Find the mean proportion between :
`7 (1)/(5), 6`

Answer

To find the mean proportion between the ratios \( \frac{7}{5} \) and \( 6 \), we can follow these steps:

### Step 1: Convert the ratio
We first convert \( 7 \frac{1}{5} \) into an improper fraction.

\[ 7 \frac{1}{5} = 7 + \frac{1}{5} = \frac{35}{5} + \frac{1}{5} = \frac{36}{5} \]

### Step 2: Set up the proportion
Let \( x \) be the mean proportion. We can set up the proportion as follows:

\[ \frac{36/5}{x} = \frac{x}{6} \]

### Step 3: Cross-multiply
Using the property of proportions, we cross-multiply:

\[ (36/5) \cdot 6 = x \cdot x \]

### Step 4: Simplify the equation
Now simplify the left side of the equation:

\[ \frac{36 \cdot 6}{5} = x^2 \]

Calculating \( 36 \cdot 6 \):

\[ 36 \cdot 6 = 216 \]

So, we have:

\[ \frac{216}{5} = x^2 \]

### Step 5: Solve for \( x \)
To find \( x \), we take the square root of both sides:

\[ x = \sqrt{\frac{216}{5}} = \frac{\sqrt{216}}{\sqrt{5}} \]

### Step 6: Simplify \( \sqrt{216} \)
We can simplify \( \sqrt{216} \):

\[ \sqrt{216} = \sqrt{36 \cdot 6} = \sqrt{36} \cdot \sqrt{6} = 6\sqrt{6} \]

So, we have:

\[ x = \frac{6\sqrt{6}}{\sqrt{5}} \]

### Step 7: Final answer
Thus, the mean proportion between \( 7 \frac{1}{5} \) and \( 6 \) is:

\[ x = \frac{6\sqrt{6}}{\sqrt{5}} \]

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