Simplification

Simplification

Overview

Simplification is the foundation of quantitative aptitude for TNPSC Group IV. Every arithmetic problem—whether it involves percentages, interest, or time-work—ultimately requires you to simplify numerical expressions accurately and quickly. This topic tests your command over the order of operations, your fluency with fractions and decimals, and your ability to handle roots.

In the exam, simplification questions appear both as standalone problems and as embedded steps within larger word problems. Mastering this topic means you solve faster with fewer errors, freeing up time for reasoning-heavy questions. The key is systematic application of BODMAS and confident manipulation of fractions, decimals, and roots.

Expect 2–4 direct simplification questions in the aptitude section, plus indirect use in nearly every quantitative problem.

Key Concepts

• BODMAS Rule: Operations must be performed in the order—Brackets, Orders (powers and roots), Division, Multiplication, Addition, Subtraction. Division and multiplication are done left to right; same for addition and subtraction.
• Brackets Hierarchy: Solve innermost brackets first. The order is: Vinculum (bar), then Parentheses ( ), then Curly Braces { }, then Square Brackets [ ].
• Fractions: A fraction a/b represents a parts out of b equal parts. To add or subtract fractions, find a common denominator. To multiply, multiply numerators and denominators. To divide, multiply by the reciprocal.
• Decimals: Decimals are fractions with denominators as powers of 10. Converting between fractions and decimals is essential—memorize common conversions like 1/4 = 0.25, 1/8 = 0.125.
• Square Roots: √a is the number which, when squared, gives a. Key values: √2 ≈ 1.414, √3 ≈ 1.732, √5 ≈ 2.236. Perfect squares up to 625 should be memorized.
• Cube Roots: ∛a is the number which, when cubed, gives a. Memorize cubes: 2³ = 8, 3³ = 27, 4³ = 64, 5³ = 125, up to 10³ = 1000.
• Simplification of Surds: √a × √b = √(ab); √a ÷ √b = √(a/b); (√a)² = a. Rationalize denominators when needed.
• Approximation Technique: When answer choices are well-separated, round numbers strategically to save time. Always check if approximation is safe before using it.

Formulas / Key Facts

Worked Examples

Example 1: BODMAS Application

Simplify: 48 ÷ 12 × 3 + 8 − 2

Step 1: No brackets or orders, so start with division and multiplication (left to right).

• 48 ÷ 12 = 4
• 4 × 3 = 12

Step 2: Now addition and subtraction (left to right).

• 12 + 8 = 20
• 20 − 2 = 18

Answer: 18

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Example 2: Nested Brackets

Simplify: 25 − [10 + {8 − (6 − 2)}]

Step 1: Solve innermost bracket first.

• (6 − 2) = 4

Step 2: Substitute and solve curly braces.

• {8 − 4} = 4

Step 3: Solve square bracket.

• [10 + 4] = 14

Step 4: Final subtraction.

• 25 − 14 = 11

Answer: 11

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Example 3: Fractions and Decimals

Simplify: 2/3 + 0.5 − 1/6

Step 1: Convert decimal to fraction.

• 0.5 = 1/2

Step 2: Find LCM of denominators (3, 2, 6) = 6.

• 2/3 = 4/6
• 1/2 = 3/6
• 1/6 = 1/6

Step 3: Perform operations.

• 4/6 + 3/6 − 1/6 = 6/6 = 1

Answer: 1

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Example 4: Square and Cube Roots

Simplify: √144 + ∛64 − √49

Step 1: Find each root.

• √144 = 12
• ∛64 = 4
• √49 = 7

Step 2: Compute.

• 12 + 4 − 7 = 9

Answer: 9

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Example 5: Mixed Simplification

Simplify: (3/4 of 48) ÷ 6 + √81

Step 1: "Of" means multiplication.

• 3/4 × 48 = 36

Step 2: Division.

• 36 ÷ 6 = 6

Step 3: Square root.

• √81 = 9

Step 4: Addition.

• 6 + 9 = 15

Answer: 15

Common Mistakes

• Ignoring left-to-right rule for division and multiplication: Students assume multiplication comes before division. Wrong—they have equal priority; go left to right. Correct: 24 ÷ 4 × 2 = 6 × 2 = 12, not 24 ÷ 8 = 3.
• Forgetting to convert decimals before operating with fractions: Adding 1/3 + 0.5 directly leads to errors. Always convert 0.5 to 1/2 first, then find common denominator.
• Wrong bracket order: Solving [ ] before ( ) is incorrect. Always start from innermost bracket regardless of type.
• Confusing square and cube roots: √64 = 8 but ∛64 = 4. Read carefully whether the question asks for square root or cube root.
• Rushing approximation on close options: If answer choices are 12.4, 12.5, 12.6, and 12.8, approximation can mislead. Compute precisely when options are close.

Quick Reference

• BODMAS: Brackets → Orders → Division/Multiplication (L→R) → Addition/Subtraction (L→R)
• "Of" = Multiplication: Solve "of" immediately after brackets
• Fraction division: Flip the second fraction and multiply
• LCM for fraction addition: Always find least common denominator before adding or subtracting
• Memorize roots: √2 ≈ 1.41, √3 ≈ 1.73, √5 ≈ 2.24; cubes up to 10³ = 1000
• Vinculum first: If there's a bar over numbers, solve that before any bracket