of 16, which is 4.
- Falls need bigger rises to recover: -20% needs +25% (1 ÷ 0.8 = 1.25), -50% needs +100%.
2Mental maths shortcuts
Zeros and units. Strip the zeros, do the core sum, put the zeros back. Thousand × thousand = million; thousand × million = billion.
e.g. 3,000 × 450,000: 3 × 45 = 135, plus 3 + 4 = 7 zeros = 1,350,000,000 = 1.35bn
Zeros when dividing. Divide the digits, then subtract the zeros below from the zeros above.
e.g. £2.4bn ÷ 80,000: 24 ÷ 8 = 3, and 8 zeros - 4 zeros = 4 zeros, so £30,000
Round and correct. Round to a friendly number, then fix the difference. Rounding both? Go in opposite directions.
e.g. 98 × 37 = 100 × 37 - 2 × 37 = 3,700 - 74 = 3,626. Estimate: 47 × 62 ≈ 50 × 60 = 3,000 (exact 2,914)
Halving and doubling. Halve one number and double the other: the product stays the same. Double both sides of a division.
e.g. 16 × 35 = 8 × 70 = 560. 4.5 × 18 = 9 × 9 = 81. 270 ÷ 15 = 540 ÷ 30 = 18
× 5, 25 and 125. × 5 = × 10 ÷ 2. × 25 = × 100 ÷ 4. × 125 = × 1,000 ÷ 8.
e.g. 25 × 368 = 36,800 ÷ 4 = 9,200. 125 × 48 = 48,000 ÷ 8 = 6,000
Percentages from 10% and 1%. 10% moves the decimal point one place, 1% two places. Build any percentage from those.
e.g. 17% of 340 = 10% (34) + 5% (17) + 2% (6.8) = 57.8
Factor the divisor. Split an awkward divisor into small factors and divide by each in turn.
e.g. 1,872 ÷ 24 = 1,872 ÷ 4 ÷ 6 = 468 ÷ 6 = 78
Changes multiply. Two percentage changes multiply. Small ones roughly add.
e.g. Price +10%, volume -10%: 1.1 × 0.9 = 0.99, so revenue falls 1%. +3% then +2% ≈ +5% (exact 5.06%)
3Core business formulas
Profit and margin
Profit = Revenue - Costs = (Price - Variable cost per unit) × Volume - Fixed costs
Revenue = Volume × Price, where Volume = Market size × Market share
Margin = Profit ÷ Revenue
Markup = Profit ÷ Cost: a 50% markup is a 33% margin
Contribution and breakeven
Contribution per unit = Price - Variable cost per unit
Contribution margin = Contribution per unit ÷ Price
Breakeven volume = Fixed costs ÷ Contribution per unit
£6m ÷ (£40 - £25) = 400,000 units
Breakeven revenue = Fixed costs ÷ Contribution margin
Market and mix
Market share = Our sales ÷ Market sales
Same units, same year. Relative share = our share ÷ largest rival's
Mix (weighted average) = Σ (share of volume × margin or price)
60% at 40% + 40% at 20% = 32%; at 50/50 it falls to 30%
Growth
Growth rate = New ÷ Old - 1
10% to 12% is +2 percentage points, or +20%
CAGR = (End ÷ Start)1 ÷ years - 1
Estimate it from the compounding table below
Years to double ≈ 72 ÷ growth rate in %
The rule of 72: at 8% a year, about 9 years
Investment
Payback = Investment ÷ Annual cash flow
If cash flow ramps up, add it year by year until it covers the investment
ROI = (Total return - Investment) ÷ Investment
Always over a stated period
NPV = -Investment + CF1 ÷ (1 + r) + CF2 ÷ (1 + r)2 + ...
Positive NPV: worth more than it costs at rate r
Perpetuity = Cash flow ÷ r. Growing: Next year's cash flow ÷ (r - g)
£10m at 10% = £100m. Growing 2%: £10m ÷ 8% = £125m. Needs r > g
Pricing
Volume rise to offset a price cut = Price cut ÷ New unit margin
£100 price, £60 cost, 10% cut: £10 ÷ £30 = +33% volume
4Compounding and discounting
| Growth a year | 3 yrs | 5 yrs | 7 yrs | 10 yrs |
|---|---|---|---|---|
| 5% | 1.16 | 1.28 | 1.41 | 1.63 |
| 10% | 1.33 | 1.61 | 1.95 | 2.59 |
| 15% | 1.52 | 2.01 | 2.66 | 4.05 |
| 20% | 1.73 | 2.49 | 3.58 | 6.19 |
| Rate | Yr 1 | Yr 2 | Yr 3 | Yr 4 | Yr 5 | Sum |
|---|---|---|---|---|---|---|
| 5% | 0.95 | 0.91 | 0.86 | 0.82 | 0.78 | 4.33 |
| 10% | 0.91 | 0.83 | 0.75 | 0.68 | 0.62 | 3.79 |
Read it backwards for CAGR: £200m to £320m in 5 years is × 1.6, about 10% a year (exact 9.9%). £1m a year for 5 years at 10% is worth about £3.8m today.
5Reference figuresRough, round numbers for sense checks
- World
- about 8bn people
- China, India
- about 1.4bn each
- European Union
- about 450m
- US
- about 340m people, 130m households
- UK
- about 69m people, 28m households
- Household
- about 2.4 (UK) to 2.5 (US) people
- Any one age
- roughly population ÷ 80
- A year
- 365 days, 52 weeks, 8,760 hours
- Working year
- about 250 days, 2,000 hours
- A week, a day
- 168 hours; 1,440 minutes
- £1m a year
- about £20k a week, £2,700 a day
- Scale
- 1bn = 1,000m; 1tn = 1,000bn