Variation

Two related variables $x$ and $y$ are said to be in direct variation (বা সরল ভেদে আছে) if an increase or decrease in $y$ causes a proportional increase or decrease in $x$ such that the ratio of their corresponding values remains constant.

• Mathematical Notation: $x \propto y$ (read as 'x varies directly as y' or 'x is proportional to y').
• Equational Form: $$x \propto y \iff \frac{x}{y} = k \iff \mathbf{x = ky}$$ where $k \neq 0$ is a non-zero real constant called the constant of variation (ভেদ ধ্রুবক).
• Finding the Constant $k$: If one set of corresponding values $(x_1, y_1)$ is known, $k = \frac{x_1}{y_1}$. This constant then governs all other pairs of values: $\frac{x_1}{y_1} = \frac{x_2}{y_2} = k$.
• Graphical Nature: The graph of $x = ky$ in the Cartesian coordinate plane is a straight line passing through the origin $(0, 0)$ with slope $k$.
• Real-World Examples:
  • Distance ($s$) traveled at constant velocity ($v$) varies directly as time ($t$): $s = vt \implies s \propto t$.
  • Cost ($C$) of purchasing identical notebooks varies directly as the quantity ($n$): $C \propto n$.
  • Circumference of a circle ($C$) varies directly as its radius ($r$): $C = (2\pi) r \implies C \propto r$ (here $k = 2\pi$).

Two related variables $x$ and $y$ are said to be in inverse variation (বা ব্যস্ত ভেদে আছে) if an increase in $y$ causes a proportional decrease in $x$, and vice versa, such that the product of their corresponding values remains constant.

• Mathematical Notation: $x \propto \frac{1}{y}$ (read as 'x varies inversely as y').
• Equational Form: $$x \propto \frac{1}{y} \iff x = k \left(\frac{1}{y}\right) \iff \mathbf{xy = k}$$ where $k \neq 0$ is the non-zero constant of variation.
• Finding the Constant: If $(x_1, y_1)$ is a known pair of values, $k = x_1 y_1$. For any other pair $(x_2, y_2)$: $$x_1 y_1 = x_2 y_2 = k \iff \frac{x_1}{x_2} = \frac{y_2}{y_1}$$
• Graphical Nature: The graph of $xy = k$ in the Cartesian plane is a rectangular hyperbola asymptotic to both axes.
• Real-World Examples:
  • Boyle's Law: At constant temperature, the pressure ($P$) of a fixed mass of gas varies inversely as its volume ($V$): $P \propto \frac{1}{V} \implies PV = k$.
  • Speed and Time: For a fixed journey distance $D$, speed ($v$) varies inversely as transit time ($t$): $v \propto \frac{1}{t} \implies vt = D$.
  • Workforce and Duration: To complete a fixed amount of construction work, the number of laborers ($M$) varies inversely as the number of days required ($D$): $M \propto \frac{1}{D} \implies MD = k$.

1. Let the initial values of the variables be $(x_1, y_1, z_1)$.
2. Suppose the variables change to new values $(x_2, y_2, z_2)$. We can conceptualize this change as occurring in two consecutive stages:
3. Stage 1: Keep $z$ constant at $z_1$, and allow $y$ to change from $y_1$ to $y_2$. Let the corresponding value of $x$ change from $x_1$ to an intermediate value $x'$.
   Since $x \propto y$ when $z$ is constant: $$\frac{x'}{x_1} = \frac{y_2}{y_1} \quad \dots \text{(Equation 1)}$$
4. Stage 2: Now keep $y$ constant at $y_2$, and allow $z$ to change from $z_1$ to $z_2$. Then $x$ changes from intermediate value $x'$ to final value $x_2$.
   Since $x \propto z$ when $y$ is constant: $$\frac{x_2}{x'} = \frac{z_2}{z_1} \quad \dots \text{(Equation 2)}$$
5. Combining Stages: Multiply Equation 1 and Equation 2: $$\frac{x'}{x_1} \times \frac{x_2}{x'} = \frac{y_2}{y_1} \times \frac{z_2}{z_1}$$ $$\frac{x_2}{x_1} = \frac{y_2 z_2}{y_1 z_1} \implies \frac{x_2}{y_2 z_2} = \frac{x_1}{y_1 z_1} = k \text{ (a constant)}$$
6. Therefore, for any simultaneous variation of $y$ and $z$: $$\frac{x}{yz} = k \implies \mathbf{x = kyz} \implies \mathbf{x \propto yz}$$

Extension to Inverse and Multiple Variations:
If $x \propto y$ (when $z$ is constant) and $x \propto \frac{1}{z}$ (when $y$ is constant), then $x \propto \frac{y}{z} \implies x = k \frac{y}{z}$.
Scientific Application (Ideal Gas Law): Charles's law gives $V \propto T$ (at constant $P$), and Boyle's law gives $V \propto \frac{1}{P}$ (at constant $T$). By joint variation, $V \propto \frac{T}{P} \implies PV = k T$!

In classic arithmetic problems solved via variation:

• Work done ($W$) varies directly as the number of workers ($M$) when days ($D$) are constant: $W \propto M$.
• Work done ($W$) varies directly as the working days ($D$) when workers ($M$) are constant: $W \propto D$.
• By the Theorem of Joint Variation: $$W \propto M \times D \implies \mathbf{W = k \cdot M \cdot D}$$
• To find the number of workers required: $M = \frac{W}{k D} \implies M \propto \frac{W}{D}$.

Many real-world expenses involve a fixed overhead plus a variable cost proportional to consumption or distance: