11.1. Introduction

Mohan prepares tea for himself and his sister. He uses 300 mL of water, 2 spoons of sugar, 1 spoon of tea leaves, and 50 mL of milk. How much quantity of each item will he need if he has to make tea for five persons?

In this chunk, we learn how to scale the ingredients for making tea from 2 servings to 5 servings. If Mohan originally uses 300 mL of water for 2 servings, to find out how much he needs for 5 servings, we scale the quantities based on the ratio of servings. This involves multiplying the original quantities by a factor of 2.5 (5 divided by 2).

Think of baking a cake. If a recipe calls for 2 eggs to make a small cake but you want to make a larger cake that serves 5, you simply multiply the number of eggs by 2.5 (the same scaling process).

If two students take 20 minutes to arrange chairs for an assembly, then how much time would five students take to do the same job?

This example illustrates the relationship between the number of workers and the time taken to complete a task. If 2 students take 20 minutes, we can use the concept of direct proportion to find out how much time it would take 5 students. More workers typically mean less time needed, in this case, the time taken will decrease as the number of students increases.

Imagine cleaning a big house. If one person takes all weekend to clean, 2 people could probably do it in half the time. Similarly, the more hands you have on a task, the quicker it can be completed.

We come across many such situations in our day-to-day life, where we need to see variation in one quantity bringing in variation in the other quantity: (i) If the number of articles purchased increases, the total cost also increases. (ii) More the money deposited in a bank, more is the interest earned. (iii) As the speed of a vehicle increases, the time taken to cover the same distance decreases. (iv) For a given job, more the number of workers, less will be the time taken to complete the work.

This section highlights various real-world instances where two quantities are related proportionally. For example, buying more items means spending more money; similarly, increased speed means decreased travel time. Each example shows how a change in one quantity leads to a predictable change in another, emphasizing the concept of proportionality.

Consider filling a gas tank—if the price of gas rises, the total cost to fill it also rises, illustrating direct proportion. Just like a water tank fills faster when there are more pipes attached, representing how inputs can affect outcomes.

How do we find out the quantity of each item needed by Mohan? Or, how do we find out the time five students take to complete the job? To answer such questions, we now study some concepts of variation.

This passage sets the stage for exploring whether quantities vary directly or inversely. It asks readers to consider methods for solving problems of proportionality and lays the groundwork for understanding variations. Key questions focus on determining how to adjust quantities as situations change.

Imagine you’re filling your car’s gas tank and you need to figure out how much it will cost to fill it up based on how empty it is. Understanding variations helps you make sense of what to expect based on different circumstances.

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