Percentages, Ratios and Growth Rates: The Maths Behind EPSO Numerical Reasoning

1. Percentage change

Formula: (new value − old value) ÷ old value × 100.

Example: a programme's budget rises from €240 million to €300 million. The change is 60 ÷ 240 = 0.25, so the budget has increased by 25%.

Common mistake: dividing by the new value instead of the old one. 60 ÷ 300 gives 20%, which is a typical distractor among the answer options. Always divide by the starting point of the comparison — usually the earlier year.

2. Percent versus percentage points

When the quantity itself is a percentage, there are two ways to describe a change. If an unemployment rate goes from 8% to 10%, it has risen by 2 percentage points, but by 25% in relative terms (2 ÷ 8).

Questions often ask for one and offer the other as a distractor. Read the question wording carefully: 'by how many percentage points' and 'by what percentage' have different answers.

3. Reverse percentages

Sometimes you know the value after a change and need the original. If a price after a 20% increase is €360, the original price is 360 ÷ 1.20 = €300 — not 360 − 20% of 360 (which would give €288).

The same logic applies to decreases: if a value after a 15% reduction is 1,700, the original was 1,700 ÷ 0.85 = 2,000.

4. Successive percentage changes

Percentage changes multiply; they do not add. A 10% increase followed by a 10% decrease gives 1.10 × 0.90 = 0.99, a net fall of 1%, not zero.

Example: exports grow by 20% in year one and fall by 5% in year two. The overall change is 1.20 × 0.95 = 1.14, an increase of 14%. Adding the two figures would give 15%, which is likely to appear as a wrong option.

5. Compound growth over several years

If a quantity grows by the same rate every year, multiply by the growth factor once for each year. Growth of 5% a year for three years gives 1.05 × 1.05 × 1.05 ≈ 1.158, an increase of about 15.8%.

The reverse question — what constant annual rate turns 100 into 121 over two years? — is answered by recognising that 1.1 × 1.1 = 1.21, so the rate is 10%. You rarely need a formula: test the answer options by multiplying.

6. Shares and proportions

A share is a part divided by the whole. If one Member State receives €1.2 billion out of a total fund of €4.8 billion, its share is 1.2 ÷ 4.8 = 25%.

Common mistake: using the wrong total. Tables often contain subtotals by region or category. Before dividing, confirm which row or column the question defines as 'the total'.

7. Ratios

To split a quantity in a ratio, add the parts and divide. A budget of 640 split in the ratio 3:5 gives 640 ÷ 8 = 80 per part, so the two shares are 240 and 400.

To compare ratios between two tables, convert them to the same base. A staff ratio of 12 officials to 3 assistants (4:1) is higher than 15 to 5 (3:1), even though the second unit has more people in total.

8. Weighted averages

When groups have different sizes, a simple average of their rates is wrong. Suppose three regions have populations of 10, 20 and 30 million and unemployment rates of 5%, 8% and 10%. The number of unemployed is 0.5 + 1.6 + 3.0 = 5.1 million out of 60 million, so the combined rate is 8.5%. The simple average of the three rates, about 7.7%, is a classic distractor.

9. Per-capita figures and unit conversions

Per-capita values divide a total by population: a GDP of €450 billion for 15 million inhabitants is €30,000 per person. Pay attention to units — billions divided by millions gives thousands.

Currency questions work the same way. If €1 = $1.10, then $550 = 550 ÷ 1.10 = €500. Decide first whether you should multiply or divide by checking which currency is larger per unit.

Estimate before you calculate

An on-screen calculator is usually available, but typing every number takes time and a single keying error produces a confident wrong answer. A quick estimate protects you. For example, 19.8% of 4,970 is close to 20% of 5,000, which is 1,000; the exact answer (about 984) must therefore be near 1,000. If your calculator shows 98 or 9,840, you know immediately that something went wrong.

Estimation also lets you skip the calculation entirely when the answer options are far apart. If the options are 12%, 25%, 38% and 51%, and your rough estimate is 'about a quarter', you can answer without precise arithmetic.

A step-by-step routine for every question

• Read the question before the table, and identify the exact quantity required (value, percentage, percentage points, ratio).
• Find only the cells you need; ignore the rest of the table.
• Check units and footnotes (thousands, millions, constant prices, per 100 inhabitants).
• Estimate the answer roughly.
• Calculate, then compare the result with your estimate and with the answer options.