Understanding the Third Proportional and the Third Proportional Formula

When we talk about the third proportional to two numbers, say 'a' and 'b', we are looking for a third number 'c' such that the relationship between 'a' and 'b' is the same as the relationship between 'b' and 'c'. In other words, a, b, and c are in continued proportion. This means that the ratio of the first number to the second number is equal to the ratio of the second number to the third number.

Mathematically, this relationship is expressed as:  

\(\frac{a}{b} = \frac{b}{c}\)

Here, 'c' is the third proportional to 'a' and 'b'.

Deriving the Third Proportional Formula

To find the third proportional 'c', we can rearrange the proportion equation:

\(\frac{a}{b} = \frac{b}{c}\)

Multiply both sides by 'c' and 'b':

\(a \times c = b \times b\)

\(ac = b^2\)

Now, isolate 'c' by dividing both sides by 'a':

\(c = \frac{b^2}{a}\)

So, the third proportional formula is \(c = \frac{b^2}{a}\). This formula is essential when you need to find the third proportion quickly.

Applying the Third Proportional Formula to 16 and 24

The question asks us to find the third proportion to 16 and 24.

Here, the first number 'a' is 16, and the second number 'b' is 24. We need to find the third proportional 'c'.

Using the third proportional formula \(c = \frac{b^2}{a}\), we substitute the values of 'a' and 'b':

\(c = \frac{24^2}{16}\)

Calculating the Third Proportion

Now, let's perform the calculation using the formula \(c = \frac{24^2}{16}\):

• First, calculate \(24^2\): \(24 \times 24 = 576\).
• Next, divide 576 by 16.

The calculation is:

\(c = \frac{576}{16}\)

To divide 576 by 16, we can perform long division or simplify the fraction.

Using division:

So, \(576 \div 16 = 36\).

Alternatively, simplifying the fraction:

\(c = \frac{576}{16} = \frac{288}{8} = \frac{144}{4} = \frac{72}{2} = 36\)

Both methods give the same result. The third proportion to 16 and 24 is 36. This confirms our application of the third proportional formula.

Conclusion on Finding the Third Proportion

To find the third proportion for any two numbers 'a' and 'b', remember the third proportional formula \(c = \frac{b^2}{a}\). This concept is important for understanding proportions, and questions like this 3rd proportion question are common in various exams, including those that might involve reasoning similar to a 3rd proportion question UPSC might pose. Whether you need to find second proportion or third proportion, understanding the relationships in proportion is key. For example, the third proportional to 8 and 16.4 is calculated using the same principle.

The calculated third proportion is 36, which matches one of the given options.