Numeracy And Arithmetic Reasoning

Numeracy and Arithmetic Reasoning form a core component of the Ghana Police Service aptitude test. This module develops the mathematical skills required for daily policing duties, such as handling money, estimating time and distance, interpreting numerical information, and making quick, accurate decisions under pressure.

The focus is not advanced mathematics, but practical numerical understanding, accuracy, and logical reasoning.

UNIT 2.1: BASIC ARITHMETIC OPERATIONS

Explanation

Basic arithmetic operations are the foundation of all numerical reasoning. Candidates are expected to perform calculations quickly and accurately, often without calculators.

The Four Basic Operations

Operation	Symbol	Meaning
Addition	+	Combining numbers
Subtraction	−	Finding the difference
Multiplication	×	Repeated addition
Division	÷	Sharing equally

Examples

Addition:
245 + 378 = 623
Subtraction:
850 − 476 = 374
Multiplication:
24 × 15 = 360
Division:
144 ÷ 12 = 12

Key Teaching Points

Always work step by step
Check for place value errors
Estimate first to confirm answers make sense

UNIT 2.2: FRACTIONS, DECIMALS, AND PERCENTAGES

Explanation

Fractions, decimals, and percentages are different ways of expressing parts of a whole. Police aptitude tests frequently assess the ability to convert between them and apply them in real-life situations such as finance and statistics.

Fractions

A fraction has two parts:

Numerator (top)
Denominator (bottom)

Example:

3/4

means 3 parts out of 4 equal parts.

Types of Fractions

Proper fraction: 2/5
Improper fraction: 7/4
Mixed number: 1 3/4

Decimals

Decimals represent fractions with denominators of 10, 100, 1000, etc.

Examples:

1/10 = 0.1
25/100 = 0.25

Percentages

A percentage means “out of 100”.

Examples:

50% = 50 out of 100
25% = 25/100

Conversions (VERY IMPORTANT)

Fraction to Decimal

Decimal = Numerator ÷ Denominator

Example:

3 ÷ 4 = 0.75

Decimal to Percentage

Percentage = Decimal × 100

Example:

0.45 × 100 = 45%

Percentage to Fraction

Percentage / 100

Example:

20% = 20/100 = 1/5

UNIT 2.3: RATIOS AND PROPORTIONS

Explanation

A ratio compares two or more quantities. In policing, ratios are used for resource allocation, population comparisons, and statistical analysis.

Ratio

Written as:

a : b

Example:

Ratio of male to female officers = 3 : 2

Simplifying Ratios

Divide both numbers by their highest common factor (HCF).

Example:

12 : 18

Divide both by 6:

2 : 3

Proportion

A proportion states that two ratios are equal.

Example:

a/b = c/d

Formula:

a × d = b × c

Example:

3/4 = x/12

3 × 12 = 4 × x
36 = 4x
x = 9

UNIT 2.4: WORD PROBLEMS (TIME, DISTANCE, MONEY, AVERAGES)

Explanation

Word problems test understanding, interpretation, and application of numerical skills. Candidates must identify what is given, what is required, and which formula to apply.

Time, Speed, and Distance

Formula:

Speed = Distance ÷ Time
Distance = Speed × Time
Time = Distance ÷ Speed

Example:
A patrol vehicle travels 120 km in 2 hours.

Speed = 120 ÷ 2 = 60 km/h

Money Problems

Includes profit, loss, and simple financial calculations.

Profit

Profit = Selling Price − Cost Price

Loss

Loss = Cost Price − Selling Price

Averages (Mean)

Formula:

Average = Total ÷ Number of items

Example:
Scores: 60, 70, 80, 90

Total = 300
Average = 300 ÷ 4 = 75

UNIT 2.5: DATA INTERPRETATION

Explanation

This unit tests the ability to read and interpret information from tables, charts, and basic statistics — a vital skill in report writing and crime data analysis.

Tables

Example:

Year	Cases Reported
2022	450
2023	520
2024	600

Possible Questions:

Which year recorded the highest cases?
What is the increase from 2022 to 2024?

Simple Statistics

Mean

Mean = Sum of values ÷ Number of values

Range

Range = Highest value − Lowest value

UNIT 2.6: BASIC NUMERICAL LOGIC & REASONING

Explanation

This section measures how well candidates can identify patterns, sequences, and relationships between numbers.

Number Sequences

Example:

2, 4, 8, 16, ?

Pattern: ×2
Answer:

Odd-One-Out

Example:

3, 5, 7, 9, 11

Answer:

9 (only composite number)

Logic-Based Calculation

Example:
If:

A = 2
B = 4
C = 6

What is:

A + B + C = 12

SUMMARY OF MODULE 2

Skill Area	Importance
Arithmetic	Accuracy in calculations
Fractions & Percentages	Financial and statistical understanding
Ratios	Comparison and allocation
Word Problems	Real-life application
Data Interpretation	Analysis and reporting
Numerical Logic	Reasoning under pressure

Great call 👍 — worked examples are what turn notes into exam-winning material. Below, I’ve added clear, step-by-step WORKED EXAMPLES for EACH UNIT in Module 2, written at teacher level, exactly how candidates should think in the exam hall.

I’ve kept:
✔ Simple language
✔ Logical steps
✔ All formulas inside code blocks
✔ Realistic Ghana Police aptitude–style questions

You can paste this directly under your existing Module 2 notes or replace sections as needed.

WORKED EXAMPLES

UNIT 2.1: BASIC ARITHMETIC OPERATIONS

Worked Examples

Example 1: Addition

Question:
Add: 2,468 + 3,759

Solution (Step-by-step):

  2468
+ 3759
------
  6227

Answer:
6,227

Teaching Note:
Always align digits according to place value (units, tens, hundreds).

Example 2: Subtraction

Question:
Subtract: 5,000 − 2,784

Solution:

  5000
- 2784
------
  2216

Answer:
2,216

Teaching Note:
Borrow carefully when subtracting larger digits from smaller ones.

Example 3: Multiplication

Question:
24 × 16

Solution:

24 × 16
= (24 × 10) + (24 × 6)
= 240 + 144
= 384

Answer:
384

Example 4: Division

Question:
180 ÷ 15

Solution:

15 × 12 = 180

Answer:
12

UNIT 2.2: FRACTIONS, DECIMALS & PERCENTAGES

Worked Examples

Example 1: Fraction to Decimal

Question:
Convert 5/8 to a decimal.

Formula:

Decimal = Numerator ÷ Denominator

Solution:

5 ÷ 8 = 0.625

Answer:
0.625

Example 2: Decimal to Percentage

Question:
Convert 0.36 to a percentage.

Formula:

Percentage = Decimal × 100

Solution:

0.36 × 100 = 36%

Answer:
36%

Example 3: Percentage of a Quantity

Question:
Find 15% of 240.

Formula:

Percentage × Quantity ÷ 100

Solution:

15 × 240 ÷ 100
= 3600 ÷ 100
= 36

Answer:
36

Police context:
Used in calculating fines, allowances, or population percentages.

UNIT 2.3: RATIOS & PROPORTIONS

Worked Examples

Example 1: Simplifying a Ratio

Question:
Simplify the ratio 18 : 24

Solution:

HCF of 18 and 24 = 6
18 ÷ 6 = 3
24 ÷ 6 = 4

Answer:

3 : 4

Example 2: Sharing in a Ratio

Question:
₵1,200 is shared between A and B in the ratio 2 : 3. How much does B receive?

Steps:

Total parts = 2 + 3 = 5
Value of 1 part = 1200 ÷ 5 = 240

B’s share:

3 × 240 = 720

Answer:
₵720

Example 3: Proportion

Question:
If 4 officers can cover an area in 6 hours, how long will 8 officers take?

Formula:

Work ∝ 1 / Number of workers

Solution:

4 × 6 = 8 × x
24 = 8x
x = 3

Answer:
3 hours

UNIT 2.4: WORD PROBLEMS

Worked Examples

Example 1: Speed, Distance & Time

Question:
A police patrol vehicle travels 180 km in 3 hours. Find its speed.

Formula:

Speed = Distance ÷ Time

Solution:

180 ÷ 3 = 60 km/h

Answer:
60 km/h

Example 2: Average

Question:
Find the average of 55, 65, 70, and 90.

Formula:

Average = Total ÷ Number of values

Solution:

Total = 55 + 65 + 70 + 90 = 280
Average = 280 ÷ 4 = 70

Answer:
70

Example 3: Profit

Question:
A radio is bought for ₵850 and sold for ₵1,000. Find the profit.

Formula:

Profit = Selling Price − Cost Price

Solution:

1000 − 850 = 150

Answer:
₵150

UNIT 2.5: DATA INTERPRETATION

Worked Examples

Example 1: Table Interpretation

Year	Robbery Cases
2022	320
2023	410
2024	480

Question:
What is the increase in cases from 2022 to 2024?

Solution:

480 − 320 = 160

Answer:
160 cases

Example 2: Mean from Data

Question:
Find the mean number of cases.

Formula:

Mean = Sum ÷ Number

Solution:

320 + 410 + 480 = 1210
1210 ÷ 3 ≈ 403.3

Answer:
403.3 cases (approx.)

UNIT 2.6: NUMERICAL LOGIC & REASONING

Worked Examples

Example 1: Number Sequence

Question:

5, 10, 20, 40, ?

Solution:

Pattern = ×2
40 × 2 = 80

Answer:
80

Example 2: Missing Number

Question:

3, 6, 11, 18, ?

Solution:

+3, +5, +7
Next = +9
18 + 9 = 27

Answer:
27

Example 3: Odd One Out

Question:

4, 9, 16, 25, 36

Solution:

All are perfect squares except 36?
No.
All are squares: 2², 3², 4², 5², 6²
So none is odd.

Teaching Insight:
Some questions test attention — not speed.

UNIT 2.1: BASIC ARITHMETIC OPERATIONS

Explanation

The Four Basic Operations

Examples

Key Teaching Points

UNIT 2.2: FRACTIONS, DECIMALS, AND PERCENTAGES

Explanation

Fractions

Types of Fractions

Decimals

Percentages

Conversions (VERY IMPORTANT)

Fraction to Decimal

Decimal to Percentage

Percentage to Fraction

UNIT 2.3: RATIOS AND PROPORTIONS

Explanation

Ratio

Simplifying Ratios

Proportion

UNIT 2.4: WORD PROBLEMS (TIME, DISTANCE, MONEY, AVERAGES)

Explanation

Time, Speed, and Distance

Money Problems

Profit

Loss

Averages (Mean)

UNIT 2.5: DATA INTERPRETATION

Explanation

Tables

Simple Statistics

Mean

Range

UNIT 2.6: BASIC NUMERICAL LOGIC & REASONING

Explanation

Number Sequences

Odd-One-Out

Logic-Based Calculation

SUMMARY OF MODULE 2

WORKED EXAMPLES

UNIT 2.1: BASIC ARITHMETIC OPERATIONS

Worked Examples

Example 1: Addition

Example 2: Subtraction

Example 3: Multiplication

Example 4: Division

UNIT 2.2: FRACTIONS, DECIMALS & PERCENTAGES

Worked Examples

Example 1: Fraction to Decimal

Example 2: Decimal to Percentage

Example 3: Percentage of a Quantity

UNIT 2.3: RATIOS & PROPORTIONS

Worked Examples

Example 1: Simplifying a Ratio

Example 2: Sharing in a Ratio

Example 3: Proportion

UNIT 2.4: WORD PROBLEMS

Worked Examples

Example 1: Speed, Distance & Time

Example 2: Average

Example 3: Profit

UNIT 2.5: DATA INTERPRETATION

Worked Examples

Example 1: Table Interpretation

Example 2: Mean from Data

UNIT 2.6: NUMERICAL LOGIC & REASONING

Worked Examples

Example 1: Number Sequence

Example 2: Missing Number

Example 3: Odd One Out

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