Civil Service Numerical Reasoning Test Word Problems and Applied Reasoning Syllabus

For 'value for money' comparisons between two products, what is the standard approach?

Reduce each to a unit cost (cost per item, per gram, per litre, etc.) using $\frac{\text{Price}}{\text{Quantity}}$, then the lower unit cost is better value.

Product A costs £6 for 500 g; Product B costs £10 for 1 kg. Which is better value per 100 g?

A: $\frac{600\text{p}}{500\text{g}} = 1.2$p/g $= \pounds 1.20$ per 100 g. B: $\frac{1000\text{p}}{1000\text{g}} = 1.0$p/g $= \pounds 1.00$ per 100 g. B is better value.

What is the formula for percentage savings when a price is reduced?

$\frac{\text{Original} - \text{New}}{\text{Original}} \times 100$.

How do you calculate the payback period for an investment with even annual savings?

$\text{Payback period} = \frac{\text{Initial cost}}{\text{Annual saving}}$, giving the number of years to recover the cost.

What is a KPI (Key Performance Indicator)?

A measurable value that shows how effectively an organisation is achieving a key objective or target.

How do you calculate the percentage of a target achieved?

$\frac{\text{Actual}}{\text{Target}} \times 100$. For example, 90 of 120 cases handled $= \frac{90}{120} \times 100 = 75\%$.

What is the formula for percentage change between two performance figures?

$\frac{\text{New} - \text{Old}}{\text{Old}} \times 100$; a positive result is an increase, negative is a decrease.

A KPI was 250 last quarter and 300 this quarter. What is the percentage increase?

$\frac{300 - 250}{250} \times 100 = \frac{50}{250} \times 100 = 20\%$ increase.

How do you calculate a mean (average) performance figure across periods?

$\text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}}$.

What is the formula to add VAT at the standard UK rate of 20% to a net price?

$\text{Gross} = \text{Net} \times 1.20$ (equivalently Net $+ 20\%$ of Net).

A net price is £80. What is the VAT amount and the gross price at 20% VAT?

VAT $= 80 \times 0.20 = \pounds 16$; Gross $= 80 + 16 = \pounds 96$.

How do you remove 20% VAT from a gross (VAT-inclusive) price to find the net price?

$\text{Net} = \frac{\text{Gross}}{1.20}$. The VAT element is Gross $-$ Net, or Gross $\times \frac{1}{6}$.

Why is the VAT fraction of a 20% VAT-inclusive price equal to $\frac{1}{6}$?

Gross represents 120% of net, so VAT (20 parts) of the gross (120 parts) is $\frac{20}{120} = \frac{1}{6}$.

What is the general formula for net pay after a single percentage deduction (e.g. tax)?

$\text{Net} = \text{Gross} \times \left(1 - \frac{\text{rate}}{100}\right)$.

If income tax is 20% and an employee earns £2,000 gross in a month, what is the deduction and take-home (ignoring other deductions)?

Deduction $= 2000 \times 0.20 = \pounds 400$; take-home $= 2000 - 400 = \pounds 1600$.

How do you apply two successive deductions, e.g. 20% tax then 8% pension, to a gross amount?

Multiply successively: $\text{Gross} \times (1 - 0.20) \times (1 - 0.08)$. The deductions are not simply added because the second applies to the already-reduced amount (unless both are on gross — read carefully).

What is the difference between a 'tax-free allowance' and 'taxable income'?

The tax-free allowance is the portion of income on which no tax is charged; taxable income $= \text{Gross income} - \text{Allowance}$, and tax is applied only to the taxable portion.

How do you calculate the elapsed time between 09:45 and 14:20?

From 09:45 to 14:45 is 5 hours; subtract 25 minutes to reach 14:20, giving 4 hours 35 minutes.

How do you convert 2 hours 45 minutes into minutes, and into decimal hours?

Into minutes: $2 \times 60 + 45 = 165$ minutes. Into decimal hours: $2 + \frac{45}{60} = 2.75$ hours.

How do you find the date 90 days after a given date?

Add days month by month using each month's length (28-31), carrying over into the next month, accounting for leap years where February has 29 days.

Which years are leap years, and why does it matter for date calculations?

Years divisible by 4 are leap years (except centuries not divisible by 400, e.g. 1900 is not, 2000 is). It matters because February then has 29 days, affecting day counts.

How do you convert a duration across time zones, e.g. a 3-hour flight departing 10:00 from a zone 2 hours ahead of the destination?

Add flight time to departure (local) then adjust for the time difference: $10\!:\!00 + 3\text{h} = 13\!:\!00$ departure-local; destination is 2 h behind, so arrival $= 11\!:\!00$ destination time.

In rota planning, how do you calculate the number of staff-hours needed to cover a shift requirement?

$\text{Staff-hours} = \text{Number of staff required} \times \text{Hours of the shift}$.

If a service must be staffed by 4 people for 12 hours a day, 7 days a week, how many staff-hours per week are needed?

$4 \times 12 \times 7 = 336$ staff-hours per week.

How do you calculate how many staff are needed to cover a workload, given each handles a fixed capacity?

$\text{Staff needed} = \frac{\text{Total workload}}{\text{Capacity per person}}$, rounding UP to the next whole person since you cannot have a fraction of a worker covering demand.

A call centre receives 480 calls in an hour and each agent handles 30 calls per hour. How many agents are needed?

$\frac{480}{30} = 16$ agents.

What is the formula relating distance, speed and time?

$\text{Distance} = \text{Speed} \times \text{Time}$; rearranged, $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$ and $\text{Time} = \frac{\text{Distance}}{\text{Speed}}$.

A journey is 150 miles at an average speed of 50 mph. How long does it take?

$\text{Time} = \frac{150}{50} = 3$ hours.

How do you calculate the fuel cost of a journey given distance, fuel efficiency (miles per gallon) and fuel price per litre?

Gallons needed $= \frac{\text{Distance}}{\text{mpg}}$; convert to litres ($\times 4.546$); cost $= \text{litres} \times \text{price per litre}$. (If efficiency is given per litre, divide distance by miles-per-litre directly.)

When comparing two journey options on cost and time, what is the standard decision approach?

Calculate the total cost and total time for each option separately, then compare directly against the stated priority (cheapest, fastest, or best balance of cost per hour saved).