Find the mean proportion between
81 and 121

Answer

To find the mean proportion between 81 and 121, we can follow these steps:

### Step 1: Set up the equation for mean proportion
The mean proportion \( x \) between two numbers \( a \) and \( b \) can be found using the formula:
\[ \frac{a}{x} = \frac{x}{b} \]
In this case, let \( a = 81 \) and \( b = 121 \).

### Step 2: Substitute the values into the equation
Substituting the values of \( a \) and \( b \) into the equation gives:
\[ \frac{81}{x} = \frac{x}{121} \]

### Step 3: Cross-multiply to eliminate the fractions
Cross-multiplying gives us:
\[ 81 \cdot 121 = x \cdot x \]
This simplifies to:
\[ x^2 = 81 \cdot 121 \]

### Step 4: Calculate the product on the right side
Now, we need to calculate \( 81 \cdot 121 \). We can rewrite \( 81 \) and \( 121 \) as squares:
\[ 81 = 9^2 \quad \text{and} \quad 121 = 11^2 \]
Thus,
\[ x^2 = 9^2 \cdot 11^2 = (9 \cdot 11)^2 \]
Calculating \( 9 \cdot 11 \):
\[ 9 \cdot 11 = 99 \]
So,
\[ x^2 = 99^2 \]

### Step 5: Take the square root of both sides
Taking the square root of both sides gives:
\[ x = \sqrt{99^2} \]
This simplifies to:
\[ x = 99 \]

### Step 6: Conclusion
Thus, the mean proportion between 81 and 121 is:
\[ \boxed{99} \]