6 Quantitative skills

Exam analysis

Most tested topics

Calculation

A percentage expresses a part relative to a whole. Percentage change compares the change with the original value, so the original—not the new value—is the denominator.

$percentage = part / whole × 100; percentage change = (new value - original value) / original value × 100$

Task	Direct route	Worked result
find 70% of 43,000 employees	43,000 × 0.70	30,100 employees
add 3% to £23,546	£23,546 × 1.03	£24,252.38
reduce 285 by 12%	285 × 0.88	250.80
change from 4.7m to 4.9m	(4.9 − 4.7) ÷ 4.7 × 100	about 4.26% increase

State whether the result is a percentage amount, a new total or a percentage change. Keep currency/people/volume units, show substitution and round only at the end to the requested precision.

A 15% discount means pay 85% of the original price. Percentage-point change differs from percentage change: a rate moving from 20% to 25% rises by 5 percentage points but by 25% relative to 20%.

An average summarises several observations as one representative value. In most calculation questions, the arithmetic mean is total of the values divided by the number of values; identify what is being averaged and keep its unit.

$mean = sum of all values / number of values$

If 9,000 vehicles are sold across 6 showrooms, average sales = 9,000 ÷ 6 = 1,500 vehicles per showroom. The denominator is showrooms, not vehicles or years.

Measure	Useful when	Limitation
mean	every value should contribute to a total-per-item result	extreme values can pull it away from a typical case
median	the middle value is needed and extremes are misleading	ignores the size of values away from the middle
mode	the most frequent category/value matters, such as popular size	may be absent or more than one mode

Use the average with spread, time period and sample context. Two shops can have the same mean sales but very different consistency, so an average alone may hide operational risk.

Do not average percentages or ratios blindly when their group sizes differ; combine the underlying totals or use an appropriate weighted mean. Always name the denominator and unit.

Revenue is income from sales; costs are resources used; profit or loss compares both for the same period and activity. Keep fixed and variable costs separate before forming total cost.

$revenue = selling price per unit × quantity sold; total variable cost = variable cost per unit × quantity; total cost = fixed cost + total variable cost; profit = revenue - total cost$

Given	Calculation	Result
fixed costs £1,150,000	—	£1,150,000
variable costs £4,250,000	—	£4,250,000
total cost	1,150,000 + 4,250,000	£5,400,000
revenue	—	£4,850,000
result	4,850,000 − 5,400,000	−£550,000: loss £550,000

A profit can rise because price or quantity raises revenue by more than added cost, or because cost falls without a larger revenue loss. Separate the arithmetic from the business explanation.

Cash inflow is not automatically revenue, and cash outflow is not automatically cost for the same period. Profit is an accounting result; it is not the closing cash balance.

A profit margin shows profit earned from each 100 of revenue, allowing businesses or periods of different size to be compared. Use the profit line named in the question and revenue from the same period.

$gross profit = revenue - cost of sales; gross profit margin = gross profit / revenue × 100; net profit margin = net profit / revenue × 100$

Statement data	Calculation	Result
revenue £200,000; cost of sales £120,000	gross profit = 200,000 − 120,000	£80,000
gross margin	80,000 ÷ 200,000 × 100	40%
net profit £30,000	30,000 ÷ 200,000 × 100	15%

The gross margin reflects pricing and direct/cost-of-sales control; the net margin also reflects further expenses included before net profit. Compare like definitions, periods and business models, then inspect the underlying figures before explaining a change.

Do not divide profit by cost: that produces a markup/return measure, not a profit margin. The specification's formula appendix labels operating profit margin separately; when a question asks net profit margin, use the exact net-profit line and formula supplied for that task.

Average rate of return (ARR) expresses average annual profit as a percentage of the initial investment. First calculate profit over the whole project, then convert it to an annual average.

$total profit = total returns - initial investment; average annual profit = total profit / project years; ARR = average annual profit / initial investment × 100$

Five-year project	Calculation	Result
initial investment	—	£100,000
total returns	—	£150,000
total profit	150,000 − 100,000	£50,000
average annual profit	50,000 ÷ 5	£10,000
ARR	10,000 ÷ 100,000 × 100	10%

A higher ARR may indicate a stronger accounting return, but compare projects using consistent assumptions and consider risk, cash timing, finance, strategic fit and the required return.

ARR uses average annual profit, not total return, total profit or one year's cash inflow. It does not show when cash arrives or discount future money, so it cannot decide an investment alone.

A cash-flow forecast tracks expected cash moving into and out of a business by period. Add every row once, preserve the sign and carry each closing balance forward as the next period's opening balance.

$total cash inflow = sum of inflows; total cash outflow = sum of outflows; net cash flow = total inflow - total outflow; closing balance = opening balance + net cash flow$

One month	Amount (£)
opening balance	20,000
total cash inflow	28,000
total cash outflow	56,000
net cash flow	28,000 − 56,000 = −28,000
closing balance	20,000 + (−28,000) = −8,000

Within the forecast, total revenue/cash sales and total costs are calculated by summing the relevant forecast rows or by the unit relationships provided. Classify each figure before adding it; do not count a subtotal again.

A negative net cash flow means outflows exceed inflows in that period; a negative closing balance means the accumulated cash position is below zero. Use forecast patterns to plan timing, finance or payments, while testing assumptions against actual results.

Net cash flow is not profit, and a negative monthly net flow does not automatically mean the closing balance is negative if the opening balance is large enough. Forecasts are estimates whose timing and values can differ from reality.

Interpretation

A graph or chart becomes decision evidence only after its labels, units, period, categories and scale are read correctly. Extract values first; then calculate or compare; only then explain and decide.

Step	Action	Landfill example
1 identify	title, source, time, unit, axes/legend and population	landfill waste; 2019–2021; metric tons
2 extract	read exact values without rounding early	100,595; 92,856; 82,617
3 compare	calculate absolute/percentage change or category difference	decrease = 100,595 − 82,617 = 17,978 tons
4 interpret	state direction, size and pattern—not just 'changed'	waste falls each year and by 17,978 tons overall
5 decide	connect the pattern to a named business choice	investigate which waste action worked before extending it
6 qualify	check scale, missing variables, source and whether pattern shows cause	three annual totals do not prove why waste fell

Use bars for category comparisons, lines for trends over time and pie charts for shares of one whole. Check whether axes start above zero, intervals are even, categories sum appropriately and values are totals, averages, percentages or indexes.

A visual pattern is not automatically causal or representative. A truncated axis can exaggerate a small difference, a rising total may reflect business growth, and two charts cannot be compared safely unless units, definitions and periods match.

Gross and net profit margins show profit per 100 of revenue. Interpretation requires more than saying 'higher is better': compare like definitions and trace the numerator and revenue to the business cause.

Pattern	Possible interpretation	Evidence to inspect before deciding
gross margin rises	prices/mix may have improved or cost of sales fell relative to revenue	selling prices, discounts, input costs, wastage and product mix
gross margin falls	cost of sales rose faster than revenue or prices weakened	supplier prices, labour/material efficiency and competition
net margin falls while gross margin is stable	expenses after gross profit have risen relative to revenue	wages, rent, promotion, finance and other operating costs
both margins rise	stronger pricing/cost control may be flowing through the statement	absolute sales/profit, one-offs, quality and sustainability

Compare several periods and similar businesses using consistent accounting treatment, currency and time span. State the size and direction, connect it to a plausible underlying figure, then identify the decision it informs—such as price, sourcing or expense control.

A high margin can coexist with low sales or cash problems, and a temporary fall may result from investment intended to support future performance. Ratios diagnose questions; they do not prove causes or replace the underlying accounts and context.

Financial data answers different questions: profit and loss measures accounting performance over a period, ARR estimates average accounting return on investment, and a cash-flow forecast tests whether cash is available when needed.

Evidence	What it can support	Key limitation/check
revenue, costs and profit/loss	pricing, cost control, product/branch performance	profit definition, period, volume and one-off items
ARR	compare average accounting return with alternatives/target	ignores timing of cash and depends on forecasts
net cash flow and closing balance	plan payment timing, short-term finance and liquidity	forecast assumptions may be wrong; negative monthly flow differs from negative balance
combined trend	test whether a profitable plan can also remain cash-solvent and earn enough return	use consistent scenarios and include risk/non-financial fit

Example: an expansion has an attractive ARR but produces negative closing balances during setup. The project may still be profitable overall, yet the business needs sufficient finance or a different timing plan before proceeding.

Use comparable periods/scenarios, calculate the change, explain the mechanism and weigh risk, finance, strategy and non-financial evidence. Test forecasts with worse/better assumptions and compare actual results once available.

Profit is not cash and ARR is not a cash-flow measure. One favourable number cannot establish affordability, return and risk at the same time; combine the measures relevant to the named decision.

Market-research data supports a decision when it represents the target customers, answers the business question and is interpreted without turning a sample pattern into certainty.

Data	Useful interpretation	Decision supported	Reliability check
quantitative counts, ratings, prices or percentages	size/frequency of a preference and differences between segments	price range, feature priority, demand/capacity estimate	sample size, denominator, question wording and representativeness
qualitative comments/reasons	why customers respond and what trade-offs matter	design, service and message modification	coding consistency, interviewer bias and minority views
averages/graphs/charts	central pattern, category comparison or trend	compare options and monitor a test	spread, axes, period, source and whether groups are comparable
social/online responses	rapid reactions and emerging themes	investigate campaign/product response	self-selection, bots/duplicates, platform algorithm and sentiment ambiguity

State evidence → inference → action → expected effect → uncertainty. For example, a representative survey shows a large price-sensitive segment → test a lower-priced version → monitor sales and margin before a full launch.

Popularity in a sample is not guaranteed market demand, correlation is not causation and a precise percentage can still come from a biased sample. Triangulate sources and distinguish what the data shows from the reason you infer.

Market data describes a business relative to customers and competitors. Market share shows its sales as a proportion of the defined market; cost and price changes help explain—but do not automatically prove—movements in demand, margin or position.

Pattern	What it may mean	Decision question
sales rise and market share rises	business grows faster than the market or competitors lose ground	which offer/channel/segment drove the gain, and is it profitable?
sales rise but share falls	total market grows faster than the business	should capacity/promotion change, or is the chosen niche still attractive?
cost rises while price stays fixed	unit margin may be squeezed	absorb, reduce cost, redesign or raise price?
price rises	revenue per unit rises but quantity may fall	how price-sensitive are target customers and how will competitors respond?
price falls	demand/share may rise but margin per unit falls	will extra contribution from volume exceed the lower unit margin?

Define the market, sales measure, place and period consistently. Separate absolute change from percentage change, and compare cost/price changes with quantity, revenue, margin and competitor evidence before attributing the result.

A decision should name the business objective and time horizon. Growing share may support scale and visibility, while sacrificing too much margin or targeting unprofitable sales can weaken cash and profit.

Market share is relative, not a direct measure of profit, loyalty or market size. Cost inflation does not mechanically require the same price increase, and a lower price does not guarantee greater revenue or share.

6 Quantitative skills

Exam analysis

Most tested topics

Calculation

Calculate a percentage and the direction of change

Choose an average that answers the business question

Build revenue, cost and profit from the same activity

Convert profit into a comparable margin

Calculate the annual return on an investment

Complete each cash-flow forecast period in order

Interpretation

Read a graph before using it for a decision

Interpret what changed behind a profit margin

Combine financial measures before choosing an action

Translate market-research data into a bounded decision

Interpret market share, cost and price changes together