What Is the Difference Between Direct and Inverse Proportion Class 8 Explained

Definition of Direct and Inverse Proportion for Class 8

In Class 8 NCERT Mathematics, direct proportion means two variables increase or decrease together at the same rate. If one quantity doubles, the other also doubles. Mathematically, two quantities $x$ and $y$ are in direct proportion if:

$$y = kx$$

where $k$ is a constant called the constant of proportionality.

Inverse proportion means when one quantity increases, the other decreases such that their product remains constant. For two variables $x$ and $y$:

$$xy = k$$

Here, $k$ is again a constant. If $x$ doubles, $y$ becomes half.

Understanding these definitions helps solve many problems in the chapter "Direct and Inverse Proportions" in Class 8 NCERT.

How to Identify Direct and Inverse Proportion in Problems

To identify whether two quantities are directly or inversely proportional, look at how they change:

• Direct Proportion: When one quantity increases, the other also increases.
• Inverse Proportion: When one quantity increases, the other decreases.

Examples:

• Direct: Speed and distance travelled in fixed time.
• Inverse: Number of workers and time taken to complete a job.

Steps to check:

1. Calculate the ratio $\frac{y}{x}$ for direct proportion; it should be constant. 2. Calculate the product $xy$ for inverse proportion; it should be constant.

Example:

If $y = 10$ when $x = 2$, and $y = 20$ when $x = 4$, check:

$$\frac{10}{2} = 5, \quad \frac{20}{4} = 5$$

Since the ratio is constant, $x$ and $y$ are directly proportional.

Comparison Table: Direct vs Inverse Proportion

Here is a simple table to compare direct and inverse proportion:

This comparison helps Class 8 students quickly understand the core differences.

Worked Example: Direct Proportion

Problem: If 5 pens cost ₹150, what will be the cost of 8 pens?

Solution: Let cost be $y$ and number of pens be $x$.

Since cost and number of pens are directly proportional,

$$y = kx$$

Find $k$ using given values:

$$150 = k \times 5 \implies k = \frac{150}{5} = 30$$

Now find cost for 8 pens:

$$y = 30 \times 8 = 240$$

Answer: ₹240

This example shows how to apply direct proportion formula in Class 8 NCERT problems.

Worked Example: Inverse Proportion

Problem: 6 workers can complete a task in 10 days. How many days will 3 workers take to complete the same task?

Solution: Let number of workers be $x$ and days taken be $y$.

Since workers and days are inversely proportional,

$$xy = k$$

Find $k$:

$$6 \times 10 = 60$$

Find days for 3 workers:

$$3 \times y = 60 \implies y = \frac{60}{3} = 20$$

Answer: 20 days

This example helps Class 8 students understand inverse proportion in real-life problems.

Graphs of Direct and Inverse Proportion

Graphs visually represent the relationship between quantities.

• Direct Proportion Graph: A straight line passing through the origin (0,0). As $x$ increases, $y$ increases linearly.

• Inverse Proportion Graph: A curve called a hyperbola. As $x$ increases, $y$ decreases such that the product $xy$ remains constant.

Example:

For direct proportion $y = 2x$:

For inverse proportion $xy = 8$:

These graphs are important for Class 8 NCERT exams to interpret proportional relationships.