Simplify Fraction Division and Multiplication Expression

To simplify the expression $\frac{1}{5} \div \left(\frac{5}{3} \times \frac{3}{25}\right)$, follow the order of operations (PEMDAS/BODMAS), performing the operation inside the parentheses first.

Step 1: Simplify Parentheses

Calculate the product inside the parentheses:

$ \frac{5}{3} \times \frac{3}{25} $

Multiply the numerators together and the denominators together:

$ \frac{5 \times 3}{3 \times 25} = \frac{15}{75} $

Simplify the resulting fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 15:

$ \frac{15 \div 15}{75 \div 15} = \frac{1}{5} $

Alternatively, cancel out common factors before multiplying:

$ \frac{\cancel{5}^{1}}{\cancel{3}^{1}} \times \frac{\cancel{3}^{1}}{\cancel{25}^{5}} = \frac{1}{5} $

Step 2: Perform Division

Now substitute the simplified value back into the original expression:

$ \frac{1}{5} \div \frac{1}{5} $

To divide by a fraction, multiply by its reciprocal:

$ \frac{1}{5} \times \frac{5}{1} $

Multiply the numerators and the denominators:

$ \frac{1 \times 5}{5 \times 1} = \frac{5}{5} $

Simplify the final fraction:

$ \frac{5}{5} = 1 $

Therefore, the simplified value of the expression is 1.