Variation: Direct and Inverse

Introduction to Variation Concepts

In mathematics and science, the concept of variation is used to describe how the value of one quantity changes in relation to the value of another quantity. When a change in one quantity consistently corresponds to a predictable change in another quantity, we say that there is a variation between them.

Understanding variation allows us to build mathematical models that represent relationships observed in the real world and solve problems based on these relationships.

The two fundamental types of variation are:

Types of Variation

• Direct Variation: This occurs when two quantities change in the same direction. If one quantity increases, the other also increases proportionally, and vice versa. The ratio of the two quantities remains constant.
• Inverse Variation: This occurs when two quantities change in opposite directions. If one quantity increases, the other decreases proportionally, and vice versa. The product of the two quantities remains constant.

In any relationship involving variation, there is a core constant that defines the specific link between the quantities. This constant is called the constant of variation or the constant of proportionality. It acts as a factor that translates the change in one quantity into the corresponding change in the other.

For example, the relationship between the distance covered by a car and the time taken, when the speed is constant, is a form of variation. Similarly, the relationship between the number of workers and the time taken to complete a job illustrates another type of variation.

Studying variation provides a powerful framework for setting up equations and solving problems across various domains, including physics (like force, mass, and acceleration), economics (like demand and price), engineering (like pressure and volume), and everyday calculations involving rates and quantities.

Direct Variation and its Applications

Direct variation describes a relationship between two quantities where one quantity is a direct multiple of the other. In simpler terms, if $y$ varies directly as $x$, then as $x$ increases, $y$ increases proportionally, and as $x$ decreases, $y$ decreases proportionally. The ratio of $y$ to $x$ remains constant throughout their change.

Definition and Formula

If a quantity $y$ varies directly as a quantity $x$, we denote this relationship using the proportionality symbol ($\propto$) as:

To convert this proportionality into an equation, we introduce a non-zero constant, typically denoted by $k$. This constant is called the constant of direct variation or the constant of proportionality.

Equation (i) is the defining equation for direct variation. From this equation, provided $x \neq 0$, we can find the constant $k$:

This formula emphasizes that in direct variation, the ratio of the two quantities is always equal to the constant $k$ for any corresponding pair of values (except when $x=0$).

Relationship between Two Pairs of Values in Direct Variation

Consider two different pairs of corresponding values for quantities $x$ and $y$ that are in direct variation. Let the first pair be $(x_1, y_1)$ and the second pair be $(x_2, y_2)$.

According to the definition of direct variation ($y = kx$), for the first pair $(x_1, y_1)$:

And for the second pair $(x_2, y_2)$:

From equation (ii), assuming $x_1 \neq 0$, we can express the constant $k$ as:

Similarly, from equation (iii), assuming $x_2 \neq 0$, we can express the constant $k$ as:

Since $k$ is the same constant for any corresponding pair of values in a direct variation, we can equate these two expressions for $k$:

Equation (iv) is a fundamental relationship used to solve problems involving direct variation. If we are given three of the four quantities $(x_1, y_1, x_2, y_2)$, we can find the fourth one using this proportion.

This can also be written in the form of a proportion as $y_1:x_1 :: y_2:x_2$ or, more commonly, $x_1:y_1 :: x_2:y_2$ after applying Alternendo ($\frac{x_1}{y_1} = \frac{x_2}{y_2}$).

Graphical Representation

When the relationship $y = kx$ is plotted on a coordinate plane with $x$ on the horizontal axis and $y$ on the vertical axis, the graph is a straight line. Since $k$ is a non-zero constant, the equation $y=kx$ represents a linear equation of the form $y=mx+c$ where the y-intercept ($c$) is 0. Therefore, the graph of a direct variation is always a straight line that passes through the origin $(0,0)$. The slope of this line is equal to the constant of variation $k$.

Applications of Direct Variation in Real Life

Direct variation is a very common relationship observed in numerous situations:

1. Cost and Quantity: If you buy more quantity of an item at a fixed price per unit, the total cost increases proportionally. E.g., The total cost of apples is directly proportional to the number of apples purchased, assuming the price per apple is constant.

2. Distance and Time (at constant speed): If you travel at a constant speed, the distance you cover is directly proportional to the time you spend traveling. E.g., A car traveling at 60 km/h covers twice the distance in twice the time.

3. Work Done and Time (by fixed number of workers): If a fixed number of workers are working at a constant rate, the amount of work completed is directly proportional to the time they spend working. E.g., 5 workers can complete more tasks in 8 hours than in 4 hours.

4. Force and Acceleration (at constant mass): As per Newton's second law of motion ($F=ma$), if the mass of an object is kept constant, the force applied to it is directly proportional to the acceleration it experiences.

5. Simple Interest: Simple interest earned on a principal amount over a fixed period and at a fixed rate is directly proportional to the principal amount.

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Inverse Variation and its Applications

Inverse variation, also known as inverse proportion, describes a relationship between two quantities where if one quantity increases, the other quantity decreases proportionally, and vice versa. The defining characteristic of inverse variation is that the product of the two related quantities remains constant.

Definition and Formula

If a quantity $y$ varies inversely as a quantity $x$, we express this relationship using the proportionality symbol ($\propto$) and the reciprocal of $x$:

To convert this proportionality into an equation, we introduce a non-zero constant, $k$, called the constant of inverse variation or the constant of proportionality:

Here, $k$ is the constant. Provided $x \neq 0$ and $y \neq 0$, we can rearrange equation (i) by multiplying both sides by $x$ to find the constant $k$:

This formula shows that for any pair of corresponding non-zero values of $x$ and $y$ that are inversely related, their product is always equal to the same constant $k$.

Relationship between Two Pairs of Values in Inverse Variation

Consider two different pairs of corresponding values for quantities $x$ and $y$ that are related by inverse variation. Let the first pair be $(x_1, y_1)$ and the second pair be $(x_2, y_2)$.

According to the definition of inverse variation ($xy = k$), for the first pair $(x_1, y_1)$, their product equals the constant $k$:

Similarly, for the second pair $(x_2, y_2)$, their product also equals the same constant $k$:

Since both $x_1 y_1$ and $x_2 y_2$ are equal to the same constant $k$, we can equate them:

Equation (iv) is the fundamental equation used to solve problems involving inverse variation. If we are given any three of the four quantities $(x_1, y_1, x_2, y_2)$, we can use this equation to find the value of the fourth quantity.

This relationship can also be rearranged to form a proportion by dividing both sides by $x_2 y_1$ (assuming $x_2 \neq 0$ and $y_1 \neq 0$):

In ratio form, this is $x_1:x_2 :: y_2:y_1$. This shows that the ratio of the first quantity's values is proportional to the *inverse* ratio of the corresponding second quantity's values.

Graphical Representation

When the relationship $y = \frac{k}{x}$ (or $xy = k$) is plotted on a coordinate plane with $x$ on the horizontal axis and $y$ on the vertical axis, the graph is a curve called a hyperbola. For a positive constant $k$, the graph lies in the first and third quadrants. For a negative constant $k$, it lies in the second and fourth quadrants. The graph gets closer and closer to the x-axis and y-axis but never actually touches them. These axes are called asymptotes.

Applications of Inverse Variation in Real Life

Inverse variation relationships are frequently encountered in various fields:

1. Speed and Time (for a fixed distance): If the distance to be covered is constant, increasing the speed of travel reduces the time taken proportionally. For example, if you double your speed, you halve the time to reach the destination.

2. Number of Workers and Time (for a fixed amount of work): If a fixed amount of work needs to be completed, increasing the number of workers (assuming they all work at the same efficiency) decreases the time required. For example, if you double the number of workers, the job gets done in half the time.

3. Pressure and Volume of a Gas (at constant temperature): According to Boyle's Law, for a fixed amount of gas at a constant temperature, the pressure ($P$) is inversely proportional to its volume ($V$). This means $PV = k$, where $k$ is a constant.

4. Current and Resistance (at constant voltage): Ohm's Law states that for a constant voltage ($V$), the current ($I$) flowing through a circuit is inversely proportional to the resistance ($R$) ($I \propto \frac{1}{R}$ or $IR = V$).

5. Intensity of Light and Distance from Source: The intensity of light from a point source is inversely proportional to the square of the distance from the source. ($I \propto \frac{1}{d^2}$).

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Problems Combining Direct and Inverse Variation

In real-world scenarios, the relationship between quantities is often more complex than simple direct or inverse variation. Many times, a quantity may depend on several other quantities, varying directly with some and inversely with others simultaneously. This type of relationship is known as combined variation.

Combined Variation

When a quantity varies directly as one or more quantities and inversely as one or more other quantities, it is a case of combined variation.

Mathematically, if a quantity $z$ varies directly as $x$ and inversely as $y$, we can write the combined proportionality as:

To express this proportionality as an equation, we introduce a non-zero constant of variation, $k$:

From equation (i), assuming $x \neq 0$ and $y \neq 0$, we can solve for the constant $k$ by multiplying both sides by $\frac{y}{x}$:

This means that for any set of corresponding non-zero values $(x, y, z)$, the value of $\frac{zy}{x}$ will always be equal to the same constant $k$.

Relationship between Two Sets of Values in Combined Variation

Consider two different sets of corresponding values for quantities $x, y,$ and $z$ that are related by the combined variation $z \propto \frac{x}{y}$. Let the first set be $(x_1, y_1, z_1)$ and the second set be $(x_2, y_2, z_2)$.

Based on the definition $k = \frac{zy}{x}$, for the first set of values $(x_1, y_1, z_1)$, we have:

And for the second set of values $(x_2, y_2, z_2)$, we have:

Since both expressions are equal to the same constant $k$, we can equate them to get the relationship for solving problems:

This equation is particularly useful when you are given one complete set of values $(x_1, y_1, z_1)$ and two values from a second set $(x_2, y_2 \text{ or } x_2, z_2 \text{ or } y_2, z_2)$, and you need to find the third unknown value.

Joint Variation

Joint variation is a special case of combined variation where a quantity varies directly as the product of two or more other quantities. There is no inverse variation involved in the basic definition of joint variation.

If a quantity $z$ varies jointly as $x$ and $y$, we write the proportionality as:

Introducing the constant of variation $k$, the equation becomes:

From equation (v), assuming $x \neq 0$ and $y \neq 0$, we can solve for the constant $k$ by dividing both sides by $xy$:

This indicates that for any set of corresponding non-zero values $(x, y, z)$, the ratio $\frac{z}{xy}$ will always be equal to the same constant $k$.

Relationship between Two Sets of Values in Joint Variation

Consider two different sets of corresponding values $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ for quantities related by joint variation $z \propto xy$.

Based on the definition $k = \frac{z}{xy}$, for the first set of values $(x_1, y_1, z_1)$, we have:

And for the second set of values $(x_2, y_2, z_2)$, we have:

Equating the expressions for $k$ from equations (vi) and (vii), we get the relationship for solving problems:

This equation is used when you are given one complete set of values $(x_1, y_1, z_1)$ and two values from a second set, and need to find the third unknown value in a joint variation scenario.

Applications of Combined and Joint Variation

These types of variations appear in many formulas and practical problems:

1. Work Done: The total work done often varies jointly with the number of workers, the time they work, and their efficiency. If work is $W$, number of workers is $N$, hours per day is $H$, and days worked are $D$, and efficiency is $E$, then $W \propto N \times H \times D \times E$. If efficiency is constant, $W \propto NHD$, which is a joint variation. If we compare two scenarios with different workers, hours, and days for the same amount of work ($W_1=W_2$), then $N_1 H_1 D_1 = N_2 H_2 D_2$ (a common application derived from $k = NHD/W$ where $W$ is fixed).

2. Area of a Rectangle: The area ($A$) of a rectangle varies jointly as its length ($L$) and breadth ($B$). $A \propto LB \implies A = kLB$. If $k=1$, $A=LB$.

3. Volume of a Cylinder: The volume ($V$) of a cylinder varies jointly as the square of its radius ($r$) and its height ($h$). $V \propto r^2 h \implies V = k r^2 h$. If $k=\pi$, $V = \pi r^2 h$.

4. Force of Attraction (Gravitation): The force of gravitational attraction ($F$) between two objects varies jointly as their masses ($m_1, m_2$) and inversely as the square of the distance ($r$) between their centers. $F \propto \frac{m_1 m_2}{r^2} \implies F = G \frac{m_1 m_2}{r^2}$, where $G$ is the gravitational constant. This is a classic example of combined variation.

5. Electrical Resistance: The resistance ($R$) of a wire varies directly as its length ($L$) and inversely as its cross-sectional area ($A$). $R \propto \frac{L}{A} \implies R = \rho \frac{L}{A}$, where $\rho$ is the resistivity of the material. This is another example of combined variation.

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