Understanding the Mathematical Expression

The problem asks us to simplify a complex fraction. This involves a numerator and a denominator, each containing operations with fractions. To solve this, we need to follow the order of operations carefully.

The given expression is:

\( \frac{{\frac{1}{4}\; \div \;\frac{1}{{4\;}}{\rm{\{ }}\frac{1}{4}{\}}}}{{\frac{1}{4}\; \div \;\frac{1}{4}\; \times \;\frac{1}{4}}} \)

We will simplify the numerator and the denominator separately first, and then perform the final division.

Simplifying the Numerator

The numerator is \( \frac{1}{4}\; \div \;\frac{1}{{4\;}}{\rm{\{ }}\frac{1}{4}{\}} \).

The curly braces \({\rm{\{ }}\frac{1}{4}{\}}\) around \( \frac{1}{4} \) next to \( \frac{1}{4} \) indicate multiplication. So the expression can be written as:

\( \frac{1}{4} \div (\frac{1}{4} \times \frac{1}{4}) \)

Following the order of operations (BODMAS/PEMDAS), we first perform the operation inside the parentheses:

• Multiply \( \frac{1}{4} \) by \( \frac{1}{4} \): \( \frac{1}{4} \times \frac{1}{4} = \frac{1 \times 1}{4 \times 4} = \frac{1}{16} \)

Now, the numerator simplifies to:

\( \frac{1}{4} \div \frac{1}{16} \)

To divide fractions, we multiply the first fraction by the reciprocal of the second fraction:

• The reciprocal of \( \frac{1}{16} \) is \( \frac{16}{1} \).
• So, \( \frac{1}{4} \div \frac{1}{16} = \frac{1}{4} \times \frac{16}{1} \).
• Multiply the fractions: \( \frac{1}{4} \times \frac{16}{1} = \frac{1 \times 16}{4 \times 1} = \frac{16}{4} \).
• Simplify the fraction: \( \frac{16}{4} = 4 \).

Thus, the simplified numerator is \( 4 \).

Simplifying the Denominator

The denominator is \( \frac{1}{4}\; \div \;\frac{1}{4}\; \times \;\frac{1}{4} \).

This involves division and multiplication. According to the order of operations, division and multiplication are performed from left to right.

• First, perform the division: \( \frac{1}{4} \div \frac{1}{4} \). Dividing a number by itself gives 1. So, \( \frac{1}{4} \div \frac{1}{4} = 1 \).
• Now, the expression becomes \( 1 \times \frac{1}{4} \).
• Perform the multiplication: \( 1 \times \frac{1}{4} = \frac{1}{4} \).

Thus, the simplified denominator is \( \frac{1}{4} \).

Final Simplification

Now we have the simplified numerator and the simplified denominator. The original expression is:

\( \frac{\text{Numerator}}{\text{Denominator}} = \frac{4}{\frac{1}{4}} \)

To divide a number by a fraction, we multiply the number by the reciprocal of the fraction:

• The reciprocal of \( \frac{1}{4} \) is \( \frac{4}{1} \).
• So, \( \frac{4}{\frac{1}{4}} = 4 \div \frac{1}{4} = 4 \times \frac{4}{1} \).
• Perform the multiplication: \( 4 \times 4 = 16 \).

Therefore, the simplified value of the expression is \( 16 \).

Revision Table: Key Steps

Additional Information: Order of Operations

The order of operations is a set of rules used to clarify which procedures should be performed first in a given mathematical expression. A common acronym to remember this order is BODMAS or PEMDAS.

• BODMAS: Brackets (Parentheses), Orders (Exponents, Roots), Division and Multiplication (from left to right), Addition and Subtraction (from left to right).
• PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

In this problem, we dealt with multiplication indicated by braces (like parentheses), division, and multiplication. We prioritized the operation within the implicit parentheses in the numerator first, then performed division and multiplication from left to right in both the numerator and the denominator before the final division of the numerator by the denominator.