Mean Proportional Calculation for 400 and 4

The question asks us to find the mean proportional of the numbers 400 and 4. The mean proportional, also known as the geometric mean, between two positive numbers $a$ and $b$ is given by the square root of their product.

Understanding Mean Proportional

Mathematically, if $x$ is the mean proportional between $a$ and $b$, then the relationship is:

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

Cross-multiplying gives:

$ x^2 = a \times b $

Taking the square root of both sides, we get the formula for the mean proportional:

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

Applying the Formula

In this problem, we have $a = 400$ and $b = 4$. Let's substitute these values into the formula:

$ x = \sqrt{400 \times 4} $

Step-by-Step Calculation

1. Multiply the two numbers: $ 400 \times 4 = 1600 $
2. Calculate the square root of the product: $ x = \sqrt{1600} $
3. The square root of 1600 is 40, since $40 \times 40 = 1600$. $ x = 40 $

Conclusion

Therefore, the mean proportional of 400 and 4 is 40. This matches option 3.