Chapter 11

11.1 Introduction

The chapter 'Direct and Inverse Proportions' begins with practical examples that students encounter in daily life, illustrating how changes in one quantity affect another. The introductory section presents two scenarios: Mohan preparing tea for himself and his sister using fixed quantities of ingredients, and students arranging chairs for an assembly. These examples highlight the concept of variation — how one quantity changes in response to another. For instance, if Mohan needs to prepare tea for five persons instead of two, how much of each ingredient is required? Similarly, if two students take 20 minutes to arrange chairs, how long will five students take? These questions prompt the study of proportional relationships. The section further provides everyday examples where quantities vary together: (i) Increase in the number of articles purchased leads to an increase in total cost. (ii) More money deposited in a bank results in more interest earned. (iii) Increase in vehicle speed decreases the time taken to cover a fixed distance. (iv) More workers reduce the time needed to complete a job. Such observations emphasize that changes in one quantity often cause changes in another. To analyze and quantify these relationships, the chapter introduces the concepts of direct and inverse proportion. Understanding these concepts enables students to solve problems involving proportional relationships efficiently.

• Variation in one quantity often causes variation in another.
• Direct and inverse proportions help quantify these relationships.
• Real-life examples include cost vs. quantity, speed vs. time, workers vs. time.
• The introductory examples set the stage for studying proportionality.
• Understanding proportions aids in solving practical problems.
• Observation of ratios and products helps identify types of proportion.
• 📌 Variation: Change in one quantity causing change in another.
• 📌 Direct Proportion: When two quantities increase or decrease together maintaining a constant ratio.
• 📌 Inverse Proportion: When one quantity increases while the other decreases such that their product remains constant.

11.2 Direct Proportion

Direct proportion describes a relationship between two quantities where both increase or decrease together such that their ratio remains constant. This means if one quantity doubles, the other also doubles; if one triples, the other triples, and so forth. The section begins with an example: If 1 kg of sugar costs ₹36, then 3 kg costs ₹108. This proportionality is tabulated to show that as the weight of sugar increases, the cost increases maintaining a constant ratio of cost to weight. Another example involves petrol consumption and distance travelled by a car. If 4 litres of petrol allow the car to travel 60 km, then 12 litres (three times) will allow it to travel 180 km (also three times). This shows that petrol consumed (x) and distance travelled (y) satisfy the relation x/y = k, a constant. Mathematically, two quantities x and y are in direct proportion if x/y = k or x = ky, where k is a positive constant. For two pairs of values (x1, y1) and (x2, y2), the relation x1/y1 = x2/y2 holds. The section also includes activities like measuring the angle turned by a clock's minute hand over time, showing that angle turned and time are directly proportional. Another activity compares the ages of a friend and their mother over time, illustrating that not all variables that increase together are in direct proportion. Students are encouraged to think of more examples and verify direct proportionality by checking if the ratio remains constant. The section concludes with solved examples involving cloth cost, shadows, weights, and uniform speed, applying the direct proportion formula to find unknown quantities. Understanding direct proportion helps in solving practical problems involving scaling, cost estimation, speed-distance-time relations, and map reading using scales.

• Direct proportion means two quantities increase or decrease together maintaining a constant ratio.
• Mathematically, x and y are directly proportional if x/y = k or x = ky.
• Examples include cost and quantity, petrol consumption and distance, angle turned and time.
• Activities help verify direct proportion by checking constant ratios.
• Not all quantities increasing together are directly proportional.
• Direct proportion formula: (x1/y1) = (x2/y2) is used to solve problems.
• 📌 Direct Proportion: Relation where ratio of two quantities is constant.
• 📌 Constant of Proportionality (k): The constant ratio x/y in direct proportion.