Proportional Reasoning

In this section, you will be learning about logical reasoning and variation — two important mathematical concepts that help us understand how ideas are connected and how values change. These skills are not new to you. In Year 1, when you worked with sets, fractions, and De Morgan’s Law, you were already using logical steps to solve problems. In Year 2, you explored ratios, rates, and proportions, which are all examples of variation — they show how one quantity changes in relation to another.

Now, we will build on what you have already learned. You will apply logical thinking to make sound decisions, identify patterns, and solve real-life problems. You will also explore how variables can affect each other, such as how distance and time relate or how changes in price affect cost.

These skills will help you reason better, both in mathematics and in everyday life.

KEY IDEAS

• Argument Analysis: Evaluating the structure and validity of arguments to draw sound conclusions.

• Conjunctions and Disjunctions: Using logical connectives (“and”, “or”) to combine or contrast statements.

• Decision Making Tools: Applying tables and diagrams to support logical conclusions.

• Proportional Reasoning: Understanding and solving problems involving ratios, proportions and variations.

• Statement Classification: Identifying types of statements (interrogative, declarative, positive, negative) for logical analysis.

• Strategic Guessing: Using logical patterns and elimination methods to make intelligent guesses in uncertain situations.

In our daily lives, we often make decisions based on reasoning. Symbolic logic is the branch of mathematics that allows us to express these reasoning patterns using symbols and variables. By applying symbolic logic, we can analyse and compare relationships between variables in various situations—such as population growth, investment returns, or business profits—and make sound conclusions or predictions.

Think about this In a certain town, the population is growing by 15% every year.

But does this mean job opportunities also grow by 15%?

Consider the following statements P and Q below:

P: “The town’s population increases by 15% every year.”

Q: “Job opportunities increase by 15% every year.”

Think about the following real-life situations

• Have you heard of a community or town where the number of people keep increasing, but the number of jobs are not?

• Can you think of any reasons why job opportunities may not grow at the same rate as the population?

• Write at least two real-life reasons in your book. For example:

• Companies are not hiring new people.

• The town has more school graduates than job openings.

Now answer the following:

• If the population increases, does it always mean that job opportunities will increase?

• Can job opportunities increase even if the population does not?

• Can you give an example of when job opportunities might decrease even if the population grows?

Now consider these statements:

• “The number of students in the school increases.”

• “The number of textbooks available increases.”

Then ask yourself:

• Does one really lead to the other?

• What might affect each one?

Write your pair of statements and explain your thinking.

Conclusion:

Even if two things are growing, they may not grow at the same rate. One does not always cause the other. Logical reasoning helps us examine these differences clearly.

Concept of Logical Reasoning

Take a moment to think about this Have you ever seen someone with a big belly?

Does it always mean the person is pregnant?

Step 1: Go through this problem step by step Do only pregnant people have big bellies?

Now, think about other possible reasons someone may have a big belly. Can you name two?

For example:

• Someone has eaten a lot.

• Someone is sick.

• Someone has fat around their stomach.

Step 2: Read this situation Ama sees a woman with a big belly. She quickly decides she is pregnant.

Ask yourself

• Is Ama correct?

• What mistake could she have made?

• What could she do before drawing a conclusion?

Step 3

Imagine you see someone with a big belly. What would you do to find out the truth?

For example:

“I would ask politely or wait until I have more information. Just looking at someone’s belly is not enough to know if they are pregnant.”

Learn something here

• A good thinker doesn’t rush to conclusions.

• Just because something looks a certain way doesn’t always mean your guess is correct.

• Logical reasoning means asking questions, checking facts, and being careful with conclusions.

What is Logical Reasoning?

Logical reasoning is the process of using clear steps, facts and evidence to arrive at a valid conclusion. It involves thinking in a clear and orderly way so that the conclusions you reach are based on facts or evidence, not just guesses or feelings. It involves analysing information, eliminating wrong possibilities and making connections between statements.

Interrogative and Declarative Statements

Have a look at the information below:

Scenario: In many Ghanaian communities, school attendance has been improving over the years.

But does this mean that the students attending school always achieve good grades?

Step 1: Identify the Two Statements

Declarative Statement (this is a statement of fact, as you are ‘declaring’ something):

More children are attending school each year.

Interrogative Statement (this is a question, as you are seeking information – you are ‘interrogating’ the statement):

Are the students getting better grades every year?

Step 2: Connect to Real-Life Situations

• Have you heard of a town where many children go to school but still perform poorly in exams?

• Can you think of any reasons why school attendance may increase, while grades do not?

Write at least two real-life reasons based on your understanding. For example:

• Some pupils are in school but lack textbooks at home.

• Teachers may not have enough time to give individual attention.

Step 3: Convert Declarative to Interrogative

• Does more school attendance always mean better performance? (Interrogative)

• Can pupils perform better even if attendance stays the same? (Interrogative)

• Can you give an example where performance drops even when attendance is high?

Step 4: Make Your Own Example

Now create your own pair of statements—one declarative and one interrogative— about two related things in real life.

For example:

Declarative Statement: The number of shops in the town is increasing.

Interrogative Statement: Does that mean customers are buying more goods?

Then ask yourself:

• Does one fact automatically answer the question?

• What could affect each situation?

Write your pair of statements and briefly explain your thinking.

Declarative statements give us information, while interrogative statements help us seek information. By comparing both, we learn to think deeply about everyday situations and the questions we need to ask to understand them better.

What is an Interrogative Statement?

An interrogative statement, also called an interrogative sentence, asks a question. Its main purpose is to gather information, clarify a point or confirm something.

Interrogative sentences usually start with words like who, what, where, when, why, how or a helping (auxiliary) verb (is, are, will, do, can) and end with a question mark (?).

What is a Declarative Statement?

A declarative statement, also known as a declarative sentence, makes a direct statement or provides information. Its purpose is to state facts, opinions, explanations, or observations.

Example 1.1:

Convert these interrogative statements into declarative statements using the

table:

a. Is Kumasi located in the Ashanti Region?

b. Do farmers in Ghana rely on the rainy season?

c. Is cocoa a major export crop in Ghana?

d. Does education improve the lives of the youth?

Solution:

Table 1.1

Item Interrogative Statement Declarative Statement

A Is Kumasi located in the Ashanti Region?

Kumasi is located in the Ashanti Region.

B Do farmers in Ghana rely on the rainy season?

Farmers in Ghana rely on the rainy season.

C Is cocoa a major export crop in Ghana? Cocoa is a major export crop in Ghana.

D Does education improve the lives of the youth?

Education improves the lives of the youth.

Positive and Negative Statements:

To deepen your understanding and appreciation of this concept, carefully analyse this scenario:

Scenario: It is a market day in Techiman. Amma wants to buy some tomatoes and onions for a stew. What she hears from different sellers influences her buying decisions.

Seller A says:

“The tomatoes today are fresh and sweet.” (This is a positive statement about the tomatoes.)

Seller B says:

“The onions are not fresh; they are starting to spoil.” (This is a negative statement about the onions.)

Amma replies:

“I will buy tomatoes from Seller A because the tomatoes are fresh and sweet.”

“I will avoid onions from Seller B because they are not fresh.”

Positive statements express a favourable condition or fact (e.g., “The tomatoes are fresh.”) Negative statements express an unfavourable or opposite condition (e.g., “The onions are not fresh.” or “The onions are spoiled.”) This scenario shows how people make decisions based on positive and negative information.

Now look at these positive statements and how they change when made negative.

This will help you understand them better.

1. Positive statement: Yaa Asantewaa led the Ashanti in the war against the British.

2. Negative statement: Yaa Asantewaa did not lead the Ashanti in the war against the British.

3. Positive statement: 5 is a prime number.

4. Negative statement: 5 is not a prime number.

5. Positive statement: Honesty is an important moral value in Ghanaian society.

6. Negative statement: Honesty is not an important moral value in Ghanaian society.

NOTE:

Mostly, we represent statements with capital letters of the English alphabet and we attach “~” to mean negation.

Example:

E: The Volta River is the longest river in Ghana.

~E: The Volta River is not the longest river in Ghana.

F: Students who revise regularly perform better in exams.

~F: Students who revise regularly do not perform better in exams.

G:

~G:

Are declarative statements always true?

Declarative statements can either be negative or positive. A negative declarative statement can be true or false or debatable.

Example1.2:

1. B: The capital city of Ghana is Accra.

2. This is a true statement because Accra is indeed the capital city of Ghana.

3. Q: All mammals can fly.

4. This is a false statement because most mammals cannot fly—only bats can.

5. R: Students perform better when they are taught with digital tools like projectors and GeoGebra. This is a debatable statement. While digital tools can enhance understanding, performance also depends on students’ interest, the teacher’s skill, and other factors.

Conjunctions and Disjunctions

Instruction:

Think about your daily school life. Write down two activities that often happen in your classroom or school environment.

For example:

• Doing homework

• Sweeping the classroom Now form two simple statements based on your list. For example:

1. P: I did my homework.

2. Q: I swept the classroom.

Combine the statements using logical connectives: Have you got something like this?:

• I did my homework and swept the classroom.

• I did my homework or swept the classroom.

Conjunctions A conjunction connects two statements and is true only if both statements are true.

A: Kofi owns a bicycle, Q: He rides it to school These simple declarative statements can be connected with words (connectives) such as and, or, not, and if ... then to form compound statements.

We can join two statements to get compound statements which can be a conjunction or a disjunction.

Examples of Conjunctions

1. “ Kofi owns a bicycle and he rides it to school.” Making the statement a conjunction.

2. “The teacher marked the scripts and recorded the scores.” Making the statement a conjunction.

3. “Yaw is a carpenter and he lives in Tamale.” Making the statement a conjunction.

A conjunction is true if both statements are true, otherwise, it is a false conjunction.

Both parts need to be true for the conjunction to be true.

Disjunctions A disjunction connects two statements and is true if at least one of the statements is true.

Examples of Disjunctions

1. “Kofi will eat rice or he will eat yam.” This is a disjunction.

2. “It is raining or the sun is shining.” This is a disjunction.

3. “You can travel by bus or you can travel by train.” This is a disjunction.

A disjunction is considered true when at least one part of the statement is true. This means that even if only one of the statements is correct, the whole disjunction is still true.

Using the symbols “∧” and “∨” Consider the use of “and” and “or” in forming compound statements. We denote “and” with “∧” and “or” with “∨”

Example 1.3

1. Determine whether each combined statement is True or False based on the facts:

A: Water boils at 100°C. (True) B: Water freezes at 0°C. (True)

Solution

A ∧B: Water boils at 100°C and water freezes at 0°C (True conjunction) A ∨ B: Water boils at 100°C or water freezes at 0°C (True disjunction)

2. Determine whether each combined statement is True or False based on the facts:

P: Kumasi is the capital of Ghana Q: Accra is the capital of Ghana

Solution

P ∧ Q: Kumasi is the capital of Ghana and Accra is the capital of Ghana (False conjunction) P ∨ Q: Kumasi is the capital of Ghana or Accra is the capital of Ghana (True disjunction)

Activity 1.1 Understanding Logical Reasoning

Copy the table below.

1. Read each sentence carefully and fill in the table.

Sentence Type of Statement

(Interrogative / Declarative) Is it True or False?

Rewrite it in the Opposite Form Do you attend school every weekday?

I attend school every weekday.

We don’t learn mathematics in school.

The headteacher is a pupil.

Are exams fun for students?

2. Write two short sentences about your school in your exercise book:

• One sentence must be true.

• One sentence must be false.

• Underline the sentence if it says something good (positive) about the school.

• Put a star (*) next to the sentence if it says something bad (negative).

3. In one or two sentences, explain how you can use logical thinking to tell which sentence is true and which one is false.

Activity 1.2 Investigating How Real-Life Variables Change and Connect Using Logic You are a data collector in a farming town in Ghana. Farmers say the price of maize goes up when the harvest is poor. But is that always true? Now it’s your turn to investigate.

Step 1: Collect or use this sample data

Table 1.2

Year Amount of Maize Harvested (bags) Price of Maize per bag (GHS) 2020 150 100 2021 130 110 2022 110 125 2023 170 95

Step 2: Answer these questions (write in full sentences):

1. Does the amount of maize harvested always increase or decrease?

2. Does the price per bag always move in the same direction as harvest amount?

3. Can you say that “If maize harvest increases, price increases”? Is this always true?

Step 3: Use logical connectors to describe what you see.

Complete these statements in your notebook:

a. The maize harvest was low, and the price went up.

b. The maize harvest increased, or the price decreased.

c. The maize harvest was poor, and the price remained the same.

d. The price of maize changed, but the harvest did not.

Which of these sentences show a clear connection? Which ones show that two things can happen together or not?

Activity 1.3 Thinking Logically About Real-Life Issues

In a social studies class discussion, two groups shared different opinions about whether plastic bags should be banned in Ghana.

The boys said: “If the government bans plastic bags, it will help reduce flooding in Accra during the rainy season.”

The girls argued: “Banning plastic bags will create problems for traders and increase waste in other forms.”

Now it’s your turn to judge using logical reasoning

Step 1: Examine each view carefully.

a. Boys’ opinion

• What is the cause they believe in?

• What effect do they expect?

• Can you think of real examples where this has happened?

• Why might it not always work the way they think?

b. Girls’ opinion

• What problem do they think the ban might create?

• In what ways could this affect market women, food sellers, or shoppers?

• Is their view certain to happen, or is it just a possibility?

Step 2: Write a short paragraph (4–6 lines):

Provide your answers to the questions below:

• Can banning plastic bags help reduce flooding?

• What challenges might arise if plastic bags are banned?

• Can people switch easily to alternatives?

• Give at least one real or imagined example from Ghana (e.g. Makola, Kejetia, Tamale Central Market, etc.) to support your thinking.

Step 3: Final Logical Question

Can two people have different opinions and still be logical?

Write your answer with an explanation. Remember: Logical reasoning doesn’t always mean only one answer is correct — it means using evidence and careful thinking to support your view.

Real-World Applications of Logical Reasoning

Logical reasoning helps us think clearly and make good decisions. We use it every day

— when solving maths problems, choosing what to buy, planning our time, or solving problems. It helps us understand situations better and find the best solutions in school, at home, and in the community.

Take a close look at the scenario below:

Doing so will help you understand how logical reasoning can be applied in daily life.

Scenario Ama and her younger brother Kwame need to buy ingredients for a family meal from the local market in Kete - Krachi. They have only GH¢50 to spend and want to buy tomatoes, onions, and some fresh fish.

Situation Description

Budget Constraint

Ama says: “We have GH¢50 total. Tomatoes cost GH¢10 per heap, onions cost GH¢7 per heap, and fresh fish costs GH¢15 a kilo.”

Decision Making Using Logical Reasoning

Kwame suggests buying 2 heaps of tomatoes (GH¢20), 3 heaps of onions (GH¢21), and 1 kilo of fish (GH¢15).

Ama calculates: 20 + 21 + 15 = GH¢56, which is over their budget.

Ama then suggests adjusting the quantities. She reasons:

“If we buy 1 heap of tomatoes (GH¢10), 2 heaps of onions (GH¢14), and 2 kilos of fish (GH¢30), we spend exactly GH¢54 — still above budget.”

Kwame proposes: “What if we buy 1 heap of tomatoes, 1 heap of onions, and 2 kilos of fish?”

Ama calculates: 10 + 7 + 30 = GH¢47, which fits the budget with 3 cedis remaining.

Logical reasoning here They decide to compare quantities and prices logically to get the best combination that meets the budget and food needs.

Conclusion: Logical reasoning helps Ama and Kwame figure out the best way to spend their money without running out of money.

Example 1.4

Consider the following statements M: A person wears a helmet N: The person is safe while riding a motorbike Tasks

1. Represent the statements M and N on a Venn diagram.

2. Write an implication from the two statements (e.g., If M, then N), and state whether the implication is true or false. Kwame wore a helmet but still got injured in an accident.

Solution

U = {all motorbike riders} M = A person wears a helmet N = The person is safe while riding a motorbike

Example 1.5

Consider the following statements:

E: People who speak Twi fluently are Ghanaians.

F: People who attend basic school in Ghana speak Twi fluently.

a. Draw a Venn diagram to illustrate the above statements.

b. Deduce which of the following statements are valid conclusions based on E and F.

State whether each statement is true or false:

i. Kojo attends basic school in Ghana ⇒ Kojo is Ghanaian.

ii. Abena speaks Twi fluently ⇒ Abena attended basic school in Ghana.

iii. Yaw is Ghanaian ⇒ Yaw speaks Twi fluently.

Solution

Let U = persons T = people who speak Twi fluently B = people who attend basic school in Ghana G = Ghanaians

Figure 1.2

i. Valid – True statement because B is a subset of G

ii. Invalid – False statement because while all basic school attendees speak Twi (Q ⊆ P), not all Twi speakers necessarily attended basic school in Ghana. So, T is not a subset of B.

iii. Invalid – False because not all Ghanaians (G) necessarily speak Twi fluently (T). There are Ghanaians who may not speak Twi, so G is not a subset of T. members of G belong to T

Activity 1.4 Investigating the Use of Logical Reasoning in the Real World Title: Be the Change: Investigating How to Save Water and Electricity in My School Background In many schools in Ghana, water and electricity are often wasted without people noticing. Logical reasoning helps us investigate the problem, find out the causes, and choose the best ways to solve it.

Task for you You are to carry out a short investigation on how water and electricity are used or wasted in your school. You will use a checklist/questionnaire to collect your data, then analyse your findings and write a short report with suggested solutions.

Steps to Follow

Step 1: Observation and Data Collection

Use the checklist below to observe five (5) different places in your school. Tick (✓) the boxes that apply.

Table 1.3: Observation Checklist / Questionnaire Location Observed (e.g., Classroom, Bathroom, Dormitory, Kitchen)

1. Are lights/fans left on when the rooms are not in use?

2. Are taps left running or dripping?

3. Are people using too much water for simple tasks (e.g., washing hands)?

4. Are electric appliances used unnecessarily (e.g., TV/radio always on)?

5. Could this be avoided?

(Yes/No) Classroom Bathroom

Dormitory Kitchen

Step 2: Analyse Your Data

Write your answers to the following questions as part of your findings.

1. Which location showed the most waste?

2. What type of resource was mostly wasted — water or electricity?

3. What was the most common cause of waste you observed?

4. Which things can be changed easily, and which need more planning?

Step 3: Write Your Findings and Suggestions

Write a short report (6–8 lines) including:

• A summary of your findings

• Two logical suggestions to help reduce the waste of water or electricity

• One example of how you will personally help make the change at school

Note: At home, try doing a similar check to find out if there’s any waste of water or electricity and think of ways to fix it.

Intelligent guessing If one ball of banku costs GH¢5, how much would 2 balls cost?

Solution: Estimated answer will be GH¢10

Example 1.6

Observe the pattern and make a guess of the next 3 numbers of cocoa seedlings planted each week:

5, 9, 13, 17, 21, … A young farmer in Ghana is expanding his cocoa farm by planting more seedlings every week. He starts with 5 seedlings and increases the number each week using a fixed pattern.

Your Task

• Guess the number of seedlings to be planted in the next 3 weeks.

• Justify your guess based on the pattern.

• Provide a general rule that can be used to predict how many seedlings he will plant in any given week.

Solution

Find the difference between consecutive terms:

So, it’s a pattern where each term increases by 4

• 9 − 5 = 4

• 13 − 9 = 4

• 17 − 13 = 4

• 21 − 17 = 4 Therefore, the number of seedlings for each of the next three weeks will be:

• 21 + 4 = 25

• 25 + 4 = 29

• 29 + 4 = 33 Justification The farmer adds 4 more seedlings each week. That is, the number of seedlings increases in equal steps.

Therefore, the number of seedlings in the n-th week can be expressed by the rule 4n + 1.

Decision making in solving problems Decision making is a logical reasoning process that helps individuals choose the best possible solution among available options when solving a problem. It involves evaluating facts, analysing options, predicting outcomes and selecting the most appropriate course of action.

Look at the scenario below carefully to understand how logical reasoning is used to make a logical decision.

Scenario Mr. Yeboah is a farmer in Ejura who wants to increase his maize yield this year. He has two fertiliser options:

• Option A: An organic fertiliser made from animal manure — it is cheaper and environmentally friendly, but takes longer to work.

• Option B: A chemical fertiliser — it acts faster and can produce higher yields, but it is more expensive and may reduce soil quality over time.

To make a logical decision, Mr. Yeboah

1. Checks the facts — how much each fertiliser costs and how fast they work.

2. Analyses his options — which one fits his budget and timeline before the rainy season starts.

3. Predicts the outcomes — more profit in the short term (Option B) vs healthier soil in the long term (Option A).

4. Chooses the best solution — based on his goal to earn more money quickly and reinvest, he selects Option B for this farming season.

Example 1.7:

Obaa Yaa Akos is at the Dambai market and feels very hungry. She has GH¢ 15 and has only one hour before she must catch her bus to Chinderi. She sees two popular chop bars nearby:

A: Ma Vida Chop Bar is very busy. She hears people say the food is tasty, but she may have to wait 20 –30 minutes to get served.

B: Auntie Esi’s Chop Bar is less crowded. There’s no queue, but her friend once said the food there gave him a stomach upset.

Using intelligent guessing and logical reasoning, what should Obaa Yaa Akos do — wait at Ma Vidas’ chop bar, eat quickly at Auntie Esi’s, or find a snack nearby instead?

Support your decision with valid arguments.

Solution:

Eating at Ma Vida’s Chop Bar means Obaa Yaa Akos gets quality and satisfying food, which would be to her advantage. However, the long queue poses a risk of delay, and she may miss her bus.

Auntie Esi’s Chop Bar offers faster service, but the risk of food poisoning could affect Obaa Yaa Akos’s comfort during her journey.

Looking for a snack nearby will save time and reduce the health risk, allowing Obaa Yaa Akos to catch her bus on time.

There are many snack options in a busy place like Dambai, so it is unlikely that she will be unable to find something affordable and satisfying.

Obaa Yaa Akos has a better chance of being safe, on time, and less stressed if she finds a clean and quick snack.

Therefore, Obaa Yaa Akos should decide to buy a snack nearby instead of eating at either chop bar.

Example 1.8

Consider the following statements x: No plantain sellers are dishonest.

y: Some dishonest traders are very persuasive.

Using logical reasoning and Venn diagrams, determine whether the following statements are valid conclusions or false based on the given statements (x and y):

i. Afia is a plantain seller ⇒ Afia is not persuasive.

ii. Kojo is a plantain seller and persuasive ⇒ Kojo is not dishonest.

iii. Adwoa is not a plantain seller ⇒ Adwoa is dishonest and persuasive.

iv. Which of these people would you prefer to buy plantain from and why?

Solution

Let U = {All traders} Let D = {Dishonest traders} Let P = {Plantain sellers} Let V = {Persuasive traders} From P ∩ D = Ø (because no plantain seller is dishonest)

i. Afia is a plantain seller ⇒ Afia is not persuasive. We cannot know if this is valid, as some plantain sellers are persuasive, as there is an intersection between P and V.

ii. Kojo is a plantain seller and persuasive ⇒ Kojo is not dishonest. We know that not all plantain sellers are dishonest, so this is true.

iii. Adwoa is not a plantain seller ⇒ Adwoa is dishonest and persuasive. This is not valid, as it is not only plantain sellers who are not dishonest, and Adwoa.

iv. I would buy from Afia as he sells plantain and there is a chance they are not persuasive, whereas, buying from Kojo, I know they are persuasive. I would not buy from Adwoa as he does not sell plantain.

Example 1.9

The school bell rings when it is break time ⇒ It is break time because the bell just rang.

Verify the statement as valid or not.

Solution

• Does the school bell only ring at break time?

• Can the bell ring for other reasons (e.g., assembly, emergency, change of periods)?

• If the bell rang for a reason other than break time, then the conclusion that “it is break time” is not valid.

Therefore, if the first part of the statement is not always true (i.e., the bell rings only during break), then we cannot confidently say it is break time just because the bell rang.

So, the statement is not valid.

Example 1.10

Every Monday the students wear their school uniforms ⇒ Today is Monday, so the students are wearing their school uniforms. Verify the statement as valid or not.

Solution

• Is it true that students always wear school uniforms on Mondays?

• Is today indeed Monday?

• Are the students wearing their uniforms today as expected?

Since all the conditions are true and consistent with the statement, the conclusion is logically valid. Therefore, the statement is valid.

Investigating Arguments

Investigating arguments involves carefully analysing statements or claims to determine whether they are logically valid, sound or false. It requires examining how conclusions are drawn from given premises and whether those premises support the conclusion.

The steps involved in solving a problem may include:

1. Clearly defining the purpose of the investigation,

2. Collecting relevant information through methods such as interviews or surveys,

3. Examining the data to uncover patterns and insights, and

4. Drawing a conclusion based on the findings.

In a similar vein, Jerome Bruner suggests that effective problem-solving involves first grasping the nature of the problem, then creating a strategy to tackle it, putting that strategy into action, and finally reflecting on the entire process (Okafor, 2019).

Some challenging situations that might require a

solution

1. You have run out of data but need to submit an assignment online before midnight.

2. Your roommate plays loud music while you are trying to study for an important test.

3. You missed an important class because of a family emergency and now feel behind.

4. There is a misunderstanding between you and a close friend, and it is affecting group work.

5. You lost your student ID and need it to access the library and exam hall.

6. Your electricity has been disconnected, and you have a major take-home test due.

7. Your class project partner is not contributing to the work, but your grades depend on teamwork.

8. Heavy rains have flooded your area, making it hard to get to school on time.

Example 1.11

It is said that students spend about GH¢2 daily on school lunch.

Investigate the statement that ‘Everyone in our class spends exactly GH¢2 on lunch each day’.

Solution

Ask 10 students how much they spent on lunch today:

Amounts (in GH¢): 2, 2.5, 1.5, 2, 2, 2, 2, 1.5, 2.5, 2

Step 1: Find the total amount spent:

2 + 2.5 + 1.5 + 2 + 2 + 2 + 2 + 1.5 + 2.5 + 2 = 20

Step 2: Find the mean amount:

Mean The average amount spent is GH¢2, which matches the general belief.

However, not all students spent exactly GH¢2 with some spending more, others less.

So, the claim that “everyone spends exactly GH¢2” is not valid, but the average spending of GH¢2 is accurate and supported by the data.

In mathematics, variation refers to how one quantity changes in relation to another. It helps us understand relationships between variables—how a change in one affects the other.

In this lesson, we are going to look at four types of variation:

1. Direct Variation

2. Inverse Variation

3. Joint Variation

4. Partial Variation

Direct Variation

Direct variation occurs when one variable increases, or decreases, in direct proportion to another. That is, as one value goes up, the other goes up too—or when one goes down, the other goes down as well.

Mathematical Representation

If y varies directly as x, then:

y x α or y = kx where k is the constant of proportionality.

Example 1.12

In your community, a sachet water vendor sells one sachet of water for GH¢0.50. The cost of the water depends directly on the number of sachets bought.

Complete the table below by calculating the cost of different quantities of sachet water.

Figure 1.4

Table 1.5

Number of Sachets (x) Cost (GH¢) (y)

1 2 5 10 20 Use your results to:

i. Identify the constant of variation .

ii. Write the equation that relates cost to quantity .

iii. Work out how much 50 sachets would cost.

iv. Determine how many sachets one can buy with GH¢7.00.

Solution:

Table 1.6: Completed table Number of Sachets (x) Cost (GH¢) (y) 1 0.50 2 1.00 5 2.50 10 5.00 20 10.00

i. Finding the constant of variation:

We know that when = 1, = 0.5, so we can substitute these values into the equation to find the constant .

ii.

iii. Cost of 50 sachets ¢ iv.

This example shows how direct variation applies in everyday life

Example 1.13

If varies directly as , and when .

Find the value of when

Solution

From the direct variation formula: y = kx

Step 1: Find the value of the constant k by putting

Step 2: Use the value of k = 3 to find y when x = 8 Inverse Variation Inverse variation (or inverse proportion) is a relationship between two variables where as one increases, the other decreases in such a way that their product remains constant.

Mathematical Representation

If y varies inversely as x, then:

1 y x α or k y x = where k is the constant of proportionality.

Example 1.14

Figure 1.5

In your school/community, fetching water from the borehole is a common chore. It usually takes one student 30 minutes to fill 10 buckets. But when more students help, the time it takes reduces.

Use the table below to calculate the time it takes based on different numbers of students working together.

Table 1.7

Number of Students (x) Time (minutes) (y)

1 30 2 3 5 6 Use the inverse variation formula:

a. Calculate the constant k, then complete the table.

b. What happens to the time when more students are involved?

c. Predict:

i. How long will it take 10 students to fill the 10 buckets?

ii. How many students are needed to complete the task in 6 minutes?

Solution

a. Step 1: find the constant k

Step 2: Fill the table using Completed table Number of Students (x) Time (minutes) (y) 1 30 2 15 3 10 5 6 6 5

b. When more students are involved, the time taken to complete the task decreases.

This is because the work is shared among more people, so each person does less, and the task is finished faster.

c.

i. For 10 students:

ii. For 6 minutes:

Example 1.15

It takes 4 students 12 minutes to clean a classroom. If the time taken varies inversely with the number of students, how long will it take 6 students to clean the same classroom?

Solution

Since time t varies inversely with the number of students s, we use the formula:

Step 1: Find the constant k:

k = 48

Step 2: Find the time for 6 students:

Therefore, it will take 6 students 8 minutes to clean the same classroom.

Using Proportional Reasoning

Proportional reasoning involves using the concept of ratios and proportions to solve problems. It helps us compare two quantities and find unknown values when one part of a relationship changes.

a. In direct variation, the ratio y x is constant.

b. In inverse variation, the product xy is constant.

Table 1.8: Table Comparison

Type of Variation Relationship General Form Constant Quantity

Direct Variation y ∝ x y = kx y x = k Inverse Variation y ∝ 1 x y = k x xy = k Generalisations from Variation By investigating different scenarios, students can generalise:

a. In direct variation, doubling one variable doubles the other.

b. In inverse variation, if one variable is doubled, the other is halved (assuming a constant product).

c. The graphs of direct variation are straight lines through the origin.

d. The graphs of inverse variation are rectangular hyperbolas, or a ‘reciprocal graph’.

Figure 1.6: Graph of direct variation

Figure 1.7: Graph of inverse variation

Example 1.16

A textbook costs GH¢ 75.00. The total cost C varies directly with the number of textbooks n.

a. Express the cost C in terms of n.

b. If a school pays GH¢ 525.00, how many textbooks did they buy?

Solution

a. Since cost C varies directly with the number of textbooks n:

C ∝ n. This implies C = kn ⇒ k = 75 Therefore C = 75n

b. Given:

Substitute into the formula:

n = 7 The school bought 7 textbooks for GH¢525.00.

Example 1.17

The time (T) taken to fill a water tank is inversely proportional to the rate (R) at which water flows into the tank. If the rate is 10 litres per minute, it takes 30 minutes to fill the tank.

a. Derive the equation and find k (the constant of proportionality).

b. Calculate the time it will take to fill the tank if the water flows at 15litres per minute.

c. Debate: Why might this inverse model fail in areas with low or fluctuating water pressure?

Solution

a. Deriving the equation Since time T is inversely proportional to the rate R, Given:

T = 30 minutes, R=10 litres/min Substitute into the equation to find k:

So, the equation is:

b. Calculate the length of time it takes to fill the tank when the water flow rate is 15 litres per minute It will take 20 minutes to fill the tank at a rate of 15 litres per minute.

c. Debate: Why might this inverse model fail in areas with low or fluctuating water pressure?

• The model assumes a constant rate of water flow.

• In reality, water pressure can drop or fluctuate, especially during peak hours or in dry seasons.

• If pressure reduces during filling, the flow rate drops, and the time increases—violating the assumption of a fixed inverse relationship.

• Therefore, in areas with unstable supply, the inverse model may not accurately predict the filling time.

Joint variation

Activity 1.5 Understanding Joint Variation

Objective: Discover how the strength of a paper bridge depends on two variables (width and thickness) to understand joint variation.

Materials needed

• 3 sheets of paper (same size)

• 10–15 coins (or small weights like paper clips)

• Ruler

• Tape (optional, to secure bridge ends) Instructions

1. Bridge A (Control)

• Fold 1 sheet to 5 cm width.

• Place between desks (15 cm gap).

• Add coins until collapse. Record:

• Max coins = _____

2. Bridge B (2x Width)

• Fold 1 sheet to 10 cm width.

• Repeat test. Record:

• Max coins = _____

3. Bridge C (2x Thickness)

• Stack 2 sheets, fold to 5 cm width.

• Repeat test. Record:

• Max coins = _____

Table 1.9: Data table Bridge Width (cm) Thickness (sheets) Max Coins Held A 5 1 B 10 1 C 5 2 Answer these questions after the experiment

1. Which bridge held the most coins? Why?

2. How did increasing width affect strength? How did increasing thickness affect it?

3. If strength depends on BOTH width and thickness, what maths term could describe this?

In joint variation, a variable depends on two or more other variables simultaneously — combining direct and sometimes inverse variation.

For example, the volume of bags of maize (V) a farmer can harvest varies jointly with the size of the farmland (F) in acres and the amount of rainfall (R) in mm during the season.

Example 1.18

The total wages (W) paid to farm workers vary jointly with the number of workers (n) and the number of days (d) worked. If 10 workers working for 6 days are paid GH¢3,000.00.

a. Find the constant of variation (k).

b. How much will be paid to 8 workers for 5 days?

Solution

a. W ∞ nd W = knd Given:

⇒ ⇒ b.

W = = GH¢2,000.00 Therefore, GH¢2,000.00 will be paid to 8 workers for 5 days Variation 2: Introduction to Partial Variation In daily life, there are many situations where the total cost or value does not start at zero but includes both a fixed charge and a variable component. This is known as partial variation. Learning how this works helps manage budgets, plan services, and analyse real-world data. Partial variation is similar to direct variation but includes an added constant value.

Understanding Partial Variation

Partial variation happens when a quantity is made up of two parts.

• A fixed amount (also called a constant or base value), and

• A variable portion that changes directly with another quantity.

This can be described with the equation: y = kx+c Where:

• y is the outcome (dependent variable),

• x is the input (independent variable),

• k is the rate at which y changes with x,

• c is the starting or fixed value.

This formula represents a straight-line graph that does not pass through the origin, showing a linear relationship with a constant starting point.

Example 1.19

In a school in Ghana, a student is charged a fixed amount of GH¢10.00 every week for maintenance, plus an additional GH¢2.00 for each hour they use the school’s computer lab.

a. Write a mathematical equation that represents the total amount (T) the student pays in a week in terms of the number of hours (h) they use the computer lab.

b. If the student uses the computer lab for 5 hours in a week, how much will they pay in total?

c. What part of this variation is constant, and what part varies with the number of hours?

Solution

a. The total amount (T) is made up of a fixed charge and a variable charge that depends on hours used.

Fixed charge = GH¢10.00 Variable charge per hour = GH¢2.00 Therefore, the equation is:

T = 10 + 2h

b. If the student uses the computer lab for 5 hours:

T = 10 + 2(5) T = 10 + 10 T = GH¢20.00 So, the student will pay GH¢20.00 in total.

c. The constant part of the variation is the GH¢10.00 fixed maintenance fee.

The part that varies is the GH¢2.00 per hour, which depends on the number of hours (h) used.

Example 1.20

A mobile money vendor in Ghana charges a fixed fee of GH¢3.00 for every withdrawal transaction, plus GH¢0.50 for each GH¢100.00 withdrawn.

a. Write an equation that models the total charge (C) for a withdrawal of GH¢x.

b. Find the total charge if a customer withdraws GH¢500.00.

c. Identify the constant part and the variable part of the variation.

Solution

a. The total charge (C) includes a fixed fee and a variable fee based on the amount withdrawn.

Fixed fee = GH¢3.00 Variable fee = GH¢0.50 for every GH¢100 withdrawn, which is Therefore, the equation is:

C = 3 + 0.50 C = 3 + 0.005x

b. For a withdrawal of GH¢500.00:

C = 3 + 0.005 × 500 C = 3 + 2.5 C = GH¢5.50

c. The constant part is the GH¢3.00 fixed fee.

The variable part is GH¢0.005 times the amount withdrawn x, which depends on the withdrawal amount.

Activity 1.10 Planning a Three-Day Inter-SHS STEM Fair at Your School Your school has been selected to host a STEM Fair with participants from 3 different SHS levels. You are in charge of the Planning and Budget Committee.

Use your knowledge from SHS 1 to SHS 3 to solve the tasks and prepare a proposal.

1. You have data on 120 students: 60 study Mathematics, 50 study ICT and 30 study both.

a. Use a Venn diagram to show the distribution.

b. How many study only Mathematics? Only ICT? Neither?

c. Write the set equation using De Morgan’s Law to represent the group not studying either subject.

d. Write two declarative and two interrogative statements based on this data.

e. Identify a positive and negative statement from your discussion.

f. Construct one valid and one faulty logical argument about students who study ICT.

2. Out of a GH¢3,000.00 sponsorship package:

will be spent on materials, 20% on refreshments, The rest on certificates and souvenirs.

a. Calculate the amount to be spent on each category.

b. Show connections between the fraction and percentage amounts.

c. Apply additive and multiplicative inverse to check allocations using model charts.

3. To create a banner, the area is measured as m2

a. Simplify the area and express it in its simplest surd form.

b. If the printing company gives a discount using the formula:

Discount , where x is the number of banners.

Calculate the discount for 100 banners.

4. You are budgeting for 3 items: printing, feeding, and transport in a ratio 4:3:3.

a. If the total cost is GH¢2,000, calculate each item’s share.

b. Calculate the rate if printing costs GH¢800 for 4 hours of service.

5. During a STEM Fair, water sachets are distributed to guests for GH¢0.50 per sachet. Also, one volunteer can fetch 10 buckets of water in 30 minutes.

Answer the following questions

a. The cost of water sachets varies directly with the number of sachets distributed.

b. Calculate the total cost of 80 sachets of water.

c. The time taken to fetch water varies inversely with the number of students fetching the water.

d. If one volunteer fetches 10 buckets of water in 30 minutes, determine the time it will take 5 students to fetch the same number of buckets.

1. Analyse these statements and label each as True, False, or Debatable.

i. Nigeria is the most populous country in Africa.

ii. If x is an even number, then x + 1 is odd.

iii. Ashanti Region borders the Upper East Region.

iv. Ghana became a republic in 1960.

v. Science is more useful than Social Studies.

vi. Every rhombus is a parallelogram.

vii. All square numbers are even.

viii. If y < 0, then y – 3 < 0.

ix. The angles on a straight line add up to 180°.

2. Write a conjunction and a disjunction for each pair of statements.

State whether each is “True” or “False”

a. P: Multiples of 5 end in 0 or 5.

b. Q: All multiples of 5 are even numbers.

c. R: All quadrilaterals have four sides.

d. S: Some quadrilaterals have equal angles.

e. T: All mammals breathe through lungs.

f. U: Whales are mammals.

3. A farmer in the Northern Region wants to grow groundnuts and yam on her 12-acre land. She must grow at least 4 acres of groundnuts and at least 3 acres of yam. Groundnuts yield 1.5 tons per acre, and yam yields 2.5 tons per acre. How many acres of each crop should she plant to maximize her total yield and why?

4. Consider these two statements:

• Statement A: Respecting elders is a sign of discipline.

• Statement B: Discipline is a core value in many Ghanaian homes.

Examine the conclusions and label each as “Definitely True”, “Possibly True”, or “False”.

a. Efua respects elders ⇒ Can we conclude that she is disciplined?

b. Kofi is disciplined ⇒ Can we conclude that he respects elders?

c. Amma is from a Ghanaian home ⇒ Can we conclude that she is disciplined?

5. A farmer buys fertiliser at GH¢60.00 for 3 bags. How much will 8 bags cost, assuming the cost varies directly with the number of bags?

6. The distance a car travels is directly proportional to the time when moving at constant speed. If it covers 120 km in 2 hours, how far will it go in 5 hours?

7. A group of workers can complete a job in 12 days. If more workers join and the job now takes 6 days, how many workers were added, assuming the number of days varies inversely with the number of workers?

8. A pipe fills a tank in 4 hours. If a second pipe is added and both work together to fill the same tank in 2 hours, how long would it take the second pipe alone to fill the tank?

9. The total cost (C) of buying yams varies jointly as the number of tubers (t) and the price per tuber (p). If 10 tubers cost GH¢80.00 when each costs GH¢8.00, find the total cost of 15 tubers at GH¢6.00 each.

10. The work done varies inversely as the number of workers (n) and the number of hours (h) worked. If 5 workers complete a task in 6 hours, how many hours will it take 3 workers to do the same task?

11. A taxi driver in Accra charges a base fare of GH¢10.00, plus GH¢2.00 per kilometre. Write an equation for the total cost (C) of a trip and calculate the cost for a 7 km ride.

12. A data bundle costs GH¢5.00 for activation plus GH¢0.10 per MB used. How much will 250 MB of data cost?