Lesson 1: Ratios, Proportions, and Their Uses

👨🏽‍🏫 Introduction

Have you ever shared food with friends, mixed a fruit drink, or split transport fare? All these situations involve ratios and proportions. A ratio compares quantities, while a proportion ensures fairness in sharing or scaling.

In this lesson, we will explore how to work with ratios and proportions, and use them to solve real-life problems such as dividing money, scaling recipes, or calculating distances on a map.

By the end of this lesson, you will be able to:

• Write and simplify ratios
• Recognize and solve proportions
• Apply ratios and proportions to everyday situations

🧠 Key Concepts and Explanations

🔢 1. What Is a Ratio?

A ratio compares two or more quantities of the same type, using the format a:b or a to b.

Example: If there are 6 boys and 4 girls in a class, the ratio of boys to girls is
, which simplifies to
.

📏 2. Simplifying Ratios

To simplify a ratio, divide both terms by their Highest Common Factor (HCF).

Example:

📐 3. Proportion

A proportion shows that two ratios are equal.

Example:
is a proportion because both sides simplify to the same value.

To solve a proportion: Use cross multiplication:

🤝 4. Sharing in a Given Ratio

To share an amount T in a ratio a : b:

1. Add the parts:
   
2. Each person’s share =
   

Example: Share ₦1200 in the ratio 3:5.

• Total parts = 3 + 5 = 8
• First share = ₦
  
• Second share = ₦
  

🚗 5. Real-Life Application: Speed and Distance

If a car travels 60 km in 45 minutes, how far will it go in 120 minutes (2 hours) at the same speed?

Step-by-step:

• Set up the proportion:
  
• Cross-multiply:
  
• Solve:
  

🧮 Sample Problem Walkthroughs

1. Write and simplify the ratio of 10 boys to 15 girls:
   
2. Solve the proportion:
   
   Cross-multiply:
   
   Divide:
   
3. If 5 oranges cost ₦20, how much will 12 oranges cost?
   

✍🏽 Practice Exercises

🅰️ A. Simplify Ratios

1. Simplify the ratio 35:49
2. Express 8 red marbles to 12 green marbles in simplest form

🅱️ B. Proportion Problems

1. Solve:
   
2. If 3 pens cost ₦18, what is the cost of 7 pens?

🅾️ C. Word Problems

1. Share ₦1500 in the ratio 2:3
2. On a map with a scale of 1:100,000, the distance between two towns is 5.5 cm. What is the actual distance in kilometers?

✅ Answers and Explanations

🅰️ A. Simplify Ratios

• A1:
  
• A2:
  

🅱️ B. Proportion Problems

• B1:
  
• B2:
  

🅾️ C. Word Problems

• C1: Total parts = 2 + 3 = 5
  • First share =
    
  • Second share =
    
• C2:
  

📌 Recap

• ✅ A ratio compares two or more quantities
• ✅ Ratios can be simplified by dividing both terms by their HCF
• ✅ A proportion shows two ratios are equal and is solved using cross-multiplication
• ✅ Real-world applications include scaling, sharing, and interpreting maps

💭 Reflection Prompt

Have you ever shared airtime, split food, or adjusted ingredients in a recipe? Think about how using ratios or proportions helped — or could have helped — make things fair or accurate.