Find the mean proportion between
`(1)/(25) and (1)/(16)`

Answer

To find the mean proportion between \( \frac{1}{25} \) and \( \frac{1}{16} \), we can follow these steps:

### Step 1: Define the Mean Proportion
Let \( X \) be the mean proportion between \( \frac{1}{25} \) and \( \frac{1}{16} \). We can express this relationship as:
\[ \frac{1}{25} : X :: X : \frac{1}{16} \]

### Step 2: Set Up the Proportion
According to the property of proportions, the product of the extremes is equal to the product of the means. Therefore, we can write:
\[ \frac{1}{25} \times \frac{1}{16} = X \times X \]
This simplifies to:
\[ \frac{1}{25} \times \frac{1}{16} = X^2 \]

### Step 3: Calculate the Left Side
Now, we calculate the left side:
\[ \frac{1}{25} \times \frac{1}{16} = \frac{1 \times 1}{25 \times 16} = \frac{1}{400} \]
So we have:
\[ X^2 = \frac{1}{400} \]

### Step 4: Solve for \( X \)
To find \( X \), we take the square root of both sides:
\[ X = \sqrt{\frac{1}{400}} = \frac{1}{\sqrt{400}} = \frac{1}{20} \]

### Conclusion
Thus, the mean proportion between \( \frac{1}{25} \) and \( \frac{1}{16} \) is:
\[ \boxed{\frac{1}{20}} \]

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