Video Lecture 1: Thinking in Proportion — Constant Ratios and the Signature of Direct Variation

# Video Lecture 1: Thinking in Proportion — Constant Ratios and the Signature of Direct Variation
**Speaker:** Prof. Arvind Swaminathan | **Duration:** ~05:50 mins

### Introduction: What is Direct Proportion?
If 1 notebook costs ₹40, how much do 2 notebooks cost? ₹80. 3 notebooks? ₹120. 10 notebooks? ₹400.
Notice what is happening: as the number of notebooks increases, the cost increases in **exact lockstep** such that their ratio remains constant:
$$\frac{\text{Cost}}{\text{Number of Notebooks}} = \frac{40}{1} = \frac{80}{2} = \frac{120}{3} = \mathbf{40}$$

### 1. Mathematical Definition of Direct Proportion:
Two quantities $x$ and $y$ are said to be in **direct proportion** if they increase or decrease together such that the ratio of their corresponding values remains constant:
$$\frac{x}{y} = k \quad \text{(a constant)}$$
Equivalently:
$$\mathbf{\frac{x_1}{y_1} = \frac{x_2}{y_2}} \quad \text{or} \quad \mathbf{\frac{x_1}{x_2} = \frac{y_1}{y_2}}$$

### 2. The Graphical Signature:
If you plot directly proportional quantities on a Cartesian coordinate grid with $x$ on the horizontal axis and $y$ on the vertical axis, the graph is ALWAYS a **straight line passing through the Origin $(0, 0)$**! The slope of the line equals the constant of proportionality $k$.