Simplification — Study Notes for UP Police Constable

Overview

Simplification is the foundation of quantitative aptitude in UP Police Constable exams. Every numerical problem—whether profit-loss, time-work, or data interpretation—requires you to simplify expressions correctly and quickly. Most questions in this section test your ability to apply BODMAS rules, handle fractions and decimals, and compute squares, cubes, and their roots accurately within 60–90 seconds per question.

Expect 3–5 direct simplification questions in the exam, plus countless others where simplification is embedded in word problems. Mastery here means fewer calculation errors across the entire paper. The key is speed with accuracy—knowing shortcuts for common calculations and avoiding the trap of rushing through bracket operations.

Students who excel at simplification typically score 5–8 marks higher overall because they finish faster and make fewer arithmetic mistakes. This topic rewards disciplined practice: solve 20–30 problems daily for two weeks to build computational fluency.

Key Concepts

• **BODMAS/BODMAS Rule**: Operations must follow the strict order — Brackets first, then Orders (powers/roots), then Division and Multiplication (left to right), finally Addition and Subtraction (left to right). Violating this order is the most common error.

• **Fraction Operations**: For addition/subtraction, convert to common denominator; for multiplication, multiply numerators and denominators directly; for division, multiply by the reciprocal. Mixed fractions must be converted to improper fractions before operations.

• **Decimal Handling**: Align decimal points for addition/subtraction. For multiplication, count total decimal places in factors and place that many in the product. For division, shift decimal points to make divisor a whole number.

• **Square and Cube Roots**: Perfect squares (1, 4, 9, 16, 25...144, 169, 196, 225, 256, 289, 324, 361, 400) and perfect cubes (1, 8, 27, 64, 125, 216, 343, 512, 729, 1000) must be memorized. Use factorization or digit methods for non-perfect roots.

• **Reciprocal and Negative Signs**: Division by a fraction equals multiplication by its reciprocal. Double negatives make positive; a negative times positive stays negative. Track sign changes carefully through multi-step problems.

• **Approximation Techniques**: In exams, don't calculate exact decimals when options differ significantly. Round strategically—but only after all operations inside brackets are complete.

Formulas / Key Facts

**BODMAS Order**: B (Brackets) → O (Of/Orders) → D/M (Division/Multiplication, left to right) → A/S (Addition/Subtraction, left to right)

**Fraction Operations**:

• a/b + c/d = (ad + bc)/bd
• a/b × c/d = ac/bd
• a/b ÷ c/d = a/b × d/c = ad/bc

**Decimal Multiplication**: 2.5 × 3.2 = 25 × 32 / 100 = 800/100 = 8.0 (count two decimal places total)

**Decimal Division**: To divide by 0.25, multiply numerator and denominator by 100 to get division by 25.

**Perfect Squares to Memorize**: 1² = 1, 2² = 4, 3² = 9... up to 20² = 400. Also know 25² = 625, 30² = 900.

**Perfect Cubes to Memorize**: 1³ = 1, 2³ = 8, 3³ = 27, 4³ = 64, 5³ = 125, 6³ = 216, 7³ = 343, 8³ = 512, 9³ = 729, 10³ = 1000.

**Square Root by Prime Factorization**: √144 = √(2² × 2² × 3²) = 2 × 2 × 3 = 12

**Cube Root by Prime Factorization**: ³√1728 = ³√(2³ × 2³ × 3³) = 2 × 2 × 3 = 12

**Unit Digit Method for Squares**: Number ending in 2 has square ending in 4; ending in 3 has square ending in 9; ending in 7 has square ending in 9; ending in 8 has square ending in 4.

Worked Examples

**Example 1: BODMAS Application**

Simplify: 48 ÷ 4 × 2 + 15 − 3 × 5

**Solution**: Step 1: No brackets, no powers. Apply Division and Multiplication left to right.

• 48 ÷ 4 = 12
• 12 × 2 = 24
• 3 × 5 = 15

Step 2: Now expression is 24 + 15 − 15 Step 3: Apply Addition and Subtraction left to right.

• 24 + 15 = 39
• 39 − 15 = **24**

**Example 2: Fractions with Brackets**

Simplify: 5/6 + {2/3 − (3/4 − 1/6)}

**Solution**: Step 1: Solve innermost bracket (3/4 − 1/6). LCM of 4 and 6 is 12.

• 3/4 = 9/12, 1/6 = 2/12
• 9/12 − 2/12 = 7/12

Step 2: Now expression is 5/6 + {2/3 − 7/12} Step 3: Solve curly bracket (2/3 − 7/12). LCM is 12.

• 2/3 = 8/12
• 8/12 − 7/12 = 1/12

Step 4: Now 5/6 + 1/12. LCM is 12.

• 5/6 = 10/12
• 10/12 + 1/12 = 11/12 = **11/12**

**Example 3: Decimal and Square Root**

Simplify: √(6.25) + 2.5 × 0.4 ÷ 0.2

**Solution**: Step 1: √(6.25) = 2.5 (since 2.5 × 2.5 = 6.25) Step 2: Solve 2.5 × 0.4 ÷ 0.2 left to right.

• 2.5 × 0.4 = 1.0
• 1.0 ÷ 0.2 = 5.0

Step 3: 2.5 + 5.0 = **7.5**

**Example 4: Cube Root by Factorization**

Find: ³√(3375)

**Solution**: Factorize 3375:

• 3375 = 3 × 1125 = 3 × 3 × 375 = 3 × 3 × 3 × 125 = 3³ × 5³
• ³√(3375) = ³√(3³ × 5³) = 3 × 5 = **15**

Common Mistakes

**Mistake 1: Ignoring Bracket Priority** Wrong thinking: In 3 + 4 × 2, students add first getting 7 × 2 = 14. Correct fix: Multiply first per BODMAS: 4 × 2 = 8, then 3 + 8 = 11.

**Mistake 2: Division/Multiplication Order Confusion** Wrong thinking: 20 ÷ 5 × 2 is solved as 20 ÷ (5 × 2) = 20 ÷ 10 = 2. Correct fix: Solve left to right: 20 ÷ 5 = 4, then 4 × 2 = 8. Division and multiplication have equal priority.

**Mistake 3: Fraction Subtraction Without Common Denominator** Wrong thinking: 5/6 − 1/4 = (5−1)/(6−4) = 4/2 = 2. Correct fix: Find LCM (12): 5/6 = 10/12, 1/4 = 3/12, so 10/12 − 3/12 = 7/12.

**Mistake 4: Decimal Point Misplacement** Wrong thinking: 1.5 × 2.4 calculated as 15 × 24 = 360 (forgetting to place decimal). Correct fix: 15 × 24 = 360, but count two decimal places total: 360 → 3.60 = 3.6.

**Mistake 5: Memorizing Wrong Perfect Squares** Wrong thinking: Assuming 15² = 205 or 18² = 354. Correct fix: 15² = 225 (not 205), 18² = 324 (not 354). Drill squares 11–25 until automatic.

**Mistake 6: Applying Square Root to Sum** Wrong thinking: √(16 + 9) = √16 + √9 = 4 + 3 = 7. Correct fix: Simplify inside first: √(16 + 9) = √25 = 5. Square root doesn't distribute over addition.

Quick Reference

• **BODMAS = Brackets → Orders → Division/Multiplication (L to R) → Addition/Subtraction (L to R)**
• **Fraction division: flip the second fraction and multiply**
• **Decimal multiplication: ignore decimals, multiply, then count total decimal places**
• **Memorize squares 1–20 and cubes 1–10 cold**
• **√(perfect square) via factorization: pair prime factors**
• **Check calculations: estimation catches 80% of errors before marking**