2.4 Simplification & Order of Operations

2.4 Simplification & Order of Operations

Simplification tests operational arithmetic speed, accuracy, and strict adherence to algebraic precedence. In SBI PO Prelims, direct simplification and approximation questions account for 5 to 10 marks. In SBI PO Mains, complex operational expressions are embedded inside Data Analysis calculations and multi-statement data sufficiency questions.

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The VBODMAS Precedence Hierarchy

Mathematical expressions containing multiple operation symbols must be evaluated according to the strict VBODMAS hierarchy. Evaluating operations out of order alters the fundamental value of the expression.

[V] Vinculum (Bar Over Numbers: 7 - 3)
  └── [B] Brackets (Inner Round -> Middle Curly -> Outer Square)
        └── [O] 'Of' Operator & Orders/Powers (15% of 400, x^2)
              └── [D] Division (Left to Right)
                    └── [M] Multiplication (Left to Right)
                          └── [A] Addition
                                └── [S] Subtraction

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Critical Distinction: "Of" Operator vs. Division

The most frequent operational mistake made by candidates is treating the word "Of" as ordinary multiplication with equal priority to division. "Of" carries higher priority than Division.

Contrast Example

Evaluate: $48 \div 6 \text{ of } 2$.

• INCORRECT Evaluation (Treating 'Of' as regular multiplication):
  $48 \div 6 \times 2 = (48 \div 6) \times 2 = 8 \times 2 = \mathbf{16}$
  (This violates VBODMAS!)

• CORRECT VBODMAS Evaluation ('Of' evaluated before Division):
  $48 \div (6 \text{ of } 2) = 48 \div (6 \times 2) = 48 \div 12 = \mathbf{4}$

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Continuous Division Shortcut Rule

When an expression features a chain of continuous division symbols, candidates often struggle with operational direction. Continuous division operates strictly left to right, which simplifies into a single fraction with the first number in the numerator and all subsequent divisors in the denominator.

$\text{General Formula: } A \div B \div C \div D = \frac{A}{B \times C \times D}$

Worked Example 1

Evaluate: $14400 \div 40 \div 6 \div 5$.

Step-by-step Solution:

• Apply continuous division formula: $\text{Result} = \frac{14400}{40 \times 6 \times 5}$
• Multiply denominator factors: $40 \times 6 \times 5 = 40 \times 30 = 1200$.
• Divide numerator by denominator: $\text{Result} = \frac{14400}{1200} = \mathbf{12}$

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Mixed Fraction Fast Addition & Subtraction

Convert mixed fractions $a \frac{b}{c}$ by separating whole integer parts from proper fraction parts: $a \frac{b}{c} \pm d \frac{e}{f} = (a \pm d) + \left(\frac{b}{c} \pm \frac{e}{f}\right)$

Worked Example 2

Evaluate: $5 \frac{1}{3} + 3 \frac{1}{2} - 2 \frac{1}{6}$.

Step-by-step Solution:

• Step 1: Group integer components: $\text{Integer Sum} = 5 + 3 - 2 = 6$
• Step 2: Group fractional components and take common denominator ($6$): $\text{Fractional Sum} = \frac{1}{3} + \frac{1}{2} - \frac{1}{6} = \frac{2 + 3 - 1}{6} = \frac{4}{6} = \frac{2}{3}$
• Step 3: Combine integer and fractional sums: $\text{Final Answer} = 6 + \frac{2}{3} = \mathbf{6 \frac{2}{3}}$

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Surds & Indices Laws

Simplification problems frequently test exponent manipulation. Master these eight core laws of indices:

1. Product Law: $a^m \times a^n = a^{m+n}$
2. Quotient Law: $a^m \div a^n = a^{m-n}$
3. Power of Power: $(a^m)^n = a^{m \times n}$
4. Fractional Exponents: $a^{p/q} = \sqrt[q]{a^p}$
5. Negative Exponent: $a^{-n} = \frac{1}{a^n}$
6. Zero Exponent: $a^0 = 1 \quad (a \neq 0)$
7. Distributive Power: $(a \times b)^n = a^n \times b^n$
8. Quotient Power: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$

Worked Example 3: Exponent Equation

Solve for $x$ in: $16^{x+1} = 64^x$.

Step-by-step Solution:

• Convert both sides to prime base $4$ (or base $2$):
  • $16 = 4^2 \implies 16^{x+1} = (4^2)^{x+1} = 4^{2(x+1)} = 4^{2x+2}$
  • $64 = 4^3 \implies 64^x = (4^3)^x = 4^{3x}$
• Equate exponents since bases are identical: $2x + 2 = 3x \implies 3x - 2x = 2 \implies \mathbf{x = 2}$

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Worked Example 4: Comprehensive Multi-Bracket VBODMAS Solution

Problem: Solve $60 \div 5 \text{ of } 3 + \overline{8 - 3} \times 4 - 2^3$.

Step-by-step Execution:

1. Vinculum Step: Evaluate $\overline{8 - 3} = 5$. Expression becomes: $60 \div 5 \text{ of } 3 + 5 \times 4 - 2^3$.
2. "Of" Step: Evaluate $5 \text{ of } 3 = 15$. Expression becomes: $60 \div 15 + 5 \times 4 - 2^3$.
3. Power Step: Evaluate $2^3 = 8$. Expression becomes: $60 \div 15 + 5 \times 4 - 8$.
4. Division Step: Evaluate $60 \div 15 = 4$. Expression becomes: $4 + 5 \times 4 - 8$.
5. Multiplication Step: Evaluate $5 \times 4 = 20$. Expression becomes: $4 + 20 - 8$.
6. Addition & Subtraction Step: $24 - 8 = \mathbf{16}$.