Calculating the Mean Proportional of 225 and 4

The mean proportional, also known as the geometric mean, between two positive numbers is found by taking the square root of their product. If we have two numbers, $a$ and $b$, their mean proportional $x$ is given by the formula:

$ x = \sqrt{a \times b} $

In this problem, we need to find the mean proportional of the numbers 225 and 4.

Step-by-Step Calculation

1. Identify the given numbers.

   The two numbers are $a = 225$ and $b = 4$.

2. Write down the formula for the mean proportional.

   Mean Proportional ($MP$) = $\sqrt{a \times b}$

3. Substitute the given numbers into the formula.

   $MP = \sqrt{225 \times 4}$

4. Calculate the product of the two numbers.

   Product = $225 \times 4 = 900$

5. Find the square root of the product.

   $MP = \sqrt{900}$

   The square root of 900 is 30, because $30 \times 30 = 900$.

6. State the final answer.

   The mean proportional of 225 and 4 is 30.

Understanding Mean Proportional

The mean proportional represents a value that maintains the same ratio between the first number and the mean, as between the mean and the second number. For numbers 225 and 4, the mean proportional 30 satisfies:

$ \frac{225}{30} = 7.5 $

$ \frac{30}{4} = 7.5 $

Since both ratios are equal (7.5), 30 is indeed the correct mean proportional.