Simplify Fraction Expression Calculation

This solution explains the step-by-step simplification of the given nested fraction expression using the order of operations (PEMDAS/BODMAS).

Order of Operations for Fraction Simplification

We simplify the expression by working from the innermost parentheses outwards:

1. Simplify the innermost parentheses:

   Calculate $\left( \frac{1}{4} + \frac{1}{84} \right)$.

   • Find a common denominator for 4 and 84. Since $84 = 4 \times 21$, the common denominator is 84.
   • Convert $\frac{1}{4}$ to an equivalent fraction with denominator 84: $\frac{1}{4} = \frac{1 \times 21}{4 \times 21} = \frac{21}{84}$.
   • Add the fractions: $\frac{21}{84} + \frac{1}{84} = \frac{22}{84}$.
   • Simplify the resulting fraction by dividing the numerator and denominator by their greatest common divisor, 2: $\frac{22}{84} = \frac{11}{42}$.

   The expression simplifies to: $\frac{10 - \left[ \frac{3}{4} + \left\{ 4\frac{1}{2} - \frac{11}{42} \right\} \right]}{4}$

2. Simplify the curly braces:

   Calculate $\left\{ 4\frac{1}{2} - \frac{11}{42} \right\}$.

   • Convert the mixed number $4\frac{1}{2}$ to an improper fraction: $4\frac{1}{2} = \frac{(4 \times 2) + 1}{2} = \frac{9}{2}$.
   • Find a common denominator for 2 and 42. Since $42 = 2 \times 21$, the common denominator is 42.
   • Convert $\frac{9}{2}$ to an equivalent fraction with denominator 42: $\frac{9}{2} = \frac{9 \times 21}{2 \times 21} = \frac{189}{42}$.
   • Subtract the fractions: $\frac{189}{42} - \frac{11}{42} = \frac{178}{42}$.
   • Simplify the resulting fraction by dividing by 2: $\frac{178}{42} = \frac{89}{21}$.

   The expression simplifies to: $\frac{10 - \left[ \frac{3}{4} + \frac{89}{21} \right]}{4}$

3. Simplify the square brackets:

   Calculate $\left[ \frac{3}{4} + \frac{89}{21} \right]$.

   • Find a common denominator for 4 and 21. The least common multiple (LCM) is $4 \times 21 = 84$.
   • Convert $\frac{3}{4}$ to an equivalent fraction with denominator 84: $\frac{3}{4} = \frac{3 \times 21}{4 \times 21} = \frac{63}{84}$.
   • Convert $\frac{89}{21}$ to an equivalent fraction with denominator 84: $\frac{89}{21} = \frac{89 \times 4}{21 \times 4} = \frac{356}{84}$.
   • Add the fractions: $\frac{63}{84} + \frac{356}{84} = \frac{419}{84}$.

   The expression simplifies to: $\frac{10 - \frac{419}{84}}{4}$

4. Simplify the numerator:

   Calculate $10 - \frac{419}{84}$.

   • Convert 10 to a fraction with denominator 84: $10 = \frac{10 \times 84}{84} = \frac{840}{84}$.
   • Subtract the fractions: $\frac{840}{84} - \frac{419}{84} = \frac{421}{84}$.

   The expression is now: $\frac{\frac{421}{84}}{4}$

5. Perform the final division:

   Divide the numerator $\frac{421}{84}$ by 4.

   • Dividing by 4 is the same as multiplying by $\frac{1}{4}$.
   • $\frac{421}{84} \div 4 = \frac{421}{84} \times \frac{1}{4} = \frac{421 \times 1}{84 \times 4} = \frac{421}{336}$.

Final Result Conversion

The simplified fraction is $\frac{421}{336}$. To express this as a mixed number:

• Divide the numerator (421) by the denominator (336): $421 \div 336$.
• The quotient is 1, and the remainder is $421 - (1 \times 336) = 421 - 336 = 85$.
• Therefore, the mixed number representation is $1\frac{85}{336}$.

The simplified expression evaluates to $1\frac{85}{336}$.