Finding the Third Proportional

To find the third proportional 'c' to two numbers 'a' and 'b', we set up a proportion where the ratio of the first two numbers is equal to the ratio of the second and third numbers.

This means: $a : b = b : c$ Or, in fractional form: $\frac{a}{b} = \frac{b}{c}$

Calculation Steps

1. Identify Given Numbers: We are given the first two numbers, $a = 4$ and $b = 9$.
2. Set up the Proportion Equation: Using the formula derived from the definition, $\frac{a}{b} = \frac{b}{c}$.
3. Substitute Values: Substitute $a=4$ and $b=9$ into the equation: $\frac{4}{9} = \frac{9}{c}$
4. Solve for 'c': Rearrange the equation to solve for 'c'. We can cross-multiply: $4 \times c = 9 \times 9$ $4c = 81$ Now, divide by 4: $c = \frac{81}{4}$
5. Convert to Mixed Fraction: Divide 81 by 4: $c = 20$ with a remainder of $1$. So, $c = 20 \frac{1}{4}$.

The third proportional to 4 and 9 is $20 \frac{1}{4}$. This corresponds to Option B.