This module develops the numerical competence required for success in the Ghana National Fire Service aptitude test. Mathematics in fire service operations is practical and applied — involving measurement, estimation, time calculation, resource allocation, and quick decision-making under pressure.

The module progresses from foundational arithmetic to applied problem-solving scenarios relevant to logistics, emergency timing, and operational efficiency.

UNIT 1: BASIC ARITHMETIC OPERATIONS

1.1 Introduction

Basic arithmetic forms the foundation of all numerical reasoning. Candidates must demonstrate speed and accuracy in performing calculations without reliance on calculators (unless stated).

The four fundamental operations are:

Addition
Subtraction
Multiplication
Division

1.2 Addition

Addition combines quantities.

Example:
45 + 78 = 123

Properties of Addition

Property	Explanation	Example
Commutative	a + b = b + a	4 + 7 = 7 + 4
Associative	(a + b) + c = a + (b + c)	(2 + 3) + 4 = 2 + (3 + 4)
Identity	a + 0 = a	9 + 0 = 9

1.3 Subtraction

Subtraction finds the difference between quantities.

Example:
100 − 37 = 63

Important:
Subtraction is NOT commutative.

8 − 3 ≠ 3 − 8

1.4 Multiplication

Multiplication is repeated addition.

7 × 5 = 35

Properties of Multiplication

Property	Explanation	Example
Commutative	a × b = b × a	3 × 6 = 6 × 3
Associative	(a × b) × c = a × (b × c)	(2 × 3) × 4
Identity	a × 1 = a	12 × 1 = 12
Zero Property	a × 0 = 0	5 × 0 = 0

1.5 Division

Division distributes quantities equally.

20 ÷ 4 = 5

Division is NOT commutative.

1.6 Order of Operations

Use BODMAS:

B – Brackets
O – Orders (powers, roots)
D – Division
M – Multiplication
A – Addition
S – Subtraction

Example:
5 + 3 × 2 = 5 + 6 = 11

UNIT 2: FRACTIONS, DECIMALS, AND PERCENTAGES

2.1 Fractions

A fraction represents part of a whole.

a/b where:
a = numerator
b = denominator

Types of Fractions

Type	Example
Proper	3/5
Improper	7/4
Mixed	1 3/4

Operations with Fractions

Addition:
a/b + c/b = (a + c)/b

If denominators differ:
Find LCM first.

Example:
1/2 + 1/3
LCM = 6
= 3/6 + 2/6 = 5/6

2.2 Decimals

Decimals represent fractions in base 10.

0.25 = 25/100 = 1/4

Converting Fractions to Decimals

Divide numerator by denominator.

3/4 = 3 ÷ 4 = 0.75

2.3 Percentages

Percentage means “per hundred”.

Percentage formula:

Percentage = (Part / Whole) × 100

Example:
If 30 out of 50 candidates passed:

Percentage = (30/50) × 100 = 60%

Percentage Increase

Percentage Increase = (Increase / Original) × 100

Percentage Decrease

Percentage Decrease = (Decrease / Original) × 100

UNIT 3: RATIOS AND PROPORTIONS

3.1 Ratio

A ratio compares two quantities.

a : b

Example:
If 4 fire trucks serve 2 districts:

Ratio = 4 : 2 = 2 : 1

3.2 Proportion

A proportion is an equation of two ratios.

a/b = c/d

Cross multiplication rule:

a × d = b × c

Example:
3/5 = x/10
3 × 10 = 5 × x
30 = 5x
x = 6

Direct Proportion

y ∝ x
y = kx

Inverse Proportion

y ∝ 1/x
y = k/x

UNIT 4: AVERAGES

Average (Mean) formula:

Mean = Sum of Values / Number of Values

Example:
Response times: 5, 6, 7 minutes

Mean = (5 + 6 + 7) / 3 = 6 minutes

Weighted Mean (Senior Level)

Weighted Mean = (Σfx) / Σf

Where:
f = frequency
x = value

UNIT 5: WORD PROBLEMS AND APPLICATIONS

Strategy for Solving Word Problems

Read carefully
Identify known values
Identify what is required
Choose correct formula
Solve step-by-step

Example:

A fire tanker carries 2,500 liters. It uses 400 liters per operation. How many operations can it perform?

2500 ÷ 400 = 6 remainder 100

It can perform 6 full operations.

Senior-Level Application:
Multi-step percentage + ratio problems.

UNIT 6: TIME, SPEED, AND DISTANCE

Core Formula:

Speed = Distance / Time

Distance = Speed × Time

Time = Distance / Speed

Example:
A fire engine travels 120 km in 2 hours.

Speed = 120 ÷ 2 = 60 km/h

Unit Conversion Table:

Unit	Conversion
1 hour	60 minutes
1 minute	60 seconds
1 km	1000 m
1 m	100 cm

Senior-Level Concept:
Average Speed (if distances equal)

Average Speed = (2 × Speed1 × Speed2) / (Speed1 + Speed2)

UNIT 7: MEASUREMENT CONVERSIONS

Length

1 km = 1000 m
1 m = 100 cm
1 cm = 10 mm

Mass

1 tonne = 1000 kg
1 kg = 1000 g

Volume

1 liter = 1000 ml

Area

Area of Rectangle = Length × Width

Volume

Volume of Cuboid = Length × Width × Height

Practical Application:
Water tank capacity calculation.

UNIT 8: SIMPLE ALGEBRAIC EXPRESSIONS

8.1 Expressions

3x + 4

8.2 Solving Linear Equations

Example:

3x + 5 = 20
3x = 15
x = 5

8.3 Formulas

Perimeter of Rectangle:

P = 2(L + W)

Area of Circle:

A = πr²

Circumference:

C = 2πr

π ≈ 3.142

Senior-Level Word Algebra:

If 3 hoses spray x liters per minute and total output is 90 liters:

3x = 90
x = 30 liters per minute

Perfect! Let’s expand Module 2: Mathematics & Numerical Aptitude with worked examples for each unit, so candidates can clearly see step-by-step solutions. I’ll keep formulas intact and add explanations suitable for both junior and senior levels.

WORKED EXAMPLES

UNIT 1: BASIC ARITHMETIC OPERATIONS

Example 1: Addition

Add 467 + 389

Solution:

  467
+ 389
------
  856

Explanation: Start from the units column: 7 + 9 = 16 → write 6, carry 1. Tens: 6 + 8 + 1 (carry) = 15 → write 5, carry 1. Hundreds: 4 + 3 + 1 (carry) = 8.

Example 2: Subtraction

Subtract 752 − 468

Solution:

  752
- 468
------
  284

Explanation: Borrow where necessary: 2 − 8 → borrow 1 from tens. Complete subtraction column by column.

Example 3: Multiplication

Multiply 36 × 24

Solution:

     36
   × 24
  -----
    144  (36 × 4)
   720   (36 × 20)
  -----
   864

Explanation: Multiply ones first, then tens, sum partial products.

Example 4: Division

Divide 945 ÷ 15

Solution:

945 ÷ 15 = 63

Explanation: 15 × 60 = 900 → remainder 45 → 15 × 3 = 45 → quotient = 63

UNIT 2: FRACTIONS, DECIMALS, AND PERCENTAGES

Example 1: Fractions

Add 3/4 + 2/5

Solution:

LCM of 4 and 5 = 20
3/4 = 15/20, 2/5 = 8/20
15/20 + 8/20 = 23/20 = 1 3/20

Explanation: Convert to common denominator, then add numerators.

Example 2: Decimals

Subtract 12.75 − 8.46

Solution:

12.75
-8.46
------
 4.29

Explanation: Align decimal points, subtract column by column.

Example 3: Percentage

What is 20% of 250 liters?

Solution:

Percentage = (Part / Whole) × 100
Part = 20/100 × 250 = 50 liters

Explanation: Multiply the total by the percentage in decimal form.

Example 4: Percentage Increase

Price of a fire extinguisher increases from 120 GHS to 150 GHS. Find % increase.

Solution:

Increase = 150 − 120 = 30
% Increase = (30 / 120) × 100 = 25%

UNIT 3: RATIOS AND PROPORTIONS

Example 1: Ratio

If 12 firemen serve 3 fire stations, find the ratio of firemen to stations.

Solution:

Ratio = 12 : 3 = 4 : 1

Example 2: Proportion

Solve 5/8 = x/32

Solution:

Cross multiply: 5 × 32 = 8 × x
160 = 8x
x = 20

UNIT 4: AVERAGES

Example 1: Mean

Response times (minutes): 5, 7, 6, 8, 9
Find the average response time.

Solution:

Mean = (5+7+6+8+9)/5 = 35/5 = 7 minutes

Example 2: Weighted Mean

Operation type | Time (min) | Frequency (f)
Extinguish Fire | 5 | 3
Rescue | 7 | 2
Inspection | 4 | 5

Weighted Mean = Σ(f×x)/Σf

Σ(f×x) = (3×5) + (2×7) + (5×4) = 15+14+20=49
Σf = 3+2+5=10
Weighted Mean = 49/10 = 4.9 minutes

UNIT 5: WORD PROBLEMS AND APPLICATIONS

Example 1: Single-Step

A tanker carries 2,500 liters and uses 400 liters per operation. How many operations?

Solution:

Operations = 2500 ÷ 400 = 6 remainder 100
Answer: 6 full operations

Example 2: Multi-Step

A fire station has 8 trucks. Each truck carries 500 liters. If 3 trucks are in use, how much water remains?

Solution:

Total capacity = 8×500 = 4000 liters
In use = 3×500 = 1500 liters
Remaining = 4000−1500 = 2500 liters

UNIT 6: TIME, SPEED, AND DISTANCE

Example 1: Speed

Fire engine travels 180 km in 3 hours. Find speed.

Solution:

Speed = Distance / Time = 180 ÷ 3 = 60 km/h

Example 2: Average Speed (Senior Level)

Fire engines travel to a fire 60 km away at 40 km/h and return at 30 km/h. Find average speed.

Solution:

Average Speed = (2 × 40 × 30)/(40 + 30) = 2400/70 ≈ 34.29 km/h

UNIT 7: MEASUREMENT CONVERSIONS

Example 1: Volume

Convert 5 m³ to liters.

Solution:

1 m³ = 1000 liters
5 m³ = 5 × 1000 = 5000 liters

Example 2: Area

Area of rectangle: length = 12 m, width = 8 m

Solution:

Area = Length × Width = 12 × 8 = 96 m²

UNIT 8: SIMPLE ALGEBRAIC EXPRESSIONS

Example 1: Solve Linear Equation

3x + 7 = 22

Solution:

3x = 22 − 7 = 15
x = 15 ÷ 3 = 5

Example 2: Practical Application

3 hoses deliver x liters per minute. Total = 90 liters. Find x.

Solution:

3x = 90
x = 90 ÷ 3 = 30 liters/min

UNIT 1: BASIC ARITHMETIC OPERATIONS

1.1 Introduction

1.2 Addition

Properties of Addition

1.3 Subtraction

1.4 Multiplication

Properties of Multiplication

1.5 Division

1.6 Order of Operations

UNIT 2: FRACTIONS, DECIMALS, AND PERCENTAGES

2.1 Fractions

Types of Fractions

Operations with Fractions

2.2 Decimals

Converting Fractions to Decimals

2.3 Percentages

Percentage Increase

Percentage Decrease

UNIT 3: RATIOS AND PROPORTIONS

3.1 Ratio

3.2 Proportion

Direct Proportion

Inverse Proportion

UNIT 4: AVERAGES

UNIT 5: WORD PROBLEMS AND APPLICATIONS

Strategy for Solving Word Problems

UNIT 6: TIME, SPEED, AND DISTANCE

UNIT 7: MEASUREMENT CONVERSIONS

Length

Mass

Volume

Area

Volume

UNIT 8: SIMPLE ALGEBRAIC EXPRESSIONS

8.1 Expressions

8.2 Solving Linear Equations

8.3 Formulas

WORKED EXAMPLES

UNIT 1: BASIC ARITHMETIC OPERATIONS

Example 1: Addition

Example 2: Subtraction

Example 3: Multiplication

Example 4: Division

UNIT 2: FRACTIONS, DECIMALS, AND PERCENTAGES

Example 1: Fractions

Example 2: Decimals

Example 3: Percentage

Example 4: Percentage Increase

UNIT 3: RATIOS AND PROPORTIONS

Example 1: Ratio

Example 2: Proportion

UNIT 4: AVERAGES

Example 1: Mean

Example 2: Weighted Mean

UNIT 5: WORD PROBLEMS AND APPLICATIONS

Example 1: Single-Step

Example 2: Multi-Step

UNIT 6: TIME, SPEED, AND DISTANCE

Example 1: Speed

Example 2: Average Speed (Senior Level)

UNIT 7: MEASUREMENT CONVERSIONS

Example 1: Volume

Example 2: Area

UNIT 8: SIMPLE ALGEBRAIC EXPRESSIONS

Example 1: Solve Linear Equation

Example 2: Practical Application

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