Understanding Mean Proportional Calculation

The question asks us to find the mean proportional of the two numbers, 81 and 36. The mean proportional (also known as the geometric mean) between two positive numbers $a$ and $b$ is a number $x$ such that the ratio $a:x$ is equal to the ratio $x:b$.

Mathematical Formula for Mean Proportional

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

Cross-multiplying gives:

To find $x$, we take the square root of the product of $a$ and $b$:

Step-by-Step Calculation

In this problem, we have $a = 81$ and $b = 36$. We need to find the mean proportional, $x$.

1. Identify the numbers: The given numbers are 81 and 36.
2. Apply the formula: Substitute these values into the formula $x = \sqrt{a \times b}$. $ x = \sqrt{81 \times 36} $
3. Calculate the product: First, multiply 81 by 36. $ 81 \times 36 = 2916 $ So, the equation becomes: $ x = \sqrt{2916} $
4. Calculate the square root: Now, find the square root of 2916. We can also simplify this by taking the square root of each number individually since multiplication is involved under the square root: $ x = \sqrt{81} \times \sqrt{36} $ We know that $\sqrt{81} = 9$ (since $9 \times 9 = 81$) and $\sqrt{36} = 6$ (since $6 \times 6 = 36$).
5. Final Multiplication: Multiply the square roots together. $ x = 9 \times 6 $ $ x = 54 $

Therefore, the mean proportional of 81 and 36 is 54.

Verification

We can check if 54 is indeed the mean proportional:

• Ratio 1: $\frac{81}{54}$
• Ratio 2: $\frac{54}{36}$

Simplify the ratios:

• $\frac{81}{54} = \frac{81 \div 27}{54 \div 27} = \frac{3}{2}$
• $\frac{54}{36} = \frac{54 \div 18}{36 \div 18} = \frac{3}{2}$

Since both ratios are equal ($\frac{3}{2}$), our calculation is correct. The mean proportional of 81 and 36 is 54.