Numerical Reasoning Formulas You Need To Know

Published 1 April 2024 · Updated 4 August 2026

Numerical reasoning tests are built from a short list of formulas: percentage change, percentages of an amount, ratio sharing, averages, speed and distance, and interest. Percentage change is by far the most common, and it is (new value - original value) ÷ original value × 100.

Every formula below is written out plainly and followed by a worked example. None of them go beyond arithmetic you can do with the on-screen calculator, because the difficulty in numerical reasoning tests comes from reading the data and working at speed, not from the maths itself.

Percentage change

Percentage change tells you how far a figure has risen or fallen compared with where it started.

Percentage change = (new value - original value) ÷ original value × 100

Worked example: a department’s sales rise from £48,000 to £54,000.

(54,000 - 48,000) ÷ 48,000 × 100 = 6,000 ÷ 48,000 × 100 = 12.5% increase.

A negative answer means a decrease. If the same department fell from £54,000 back to £48,000, the calculation would be (48,000 - 54,000) ÷ 54,000 × 100 = 11.1% decrease to one decimal place. The same £6,000 gives a different percentage because the starting figure has changed, and that asymmetry is a favourite trap in test questions.

Percentage of an amount

Percentage of an amount = amount × percentage ÷ 100

Worked example: 15% of 340 is 340 × 15 ÷ 100 = 51.

To increase or decrease an amount by a percentage, use a multiplier instead, which is quicker under time pressure. Increasing 340 by 15% is 340 × 1.15 = 391, and decreasing it by 15% is 340 × 0.85 = 289.

Reverse percentages

When you are given a figure that already includes an increase, you cannot simply take the percentage back off it. Divide instead.

Original value = final value ÷ (1 + percentage increase ÷ 100)

Worked example: a product costs £96 including 20% VAT. The price before VAT is 96 ÷ 1.2 = £80.

Check it: £80 × 1.2 = £96. Taking 20% off £96 would have given £76.80, which is the wrong answer and usually one of the options offered.

Ratios and proportion

Value of one part = total ÷ sum of the ratio parts

Worked example: £4,800 is split between two departments in the ratio 5:3. The parts add up to 5 + 3 = 8, so one part is 4,800 ÷ 8 = £600. The shares are 5 × 600 = £3,000 and 3 × 600 = £1,800, which add back to £4,800.

Direct proportion questions work the same way through a unit rate. If 6 machines produce 810 units an hour, one machine produces 810 ÷ 6 = 135, so 10 machines produce 1,350 an hour.

Averages: mean, median, mode and range

Mean = sum of the values ÷ how many values there are
Median = the middle value once they are in order
Mode = the value that appears most often
Range = largest value - smallest value

Worked example, using the dataset 4, 7, 7, 9, 13:

The mean is (4 + 7 + 7 + 9 + 13) ÷ 5 = 40 ÷ 5 = 8.

The median is the third of five values in order, which is 7.

The mode is 7, the only value that appears twice.

The range is 13 - 4 = 9.

When there is an even number of values, the median is the average of the middle two. For 12, 15, 21, 24 the median is (15 + 21) ÷ 2 = 18.

Weighted average

When groups are different sizes, averaging the averages gives the wrong answer. Weight each average by the size of its group.

Weighted average = (each value × its weight, added together) ÷ total of the weights

Worked example: 30 staff earn an average of £28,000 and 20 staff earn an average of £33,000.

(30 × 28,000) + (20 × 33,000) = 840,000 + 660,000 = 1,500,000.

1,500,000 ÷ 50 = £30,000.

Averaging the two figures on their own would have given £30,500, which is the distractor the question is fishing for.

Speed, distance and time

Speed = distance ÷ time
Distance = speed × time
Time = distance ÷ speed

Worked example: a train covers 210 km in 2.5 hours, so its speed is 210 ÷ 2.5 = 84 km per hour. At that speed, 315 km takes 315 ÷ 84 = 3.75 hours, which is 3 hours and 45 minutes.

Convert minutes into decimals of an hour before you divide. Two hours 30 minutes is 2.5 hours, not 2.3, and getting that wrong is the single most common error in this question type.

Simple and compound interest

Simple interest = principal × rate ÷ 100 × number of years
Compound total = principal × (1 + rate ÷ 100) to the power of the number of years

Worked example: £4,000 invested at 3% a year for 5 years.

With simple interest, 4,000 × 0.03 × 5 = £600 of interest, giving £4,600 in total.

With compound interest, 4,000 × 1.03 to the power of 5 = 4,000 × 1.159274 = £4,637.10, so the interest is £637.10.

Compound growth always beats simple interest over more than one period, and the same multiplier method covers questions about population growth, inflation and depreciation. For depreciation, use a multiplier below 1: a value falling 8% a year for 3 years is multiplied by 0.92 to the power of 3, which is 0.778688.

Currency and unit conversion

Converted amount = original amount × the rate

Worked example: at 1.15 euros to the pound, £250 converts to 250 × 1.15 = 287.50 euros. Going the other way, divide: 287.50 ÷ 1.15 = £250.

The trap here is the direction of the rate. Write down which unit you are starting in before you touch the calculator.

Standard deviation and variance

These appear less often, and usually in questions that ask you to interpret them rather than calculate them, but the formulas are worth knowing.

Variance = the sum of (each value - the mean) squared ÷ how many values there are
Standard deviation = the square root of the variance

Worked example, using 4, 7, 7, 9, 13 again, which has a mean of 8.

The differences from the mean are -4, -1, -1, 1 and 5. Squared, they are 16, 1, 1, 1 and 25, which add up to 44.

The variance is 44 ÷ 5 = 8.8, and the standard deviation is the square root of 8.8, which is 2.97 to two decimal places.

A low standard deviation means the values sit close to the mean. A high one means they are spread widely around it.

Numerical reasoning formulas at a glance

Using the formulas under time pressure

Knowing the formulas is only half of it, because these tests are scored on speed as much as accuracy. Three habits make the difference.

Read the units and labels on the table first. Figures given in thousands or millions, or a column covering a quarter rather than a year, decide the whole answer before any arithmetic starts.

Estimate before you calculate. Rounding to the nearest whole number usually rules out two or three of the options, and sometimes it identifies the answer outright.

Only use the data in front of you. Numerical reasoning questions are answerable from the table or chart alone, so if you find yourself relying on outside knowledge, you have misread something.

Practise applying these formulas

Formulas stick when you use them against the clock rather than reread them. Work through full-length numerical reasoning tests to see them in the context of real tables and charts, drill the arithmetic on its own with basic numeracy tests, and review worked solutions in our aptitude test sample questions and answers.