Calculating the Third Proportional C

This problem requires us to find the value of C, which is defined as the third proportional to 29 and another number, B. We first need to determine the value of B based on the sum of the first three even natural numbers.

Determining the Value of B

The natural numbers are 1, 2, 3, 4, ... The even natural numbers are 2, 4, 6, 8, ... The first three even natural numbers are 2, 4, and 6. The problem states that B is the sum of these numbers.

• First even natural number = 2
• Second even natural number = 4
• Third even natural number = 6

Calculating B:

$B = 2 + 4 + 6$

$B = 12$

Understanding Third Proportional

In mathematics, if a number 'a' relates to 'b' in the same way that 'b' relates to 'c', then 'c' is called the third proportional to 'a' and 'b'. This relationship can be expressed using ratios:

$a : b :: b : c$

This proportion implies the following equation:

$\frac{a}{b} = \frac{b}{c}$

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

$c = \frac{b^2}{a}$

Calculating the Value of C

We are given that C is the third proportional of 29 and B. Here, $a = 29$ and $b = B = 12$. Using the formula derived above:

$C = \frac{B^2}{29}$

Substitute the value of B:

$C = \frac{12^2}{29}$

Calculate the square of B:

$C = \frac{144}{29}$

Now, perform the division:

$C \approx 4.965517...$

The question asks to round the value of C off to two decimal places. Looking at the third decimal place (5), we round up the second decimal place.

$C \approx 4.97$