Understanding Direct Proportionality

When we say that one quantity is directly proportional to another, it means that as one quantity increases, the other quantity increases at the same rate, and as one quantity decreases, the other quantity decreases at the same rate. Mathematically, if P is directly proportional to Q, we write this as:

$\text{P} \propto \text{Q}$

This relationship can be expressed as an equation by introducing a constant, often denoted by $k$. This constant is called the proportionality constant.

$\text{P} = k\text{Q}$

Here, $k$ is a fixed number. To solve problems involving direct proportionality, we typically use the given information to first find the value of this constant $k$, and then use $k$ to find the unknown value.

Solving the Direct Proportionality Problem

The question states that P is directly proportional to Q. We are given that Q = 7 when P = 15. We need to find the value of P when Q = 14.

Step 1: Find the Proportionality Constant ($k$)

We use the initial pair of values (P = 15, Q = 7) and the direct proportionality equation $\text{P} = k\text{Q}$ to find the value of $k$.

Substitute the given values into the equation:

$15 = k \times 7$

To find $k$, we need to isolate $k$ by dividing both sides of the equation by 7:

$k = \frac{15}{7}$

So, the proportionality constant $k$ is $\frac{15}{7}$. The relationship between P and Q is $\text{P} = \frac{15}{7}\text{Q}$.

Step 2: Find P when Q = 14

Now that we know the value of $k$, we can use the equation $\text{P} = k\text{Q}$ to find P when Q = 14.

Substitute the value of $k$ and the new value of Q into the equation:

$\text{P} = \left(\frac{15}{7}\right) \times 14$

We can simplify the calculation:

$\text{P} = 15 \times \left(\frac{14}{7}\right)$

Since $\frac{14}{7} = 2$, we have:

$\text{P} = 15 \times 2$

$\text{P} = 30$

Therefore, when Q is 14, P is 30.

Summary of Calculation

Comparing with Options

We found that P = 30 when Q = 14. Let's check the given options:

• Option 1: 10
• Option 2: 20
• Option 3: 30
• Option 4: 22

Our calculated value, 30, matches Option 3.

Revision Table: Key Concepts in Direct Proportionality

Additional Information: Types of Variation

Besides direct proportionality, there are other types of variation you might encounter in mathematics:

• Inverse Proportionality: P is inversely proportional to Q if P is proportional to the reciprocal of Q. The equation is $\text{P} = \frac{k}{\text{Q}}$, or PQ = $k$. As one quantity increases, the other decreases.
• Joint Variation: P varies jointly as Q and R if P is directly proportional to the product of Q and R. The equation is $\text{P} = k\text{QR}$.
• Combined Variation: This involves a combination of direct, inverse, and joint variations. For example, P might vary directly as Q and inversely as R, which would be $\text{P} = \frac{k\text{Q}}{\text{R}}$.

Understanding these different types of variation helps in solving various problems in physics, engineering, and other fields.