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The concept of proportions is fundamental in mathematics, especially in topics like arithmetic, algebra, and geometry. among the various types of proportions, the third proportional plays a critical role in solving problems related to ratios and proportions. understanding the third proportional formula is essential for students in Class 10, Class 11, and Class 12 as it lays the groundwork for advanced mathematical concepts, including progression, similarity, and real-world applications.

What is the Third Proportional?

The third proportional is a value that is related to two given numbers in such a way that the ratio of the first number to the second number is equal to the ratio of the second number to the third number. In mathematical terms, if a, b, and c are three numbers, then c is the third proportional of a and b if:

a : b = b : c

This can also be written in fraction form as:

a / b = b / c

From this relationship, we can derive the formula for the third proportional, which will be discussed in detail in the next section.

Third Proportional Formula

The formula for finding the third proportional of two numbers is derived from the definition of proportion. When a, b, and c are in proportion such that a : b = b : c, the third proportional c can be calculated using the formula:

c = (b × b) / a

Here:

• a is the first number.
• b is the second number.
• c is the third proportional, which is to be determined.

This formula is derived by cross-multiplying the proportion a : b = b : c and solving for c.

How to Derive the Third Proportional Formula?

Let us derive the third proportional formula step by step:

1. Start with the proportional relationship: a : b = b : c.
2. Write this as a fraction: a / b = b / c.
3. Cross-multiply to eliminate the fractions: a × c = b × b.
4. Rearrange the equation to solve for c: c = (b × b) / a.

This formula shows that the third proportional depends on the square of the second number divided by the first number.

Properties of the Third Proportional

The third proportional has some unique properties that make it useful in various mathematical applications:

• The third proportional is always positive if the given numbers a and b are positive.
• If a, b, and c are in proportion, then b is the geometric mean of a and c, i.e., b² = a × c.
• The ratio of the first number to the second number is equal to the ratio of the second number to the third number: a : b = b : c.

Examples of Third Proportional

Let us solve some examples to understand how to calculate the third proportional using the formula:

Example 1:

Find the third proportional of 4 and 8.

Solution:

We know the formula for the third proportional is:

c = (b × b) / a

Here, a = 4 and b = 8. Substituting the values:

c = (8 × 8) / 4 = 64 / 4 = 16

Therefore, the third proportional of 4 and 8 is 16.

Example 2:

Find the third proportional of 5 and 15.

Solution:

Using the formula:

c = (b × b) / a

Here, a = 5 and b = 15. Substituting the values:

c = (15 × 15) / 5 = 225 / 5 = 45

Therefore, the third proportional of 5 and 15 is 45.

Applications of the Third Proportional

The third proportional is widely used in various areas of mathematics and real-life applications, such as:

• Geometry: The third proportional is used in problems involving similar triangles, where the sides of the triangles are proportional.
• Physics: Proportions are often used in physics to calculate quantities that are related by ratios, such as speed, force, and time.
• Construction: The third proportional helps in scaling models and designs, ensuring that the proportions of the original design are maintained.
• Ratio and Proportion Problems: It is used to solve problems that involve three quantities in a proportional relationship.
• Economics: The concept of proportionality is used in financial modeling and calculations involving ratios.

Difference Between Third Proportional and Mean Proportional

While both the third proportional and the mean proportional involve ratios, they are not the same. Here is a comparison:

Third Proportional	Mean Proportional
The third proportional is the value c such that a : b = b : c.	The mean proportional is the value b such that a : b = b : c, and b is the geometric mean of a and c.
Formula: c = (b × b) / a	Formula: b = √(a × c)
Used to find the third value in a proportional relationship.	Used to find the middle value in a proportion.

Common Mistakes to Avoid

Here are some common errors students make while solving third proportional problems:

• Confusing the formula for the third proportional with the mean proportional formula.
• Not simplifying the values correctly before applying the formula.
• Forgetting to check the units of the given numbers, especially in real-world problems.
• Incorrectly writing the proportion as a : b = c : b instead of a : b = b : c.

Conclusion

The third proportional is a crucial concept in mathematics, with applications in geometry, physics, and real-world problem-solving. By understanding the third proportional formula and practicing its application, students can strengthen their problem-solving skills and build a strong foundation in ratios and-solving. The formula for the third proportional, c = (b × b) / a, is straightforward but highly useful in solving proportionality problems efficiently. By mastering this concept, students can confidently tackle questions in exams and apply it to real-life scenarios.

Whether you are preparing for board exams, competitive exams, or simply enhancing your mathematical understanding, the third proportional is a topic worth focusing on. Practice regularly, avoid common mistakes, and ensure you understand the underlying concepts behind the formula. With consistent effort, you can excel in this area of mathematics and apply it effectively in various fields of study.

Proportions are not just abstract mathematical concepts—they form the basis of numerous practical applications in science, engineering, architecture, and even art. Start applying the concept of the third proportional in your calculations today, and you’ll see how understanding this simple formula opens up a world of possibilities!