Question

If \[ \frac{1}{5}\div\frac{1}{x}=\frac{1}{x}\div\frac{1}{125}, \] then the value of \(x\) is

• A. \(\dfrac{1}{25}\)
• B. \(25\)
• C. \(\dfrac{1}{625}\)
• D. \(5\)

Hint:

Remember: \[ \frac{a}{b}\div\frac{c}{d} = \frac{a}{b}\times\frac{d}{c} \] Always convert division of fractions into multiplication by taking the reciprocal.

Answer 1

The Correct Option is B

Concept: Division of fractions is performed by multiplying the first fraction by the reciprocal of the second fraction. \[ \boxed{\frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\times\frac{d}{c}} \]

Step 1: Simplify the left-hand side.
\[ \frac{1}{5}\div\frac{1}{x} =\frac{1}{5}\times\frac{x}{1} =\frac{x}{5} \]

Step 2: Simplify the right-hand side.
\[ \frac{1}{x}\div\frac{1}{125} =\frac{1}{x}\times\frac{125}{1} =\frac{125}{x} \] Hence, \[ \frac{x}{5}=\frac{125}{x} \]

Step 3: Solve the equation.
Cross-multiplying, \[ x^2=125\times5 \] \[ x^2=625 \] Taking the positive value (as given in the options), \[ x=25. \]

Step 4: Verify the answer.
Substitute \(x=25\): \[ \frac{1}{5}\div\frac{1}{25}=5, \] and \[ \frac{1}{25}\div\frac{1}{125}=5. \] Both sides are equal.

Step 5: Final conclusion.
Therefore, \[ \boxed{x=25} \]