If we multiply a fraction by itself and divide to product by its reciprocal, the fraction thus obtained is `18(26)/(27)`. What is the origianl fraction ?

Answer

To solve the problem step by step, let's denote the original fraction as \( x \).

### Step 1: Set up the equation
According to the problem, when we multiply the fraction \( x \) by itself, we get \( x^2 \). Then, we divide this product by its reciprocal, which is \( \frac{1}{x} \). Therefore, we can set up the equation:

\[ \frac{x^2}{\frac{1}{x}} = x^2 \cdot x = x^3 \]

This gives us the equation:

\[ x^3 = \frac{18 \frac{26}{27}}{1} \]

### Step 2: Convert the mixed number to an improper fraction
Next, we need to convert the mixed number \( 18 \frac{26}{27} \) into an improper fraction. To do this, we multiply the whole number part (18) by the denominator (27) and add the numerator (26):

\[ 18 \cdot 27 + 26 = 486 + 26 = 512 \]

So, we can express \( 18 \frac{26}{27} \) as:

\[ \frac{512}{27} \]

### Step 3: Substitute back into the equation
Now we substitute this back into our equation:

\[ x^3 = \frac{512}{27} \]

### Step 4: Solve for \( x \)
To find \( x \), we take the cube root of both sides:

\[ x = \sqrt[3]{\frac{512}{27}} \]

### Step 5: Simplify the cube root
We can simplify this expression by taking the cube root of the numerator and the denominator separately:

\[ x = \frac{\sqrt[3]{512}}{\sqrt[3]{27}} = \frac{8}{3} \]

### Step 6: Convert to a mixed number
The fraction \( \frac{8}{3} \) can be converted to a mixed number:

\[ \frac{8}{3} = 2 \frac{2}{3} \]

### Final Answer
Thus, the original fraction is:

\[ \boxed{2 \frac{2}{3}} \]