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Finance numeracy

Financial Mathematics Guide

Learn the calculations behind commercial decisions, money over time, and financial data. Each lesson connects a question cue to one method, then tests it with a fresh worked example.

14 lessons 3 sections About 50 minutes

Tax and accounting examples teach assessment arithmetic, not legal or investment advice. Real rules depend on jurisdiction, reporting standards, and the facts of a transaction.

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Section 1

Commercial Mathematics

Turn prices, costs, taxes, and cash movements into comparable business measures.

Multiply for totals. Divide for per-unit figures.

Revenue and Unit Economics

Foundation

Revenue measures sales earned. Unit economics converts totals into per-unit amounts so products of different scale can be compared fairly.

In plain words

Multiply selling price by quantity for revenue. Divide total cost by quantity for unit cost. Subtract unit cost from selling price for unit profit.

Worked example

An EU training office sells 480 digital practice licences for €27 each. Platform access costs €3,360, and tutor support costs €5 per licence.

What are the revenue, total unit cost, and profit per licence?

  1. 1 Revenue: 480 x €27 = €12,960.
  2. 2 Tutor support: 480 x €5 = €2,400. Total cost: €3,360 + €2,400 = €5,760.
  3. 3 Unit cost: €5,760 / 480 = €12. Unit profit: €27 - €12 = €15.

Revenue is €12,960, unit cost is €12, and profit per licence is €15.

Trap to avoid

Dividing only the €3,360 platform cost by 480 ignores the support cost and overstates unit profit.

Common mistakes

  • Multiplying a total by quantity when it is already a total
  • Mixing units, such as monthly cost with annual sales
  • Ignoring a variable cost when calculating unit cost

Mathematical foundations

Multiplication Division Scaling
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Markup starts from cost. Margin ends at selling price.

Markup, Markdown, and Margin

Applied

Markup is measured from cost, markdown from the current selling price, and profit margin from the final selling price. Their denominators are different.

In plain words

Apply each percentage to the amount that exists at that step. A markup and a later discount are successive multipliers, not percentages that can simply be added or subtracted.

Worked example

A language-test headset costs a supplier €80. It is marked up by 25%, then discounted by 10% during a campaign.

What is the final price and the profit margin?

  1. 1 Marked price: €80 x 1.25 = €100.
  2. 2 Discounted price: €100 x 0.90 = €90.
  3. 3 Profit: €90 - €80 = €10. Margin: €10 / €90 x 100 = 11.1%.

The final price is €90 and the profit margin is 11.1%.

Trap to avoid

Treating 25% markup minus 10% markdown as a single 15% increase gives €92. The two rates act on different bases.

Common mistakes

  • Using selling price as the denominator for markup
  • Using cost as the denominator for profit margin
  • Adding successive percentage changes instead of multiplying factors

Mathematical foundations

Percentages Multiplication Comparison
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Contribution pays fixed costs first, then creates profit.

Costs, Profit, and Break-even

Applied

Fixed costs do not change with output within the relevant range. Variable costs change with output. Break-even occurs when contribution covers fixed costs exactly.

In plain words

Each sale contributes selling price minus variable cost. Divide fixed costs by that contribution to find the break-even quantity.

Worked example

A certification workshop has fixed venue and production costs of €7,200. Each seat sells for €95 and creates €35 of variable cost.

How many seats break even, and what is the profit at 150 seats?

  1. 1 Contribution per seat: €95 - €35 = €60.
  2. 2 Break-even seats: €7,200 / €60 = 120 seats.
  3. 3 Profit at 150 seats: 150 x €60 - €7,200 = €1,800.

Break-even is 120 seats. Profit at 150 seats is €1,800.

Trap to avoid

Dividing fixed costs by the €95 selling price gives 75.8 seats but ignores the €35 variable cost attached to every sale.

Common mistakes

  • Dividing fixed costs by price instead of contribution
  • Adding variable cost only once instead of once per unit
  • Rounding break-even quantity down when whole units are required

Mathematical foundations

Multiplication Division Comparison
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Revenue falls through cost layers to net profit.

The Profit Ladder

Applied

Gross profit, operating profit, profit before tax, and net profit sit at different levels. Each level removes a defined layer of cost.

In plain words

Start with revenue. Remove direct production or purchase costs for gross profit. Remove operating expenses for operating profit. Then account for other income, interest, and tax to reach net profit.

Worked example

A small assessment publisher reports revenue of €240,000, cost of goods sold of €138,000, operating expenses of €61,000, interest expense of €3,000, and tax of €9,500.

Find gross profit, operating profit, profit before tax, and net profit.

  1. 1 Gross profit: €240,000 - €138,000 = €102,000.
  2. 2 Operating profit: €102,000 - €61,000 = €41,000.
  3. 3 Profit before tax: €41,000 - €3,000 = €38,000.
  4. 4 Net profit: €38,000 - €9,500 = €28,500.

Gross profit is €102,000, operating profit €41,000, profit before tax €38,000, and net profit €28,500.

Trap to avoid

Calling all pre-tax profit gross profit hides cost of goods sold and operating expenses, which belong to different levels.

Common mistakes

  • Treating revenue as profit
  • Subtracting operating expenses twice
  • Calling profit before tax net profit

Mathematical foundations

Summation Parts And Wholes Comparison
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Add tax to net. Divide tax out of gross.

Tax-inclusive and Tax-exclusive Prices

Applied

A tax-exclusive price is the base before tax. A tax-inclusive price already contains tax, so the base must be divided out. VAT is charged on taxable value, not simply on accounting profit.

In plain words

Multiply a net price by one plus the rate to add tax. If tax is already included, divide the gross price by one plus the rate. The difference is the tax amount.

Worked example

An invoice totals €2,904 including VAT at 21%.

What are the tax-exclusive price and the VAT amount?

  1. 1 Convert 21% to 0.21, so the gross factor is 1.21.
  2. 2 Net price: €2,904 / 1.21 = €2,400.
  3. 3 VAT: €2,904 - €2,400 = €504.

The tax-exclusive price is €2,400 and the VAT amount is €504.

Trap to avoid

Taking 21% of the €2,904 gross total gives €609.84. The 21% rate applies to the net base, not the already taxed total.

Common mistakes

  • Subtracting the tax percentage directly from a gross price
  • Applying the tax rate to accounting profit
  • Confusing tax collected with revenue retained by the seller

Mathematical foundations

Percentages Fractions Division
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Same profit, different denominator.

Profit Margin versus ROI

Applied

Profit margin measures profit against revenue. Return on investment measures net benefit against the capital committed to the investment. The same profit can produce different percentages.

In plain words

Use revenue below profit for margin. Use the defined investment base below net benefit for ROI. State the period and investment base when comparing projects.

Worked example

A new online course earns €90,000 of revenue and €18,000 of profit in its first year. The company committed €40,000 of launch capital.

What are the first-year profit margin and ROI?

  1. 1 Profit margin: €18,000 / €90,000 x 100 = 20%.
  2. 2 ROI: €18,000 / €40,000 x 100 = 45%.
  3. 3 The figures differ because revenue and invested capital are different bases.

Profit margin is 20%. First-year ROI is 45%.

Trap to avoid

Dividing profit by operating expenses and labelling the result ROI uses an undefined denominator unless those expenses are explicitly the investment base.

Common mistakes

  • Using revenue as the ROI denominator
  • Using investment cost as the margin denominator
  • Comparing ROI values calculated over different periods

Mathematical foundations

Fractions Percentages Comparison
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Count cash when it moves.

Cash Flow

Foundation

Cash flow records money when it enters or leaves an account. It differs from accounting profit, which can include unpaid invoices and non-cash expenses.

In plain words

Add all cash received. Subtract all cash paid. Add the net movement to opening cash to obtain closing cash.

Worked example

A training provider opens the month with €18,000. It receives €74,000 from customers, pays €62,000 to staff and suppliers, and buys equipment for €14,000 in cash.

What are net cash flow and closing cash?

  1. 1 Total inflows: €74,000.
  2. 2 Total outflows: €62,000 + €14,000 = €76,000.
  3. 3 Net cash flow: €74,000 - €76,000 = -€2,000. Closing cash: €18,000 - €2,000 = €16,000.

Net cash flow is -€2,000 and closing cash is €16,000.

Trap to avoid

Ignoring the equipment purchase because it is not a normal operating expense would overstate cash by €14,000.

Common mistakes

  • Treating an unpaid customer invoice as a cash inflow
  • Ignoring investing or financing cash movements
  • Assuming positive profit guarantees positive cash flow

Mathematical foundations

Summation Comparison Parts And Wholes
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Section 2

Money Over Time

Measure purchasing power, borrowing cost, and repeated change across time periods.

Deflate nominal money with the CPI ratio.

Inflation, CPI, and Real Value

Advanced

A price index tracks the cost of a defined basket. Its percentage change estimates inflation. Dividing a nominal value by the index change expresses it in base-period purchasing power.

In plain words

First calculate how much the index changed. To remove inflation, multiply the later nominal amount by the old index divided by the new index. Real growth compares purchasing power, not printed euro amounts.

Worked example

A salary rises from €40,000 to €42,840 while the CPI rises from 120 to 126.

What are inflation, the later salary in start-period euros, and real salary growth?

  1. 1 Inflation: 126 / 120 - 1 = 0.05 = 5%.
  2. 2 Later salary in start-period euros: €42,840 x 120 / 126 = €40,800.
  3. 3 Real salary growth: €40,800 / €40,000 - 1 = 2%.

Inflation is 5%. The later salary is worth €40,800 in start-period euros, a real increase of 2%.

Trap to avoid

Subtracting 5% inflation from the 7.1% nominal raise gives 2.1%. Exact real growth divides the two growth factors and equals 2%.

Common mistakes

  • Subtracting rates when an exact factor calculation is required
  • Multiplying by CPI_end / CPI_start when deflating
  • Comparing nominal values from different years without adjustment

Mathematical foundations

Percentages Division Comparison Time Value
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Interest stays on the original principal.

Simple Interest

Foundation

Simple interest is calculated only on the original principal. The time unit must match the interest-rate period.

In plain words

Multiply principal by the annual rate and the fraction of a year. Add the interest to principal for the final amount.

Worked example

An organisation borrows €18,000 for eight months at 4.5% simple annual interest.

How much interest is due, and what is the repayment amount?

  1. 1 Convert eight months to years: 8 / 12 = 2 / 3.
  2. 2 Interest: €18,000 x 0.045 x 2 / 3 = €540.
  3. 3 Repayment: €18,000 + €540 = €18,540.

Interest is €540 and the repayment amount is €18,540.

Trap to avoid

Using t = 8 with an annual rate treats eight months as eight years.

Common mistakes

  • Failing to convert months into a fraction of a year
  • Using the final amount instead of principal as the interest base
  • Compounding when the question explicitly states simple interest
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Each rate changes the base for the next rate.

Compound Interest and Repeated Growth

Advanced

Compounding makes every period start from the previous period result. Constant rates use an exponent. Changing rates use a product of period factors.

In plain words

Convert each percentage change to a multiplier. Multiply the starting value by every period multiplier in sequence. Use a factor below one for a decrease.

Worked example

A programme reserve starts at €25,000. It grows by 3% in year one, 5% in year two, then falls by 2% in year three.

What is the reserve after three years?

  1. 1 Convert rates to factors: 1.03, 1.05, and 0.98.
  2. 2 Apply them in sequence: €25,000 x 1.03 x 1.05 x 0.98.
  3. 3 The result is €26,496.75.

The reserve after three years is €26,496.75.

Trap to avoid

Adding 3% + 5% - 2% and applying 6% once gives €26,500. It is close, but it ignores changing bases.

Common mistakes

  • Adding repeated percentage changes
  • Using the number of year labels instead of elapsed intervals
  • Writing a decrease as 1.02 instead of 0.98
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Section 3

Markets and the Economy

Interpret market shares, company securities, trading data, and macroeconomic comparisons.

Company part over market whole.

Market Share

Foundation

Market share is one company measure divided by the matching total market measure. Both values must cover the same product, geography, and period.

In plain words

Divide company sales by total market sales for market share. Multiply the market total by a target share to find required company sales.

Worked example

A digital-learning market is worth €24.6 million. One provider records €3.69 million of sales. Next year the market is forecast at €27 million.

What is the current share, and what sales would produce an 18% share next year?

  1. 1 Current share: €3.69m / €24.6m x 100 = 15%.
  2. 2 Convert the target to a decimal: 18% = 0.18.
  3. 3 Required sales: 0.18 x €27m = €4.86m.

Current market share is 15%. Sales of €4.86 million would produce an 18% share next year.

Trap to avoid

Applying 18% to the old €24.6 million market ignores the forecast change in the denominator.

Common mistakes

  • Dividing total market sales by company sales
  • Mixing units sold with market revenue
  • Comparing different periods or geographical markets

Mathematical foundations

Parts And Wholes Percentages Proportions
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Ownership uses current shares. EPS uses weighted-average shares.

Shares, Dividends, and EPS

Advanced

Ownership uses shares held over shares outstanding. Dividends distribute declared amounts to eligible shares. Basic EPS uses income available to common shareholders and weighted-average common shares.

In plain words

Divide holdings by outstanding shares for ownership. Divide common dividends by eligible shares for dividend per share. For basic EPS, subtract preferred dividends from net income and divide by weighted-average common shares.

Worked example

A company has 2.4 million common shares at the dividend record date. An investor owns 36,000. The company declares €600,000 of common dividends. Net income is €1.92 million, preferred dividends are €120,000, and weighted-average common shares are 1.8 million.

Find the ownership percentage, investor dividend, and basic EPS.

  1. 1 Ownership: 36,000 / 2,400,000 x 100 = 1.5%.
  2. 2 Dividend per share: €600,000 / 2,400,000 = €0.25. Investor dividend: 36,000 x €0.25 = €9,000.
  3. 3 Basic EPS: (€1,920,000 - €120,000) / 1,800,000 = €1.00.

Ownership is 1.5%, the investor dividend is €9,000, and basic EPS is €1.00.

Trap to avoid

Using the 2.4 million closing shares for EPS ignores the weighted-average denominator supplied for the reporting period.

Common mistakes

  • Using authorised shares instead of outstanding shares
  • Ignoring preferred dividends in basic EPS
  • Using closing shares instead of weighted-average shares for EPS

Mathematical foundations

Parts And Wholes Percentages Division Averages
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High minus low. Close minus open.

OHLC and Price Return

Applied

OHLC data records the opening, highest, lowest, and closing prices for one period. Open and close must lie between low and high. Price return excludes dividends unless stated otherwise.

In plain words

Subtract low from high for the trading range. Subtract open from close and divide by open for the open-to-close price return.

Worked example

A share opens at €45.20, reaches a high of €47.10, falls to a low of €44.80, and closes at €46.33.

What are the trading range and open-to-close return?

  1. 1 Trading range: €47.10 - €44.80 = €2.30.
  2. 2 Price change: €46.33 - €45.20 = €1.13.
  3. 3 Return: €1.13 / €45.20 x 100 = 2.5%.

The trading range is €2.30 and the open-to-close return is 2.5%.

Trap to avoid

Using the high of €47.10 as the final price calculates a movement that was not retained at the close.

Common mistakes

  • Treating the high as the closing price
  • Dividing the price change by the closing price
  • Accepting OHLC data where open or close lies outside the low-high interval

Mathematical foundations

Comparison Percentages Division
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Absolute change and percentage growth answer different questions.

GDP and Index Comparisons

Applied

GDP measures the market value of final goods and services produced within an economy during a period. Use real GDP for volume growth. A decline signals contraction, but does not by itself settle every formal definition of recession.

In plain words

Subtract values for absolute change. Divide that change by the starting value for growth. An index sets a base period to 100, so index changes are interpreted with the same percentage-change formula.

Worked example

Region A real GDP rises from €260 billion to €273 billion. Region B real GDP rises from €90 billion to €97.2 billion.

Which region adds more output, and which grows faster?

  1. 1 Region A absolute increase: €273bn - €260bn = €13bn. Growth: €13bn / €260bn x 100 = 5%.
  2. 2 Region B absolute increase: €97.2bn - €90bn = €7.2bn. Growth: €7.2bn / €90bn x 100 = 8%.
  3. 3 Compare absolute increases separately from percentage rates.

Region A adds more output (€13 billion), but Region B grows faster (8% versus 5%).

Trap to avoid

Choosing Region A as faster because its euro increase is larger confuses absolute scale with percentage growth.

Common mistakes

  • Dividing change by the final value
  • Confusing nominal GDP growth with real GDP growth
  • Ranking percentage growth by absolute change

Mathematical foundations

Percentages Comparison Division Scaling
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Turn method into speed

Apply the formulas under exam conditions

Start with untimed calculation, then repeat with tables and charts until you can identify the denominator, rate, and time period without hesitation.

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