Simplify the expression:

[tex]\[
\frac{\frac{x}{25} - \frac{1}{x}}{1 + \frac{5}{x}}
\][/tex]

Options:
A. [tex]\(\frac{x+5}{25}\)[/tex]
B. [tex]\(\frac{25}{x+5}\)[/tex]
C. [tex]\(\frac{x-5}{25}\)[/tex]
D. [tex]\(\frac{25}{x-5}\)[/tex]
E. [tex]\(\frac{x-5}{6}\)[/tex]
F. None of the above



Answer :

To solve the given problem, we need to simplify the expression:

[tex]\[ \frac{\frac{x}{25} - \frac{1}{x}}{1 + \frac{5}{x}} \][/tex]

and then determine which of the given choices it matches.

Let's first rewrite the given expression in a more manageable form:

1. Simplify the numerator [tex]\(\frac{x}{25} - \frac{1}{x}\)[/tex]:

[tex]\[ \frac{x}{25} - \frac{1}{x} = \frac{x^2 - 25}{25x} \][/tex]

2. Simplify the denominator [tex]\(1 + \frac{5}{x}\)[/tex]:

[tex]\[ 1 + \frac{5}{x} = \frac{x + 5}{x} \][/tex]

Now, the original expression becomes:

[tex]\[ \frac{\frac{x^2 - 25}{25x}}{\frac{x + 5}{x}} \][/tex]

To divide two fractions, we multiply by the reciprocal of the denominator:

[tex]\[ \frac{x^2 - 25}{25x} \times \frac{x}{x + 5} \][/tex]

Multiply the fractions:

[tex]\[ \frac{(x^2 - 25)x}{25x (x + 5)} = \frac{x^2 - 25}{25 (x + 5)} \][/tex]

Notice that the numerator [tex]\(x^2 - 25\)[/tex] can be factored as a difference of squares:

[tex]\[ x^2 - 25 = (x - 5)(x + 5) \][/tex]

So, we have:

[tex]\[ \frac{(x - 5)(x + 5)}{25 (x + 5)} \][/tex]

We can cancel out the common term [tex]\((x + 5)\)[/tex]:

[tex]\[ \frac{(x - 5)(\cancel{x + 5})}{25 (\cancel{x + 5})} = \frac{x - 5}{25} \][/tex]

Thus, the simplified expression is:

[tex]\[ \frac{x - 5}{25} \][/tex]

Comparing this result with the given choices, we see that it matches:

[tex]\[ \frac{x - 5}{25} \][/tex]

Therefore, the correct choice is:

[tex]\[ \boxed{3} \][/tex]