Calculating the Third Proportional of 25 and 35

Concept of Third Proportional

The third proportional of two numbers, say a and b, is a number x such that the ratio of the first number to the second number is equal to the ratio of the second number to the third proportional. This can be expressed as:

$a : b = b : x$

This proportion implies the equation:

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

Rearranging the formula to solve for x, we get:

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

Calculation Steps

Given the numbers a = 25 and b = 35, we need to find the third proportional, x.

1. Identify the given numbers: $a = 25$, $b = 35$.
2. Use the formula for the third proportional: $x = \frac{b^2}{a}$.
3. Substitute the values into the formula:

   $x = \frac{35^2}{25}$

4. Calculate the square of 35: $35^2 = 35 \times 35 = 1225$.
5. Perform the division:

   $x = \frac{1225}{25}$

6. Simplify the fraction. We can simplify by dividing both numerator and denominator by 25:

   $x = \frac{1225 \div 25}{25 \div 25} = \frac{49}{1}$

   Alternatively, simplify using factors:

   $x = \frac{(35) \times (35)}{25} = \frac{(5 \times 7) \times (5 \times 7)}{5 \times 5} = \frac{25 \times 49}{25} = 49$

7. The value of x, the third proportional, is 49.

Final Answer

The third proportional of 25 and 35 is 49.