Numerical reasoning test formulas, with examples
Numerical reasoning test formulas help you calculate percentage changes, ratios, rates, averages, growth and break-even. This reference covers twelve of them: for each, why the operation fits, which quantities go into the formula and a worked calculation, step by step. Use it to check your method, then practise that skill with independent questions that are free to use.
Choose the calculation you need from the list below. The page works as a numerical reasoning cheat sheet to keep open while you practise. You may also see these called numerical aptitude formulas; the arithmetic is the same.
For a timed test, or a drill on any one skill, go to our numerical practice section.
Percentages
Share of a total: what percentage a part is
A share compares a part with the whole it belongs to, and that whole includes the part itself.
Formulas
To find the share
Share = (Part) divided by (Whole) × 100
To find the part
Part = (Whole × Share) divided by (100)
Worked examples
Finding the share: 40 of the 160 tickets logged last week were support tickets. What share is that?
Dividing 40 by the other 120 tickets would give 33.3%, which compares the part with the rest rather than with all 160.
- Divide the part by the whole: 40 ÷ 160 = 0.25
- Multiply by 100: 0.25 × 100 = 25%
Finding the part: 25% of 160 tickets were support tickets. How many is that?
- Multiply the whole by the share: 160 × 25 = 4,000
- Divide by 100: 4,000 ÷ 100 = 40 tickets
Support tickets were 25% of all tickets.
Percentage change: how much something rose or fell
A change is measured from where it started, so the starting figure is what you divide by.
Formula
Change = (New − Old) divided by (Old) × 100
- Old is the starting figure, and it cannot be zero.
- A result below zero is a fall.
Worked example
Orders rose from 250 in March to 320 in April. What is the percentage change?
Dividing the rise by the new figure, 320, would give 21.9%, which measures the change from where it ended.
- Find the change: 320 − 250 = 70
- Divide by the starting figure: 70 ÷ 250 = 0.28
- Multiply by 100: 0.28 × 100 = 28%
Orders rose by 28%.
Start the percentage change drillLearn and practise percentage change
Reverse percentage: how to find the original number
A percentage rise or fall is worked out on the original amount, so the final amount is that original times a multiplier. Dividing by the multiplier takes you back.
Formulas
After a rise
Original = (Final) divided by (1 + r)
After a fall
Original = (Final) divided by (1 − r)
- r is the percentage as a decimal, so 20% is 0.20.
Worked examples
After a rise: A price is 120 after a 20% rise. What was it before?
Taking 20% off 120 would give 96, because it works the 20% out on 120 instead of on the original price.
- Find the multiplier: 120 is 120% of the original, so 1 + 0.20 = 1.20
- Divide the final price by it: 120 ÷ 1.20 = 100
After a fall: A price is 80 after a 20% fall. What was it before?
- Find the multiplier: 80 is 80% of the original, so 1 − 0.20 = 0.80
- Divide the final price by it: 80 ÷ 0.80 = 100
Both prices started at 100.
Start the reverse percentage drillLearn and practise reverse percentage
Nested proportion: a share of a share
The second percentage is a share of the first group only, so the two percentages are multiplied, not added.
Formula
Result = (a × b) divided by (100)
- a is the group's share of the whole, in %.
- b is the share inside that group, in %.
Worked example
25% of the staff are engineers, and 60% of the engineers are senior. What share of all staff are senior engineers?
Adding 25% and 60% would give 85%, which is a share of nothing.
- Multiply the two percentages: 60 × 25 = 1,500
- Divide by 100: 1,500 ÷ 100 = 15%
Senior engineers are 15% of all staff.
Ratios, rates and conversions
Ratio sharing: splitting an amount in a ratio
A ratio splits the total into equal parts, so work out one part first, then take as many parts as the question names.
Formulas
One part = (Total) divided by (Sum of parts)
Share = One part × Parts
- Parts is the number of parts in the share you need.
Worked example
4,800 is shared in the ratio 5 : 3. How much is the 3-part share?
Reading the wrong side of 5 : 3 would give 3,000, the 5-part share.
- Add the parts: 5 + 3 = 8
- Find one part: 4,800 ÷ 8 = 600
- Multiply by the parts you need: 600 × 3 = 1,800
The 3-part share is 1,800.
Start the ratio sharing drillLearn and practise ratio sharing
Proportional scaling: from one quantity to another
When every unit costs the same, the cost grows in step with the quantity, so find the cost of one unit first.
Formulas
Unit rate = (Cost) divided by (Units)
Cost = Unit rate × Units
Worked example
12 units cost 18 euros. What do 20 units cost?
Costing only the 8 extra units answers a different question: the order is all 20.
- Find the unit rate: 18 ÷ 12 = 1.50 euros a unit
- Multiply by the new number of units: 1.50 × 20 = 30 euros
20 units cost 30 euros.
This holds only while the price per unit stays the same. A bulk discount breaks it.
Start the proportional scaling drillLearn and practise proportional scaling
Rate conversion: changing the units of a rate
Put the amount and the time into the units the question asks for before you divide.
Formula
Rate = (Amount) divided by (Time)
Worked example
Convert 72 km/h into metres a second.
Dividing the metres by 60 would give metres a minute, because an hour has 60 minutes but 3,600 seconds.
- Convert the distance: 72 × 1,000 = 72,000 metres
- Convert the time: 60 × 60 = 3,600 seconds
- Divide: 72,000 ÷ 3,600 = 20 metres a second
72 km/h is 20 metres a second.
The shortcut for km/h to m/s is dividing by 3.6. The drill uses the same step on output: a count every so many minutes, carried across a shift of several hours.
Unit price: which option is cheaper
Packs of different sizes compare only at the same unit, so divide each price by its quantity.
Formula
Unit price = (Price) divided by (Quantity)
Worked example
A 3 kg bag costs 12 euros, and a 2 kg bag costs 9 euros. Which is cheaper per kilogram?
The 9-euro bag looks cheaper, but it also holds less.
- Big bag: 12 ÷ 3 = 4 euros per kg
- Small bag: 9 ÷ 2 = 4.50 euros per kg
The big bag is cheaper: 4 euros a kilogram, against 4.50.
Currency conversion: converting, then taking off a fee
Convert first, then take the fee off the converted amount, because a fee makes what arrives smaller.
Formulas
Converted = Sent × Rate
Net = Converted × (1 − f)
- Rate is the euros you get for each pound sent.
- f is the fee as a decimal, so 4% is 0.04.
- Net is what arrives after the fee.
Worked example
2,400 pounds are converted at 1.25 euros to the pound, and a 4% fee comes off the converted amount. How many euros arrive?
- Convert: 2,400 × 1.25 = 3,000 euros
- Turn the fee into a multiplier: 1 − 0.04 = 0.96
- Apply it: 3,000 × 0.96 = 2,880 euros
2,880 euros arrive. Adding the 4% instead of taking it off would give 3,000 × 1.04 = 3,120 euros.
The rate here is an example, not a current quotation. Check which way the rate is stated before you multiply.
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Averages, growth and break-even
Weighted average: when groups differ in size
Each group counts by its size, so turn each average back into a total first.
Formulas
Group total = Value × Size
Average = (Sum of totals) divided by (Total size)
- Value is a group's average, and size is how many people it has.
Worked example
10 people average 6, and 30 people average 8. What is the average of all 40?
Averaging 6 and 8 would give 7, as if the group of 10 counted as much as the group of 30.
- Total for the first group: 6 × 10 = 60
- Total for the second group: 8 × 30 = 240
- Add the totals: 60 + 240 = 300
- Divide by everyone: 300 ÷ 40 = 7.5
The average of all 40 people is 7.5, pulled towards 8 by the larger group.
Start the weighted average drillLearn and practise weighted average
Compound growth: growth on what has already grown
Each year's growth is worked out on what the amount has already become, so the base changes every period.
Formula
End = Start × (1 + r) to the power of n
- r is the rate as a decimal, so 10% is 0.10.
- n is the number of periods.
- The rate is the same in every period.
Worked example
1,000 grows by 10% a year for 2 years. What is it worth at the end?
- Find the growth factor: 1 + 0.10 = 1.10
- Year 1: 1,000 × 1.10 = 1,100
- Year 2: 1,100 × 1.10 = 1,210
So 1,000 grows to 1,210 in 2 years, the same as 1,000 × 1.10² in one step. Adding 10% twice would give 1,200: the extra 10 is the second year's growth on the first year's 100.
Start the compound growth drillLearn and practise compound growth
Break-even: how many units cover the costs
Each unit pays towards the fixed costs only with what is left after its own cost.
Formulas
Margin = Price − Cost
Units = (Fixed costs) divided by (Margin)
- Cost is what one unit costs to make, apart from the fixed costs.
Worked example
Fixed costs are 500 euros. Each unit sells for 12 euros and costs 7 euros to make. How many units cover the fixed costs?
Dividing by the price instead would give 500 ÷ 12 = 41.7 units.
- Find the margin per unit: 12 − 7 = 5 euros
- Divide the fixed costs by it: 500 ÷ 5 = 100 units
The business breaks even at 100 units.
The price has to be above the cost of making one unit. Where units cannot be split, round a fractional result up, because part of a unit covers nothing.
When you want a mixed set rather than one skill, the button below opens the numerical practice page. Its 12-question diagnostic runs on a 12-minute clock, and the clock starts only when you press start on that page.
How we write, check and mark the practice questions is set out on the Methodology page.