Police Initial Recruitment Test (PIRT/SEARCH) Numerical Reasoning and Work-Based Numeracy Syllabus

Numerical Reasoning and Work-Based Numeracy syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Numerical Reasoning and Work-Based Numeracy in Police Initial Recruitment Test (PIRT/SEARCH), not a summary of it.

1. 1

   Core Arithmetic Foundations

   4 topics

   • The Four Operations Under Time Pressure
     • Mental addition and subtraction with carrying and borrowing
     • Short and long multiplication of multi-digit figures
     • Division with remainders and exact quotients
     • Order of operations (BIDMAS) in multi-step sums
   • Working with Decimals and Money
     • Adding and subtracting sterling amounts to two decimal places
     • Rounding to the nearest pound and nearest penny
     • Estimating totals to sanity-check answers quickly
   • Fractions, Ratio and Proportion
     • Simplifying and comparing fractions
     • Splitting quantities in a given ratio (e.g. patrol allocations)
     • Direct and inverse proportion in resourcing scenarios
   • Negative Numbers and Number Lines
     • Temperature, time-zone and balance differences
     • Sequencing and ordering signed values
2. 2

   Percentages and Practical Calculations

   4 topics

   • Percentage Fundamentals
     • Converting between fractions, decimals and percentages
     • Finding a percentage of a quantity without a calculator
   • Percentage Change in Crime Data
     • Increase and decrease in offence figures year on year
     • Expressing one figure as a percentage of another (clear-up rates)
   • Reverse Percentages and Original Values
   • Percentage Points versus Percentage Change
3. 3

   Measures, Time and Conversion

   4 topics

   • Time and Shift Calculations
     • 24-hour clock and incident timelines
     • Calculating elapsed time across midnight
     • Response-time and duration problems
   • Speed, Distance and Time
     • Average speed for pursuit and patrol scenarios
     • Converting mph, km/h and metres per second
   • Units and Conversions
     • Metric and imperial length, weight and capacity
     • Currency conversion for cross-border enquiries
   • Scale, Distance and Map Reading
4. 4

   Data Interpretation and Statistics

   4 topics

   • Reading Tables and Charts
     • Extracting values from frequency tables and rotas
     • Bar charts, pie charts and line graphs of incident data
   • Summary Statistics
     • Mean, median, mode and range of caseloads
     • Choosing the appropriate average for a dataset
   • Multi-Source Data Synthesis
   • Spotting Misleading Presentation of Figures
5. 5

   Numerical Test Strategy

   4 topics

   • Time Allocation per Question and Pacing
   • Estimation and Elimination Techniques
   • Avoiding Common Calculation Traps
   • Managing the No-Calculator Constraint

Numerical Reasoning and Work-Based Numeracy flashcards for Police Initial Recruitment Test (PIRT/SEARCH)

19 of 50 cards from the Numerical Reasoning and Work-Based Numeracy deck — real questions with worked answers.

1. In the PIRT/SEARCH numerical test, what tool is permitted and what is the key implication for your method?

   No calculator is allowed. You must rely on mental and written (paper) arithmetic, so methods must favour speed, estimation, and clean cancellation rather than long manual computation.

2. State the correct order of operations (BODMAS/BIDMAS) for evaluating an expression.

   Brackets, then Orders (indices/powers and roots), then Division and Multiplication (left to right), then Addition and Subtraction (left to right). E.g. $2 + 3 \times 4 = 2 + 12 = 14$.

3. What is a fast mental method to multiply a number by 5 without a calculator?

   Multiply by 10 and then halve. E.g. $46 \times 5 = \frac{460}{2} = 230$.

4. Give a quick mental shortcut for multiplying a whole number by 25.

   Multiply by 100 and divide by 4. E.g. $32 \times 25 = \frac{3200}{4} = 800$.

5. When adding or subtracting decimals by hand, what is the essential alignment rule?

   Line up the decimal points (and hence place values) vertically, padding with trailing zeros as needed, then add or subtract column by column carrying/borrowing as normal.

6. How do you multiply two decimals such as $0.4 \times 0.3$ without a calculator?

   Multiply ignoring the points ($4 \times 3 = 12$), then place the decimal point so the answer has as many decimal places as the two factors combined (here $1+1=2$), giving $0.12$.

7. To divide by a decimal, such as $\frac{6}{0.2}$, what is the standard technique?

   Multiply both numbers by the same power of 10 to make the divisor a whole number: $\frac{6}{0.2} = \frac{60}{2} = 30$.

8. When working with money, how many decimal places should a final answer in pounds have, and what does $\pounds 0.5$ really mean?

   Two decimal places (pence). $\pounds 0.5$ should be written $\pounds 0.50$, i.e. 50 pence, not 5 pence.

9. Convert the fraction $\frac{3}{4}$ to a decimal and a percentage.

   $\frac{3}{4} = 0.75 = 75\%$.

10. What is the process to add two fractions with different denominators, e.g. $\frac{1}{3} + \frac{1}{4}$?

    Find a common denominator (here 12), convert each fraction ($\frac{4}{12} + \frac{3}{12}$), add the numerators, then simplify: $\frac{7}{12}$.

11. How do you multiply two fractions, and how is dividing fractions different?

    To multiply, multiply numerators together and denominators together: $\frac{2}{3} \times \frac{4}{5} = \frac{8}{15}$. To divide, multiply by the reciprocal of the second fraction: $\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}$.

12. Define a ratio and explain how to split a quantity in a given ratio, e.g. share $\pounds 60$ in the ratio $2:3$.

    A ratio compares parts of a whole. Add the parts ($2+3=5$), divide the total by the number of parts ($\frac{60}{5}=12$), then multiply: $2 \times 12 = \pounds 24$ and $3 \times 12 = \pounds 36$.

13. What is direct proportion, and how do you use the unitary method to solve it?

    In direct proportion, as one quantity increases the other increases at the same rate. The unitary method finds the value of one unit first, then scales. E.g. if 4 items cost $\pounds 20$, one costs $\pounds 5$, so 7 cost $\pounds 35$.

14. How do you simplify the ratio $12:18$ to its lowest terms?

    Divide both sides by their highest common factor (6): $12:18 = 2:3$.

15. On a number line, how do you calculate the difference between $-3$ and $5$?

    Count the distance across zero: from $-3$ to $0$ is 3, and $0$ to $5$ is 5, so the difference is $3 + 5 = 8$. Equivalently $5 - (-3) = 8$.

16. State the sign rules for multiplying and dividing negative numbers.

    Same signs give a positive result ($-\times- = +$, $+\times+ = +$); different signs give a negative result ($-\times+ = -$). E.g. $-4 \times -3 = 12$ but $-4 \times 3 = -12$.

17. What is the rule for subtracting a negative number, e.g. $7 - (-2)$?

    Subtracting a negative is the same as adding its positive: $7 - (-2) = 7 + 2 = 9$.

18. Define 'per cent' and convert $\frac{1}{5}$ into a percentage.

    Per cent means 'per hundred', i.e. a fraction out of 100. $\frac{1}{5} = \frac{20}{100} = 20\%$.

19. How do you find a percentage of an amount without a calculator, e.g. $15\%$ of $\pounds 80$?

    Break it into easy parts: $10\%$ of $80 = 8$, and $5\%$ is half of that $= 4$, so $15\% = 8 + 4 = \pounds 12$.

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