Simplification — Study Notes
Maharashtra Police Bharti (Constable)
Overview
Simplification is the backbone of the Mathematics section in Maharashtra Police Bharti exams. Every question paper contains 3–5 direct simplification problems, and the skill is essential for solving Percentage, Profit-Loss, SI/CI and other applied topics quickly.
The core idea is straightforward: reduce a complex arithmetic expression to a single numerical value using the correct sequence of operations. Mastery here means speed — most candidates know the rules, but toppers apply shortcuts that save 30–60 seconds per question. Since the exam is time-bound, even small efficiencies compound into significant score gains.
Focus areas: BODMAS rule, fraction-decimal conversions, squaring/cubing shortcuts, and approximation techniques for lengthy calculations.
Key Concepts
- BODMAS Rule — The universal order of operations: Brackets → Of (meaning "of" or multiplication) → Division → Multiplication → Addition → Subtraction. Never deviate.
- Brackets Hierarchy — Solve innermost first: Vinculum (bar) → Round ( ) → Curly { } → Square [ ].
- "Of" means Multiplication — "Half of 40" = ½ × 40. Treat "of" as × but solve it before normal ÷ and ×.
- Division and Multiplication — Left to Right — When both appear without brackets, process whichever comes first from left to right; same rule for Addition and Subtraction.
- Fraction to Decimal Conversions — Memorise common values (see Quick Reference) to avoid repeated division during the exam.
- Negative Number Rules — (−) × (−) = (+); (−) × (+) = (−); same logic for division.
- Approximation — When options are well-spaced, round numbers to nearest easy value, compute, and pick closest answer.
Formulas / Key Facts
| Concept | Formula / Fact |
|---|---|
| Square of (a + b) | a² + 2ab + b² |
| Square of (a − b) | a² − 2ab + b² |
| Difference of squares | a² − b² = (a + b)(a − b) |
| Cube shortcut | (a + b)³ = a³ + 3a²b + 3ab² + b³ |
| Percentage ↔ Fraction | 25% = ¼, 33⅓% = ⅓, 50% = ½, 66⅔% = ⅔, 75% = ¾ |
| Multiplying by 5 | Number × 5 = Number × 10 ÷ 2 |
| Multiplying by 25 | Number × 25 = Number × 100 ÷ 4 |
| Dividing by 5 | Number ÷ 5 = Number × 2 ÷ 10 |
| Square of numbers ending in 5 | n5² = n × (n+1) followed by 25. Example: 35² = 3×4 = 12, so 1225 |
Worked Examples
Example 1 — Pure BODMAS
Problem: Simplify 18 + 12 ÷ 3 × 2 − 4
Step-by-step:
- No brackets or "of." Start with ÷ and × left to right.
- 12 ÷ 3 = 4
- 4 × 2 = 8
- Expression becomes 18 + 8 − 4
- Addition and subtraction left to right: 18 + 8 = 26; 26 − 4 = 22
Example 2 — Brackets with "Of"
Problem: Simplify 36 ÷ (12 − 6) + 2 of 5
Step-by-step:
- Brackets first: 12 − 6 = 6
- Expression now: 36 ÷ 6 + 2 of 5
- "Of" next: 2 of 5 = 2 × 5 = 10
- Division: 36 ÷ 6 = 6
- Addition: 6 + 10 = 16
Example 3 — Fractions and Mixed Numbers
Problem: Simplify 3¼ + 2½ × 4 − 1
Step-by-step:
- Convert to improper fractions: 3¼ = 13/4, 2½ = 5/2
- Multiplication first: 5/2 × 4 = 20/2 = 10
- Expression becomes: 13/4 + 10 − 1
- Convert to common base (or decimals): 13/4 = 3.25; 10 = 10; 1 = 1
- 3.25 + 10 − 1 = 12.25 or 49/4
Example 4 — Nested Brackets
Problem: Simplify 50 − [20 − {10 − (7 − 3)}]
Step-by-step:
- Innermost ( ): 7 − 3 = 4
- Next { }: 10 − 4 = 6
- Next [ ]: 20 − 6 = 14
- Outer expression: 50 − 14 = 36
Common Mistakes
| Wrong Thinking | Correct Fix |
|---|---|
| Performing addition before division because it appears earlier in the expression. | Always follow BODMAS order, not the left-to-right position of the first symbol you see. |
| Treating "of" as a separate low-priority operation. | "Of" = multiplication and ranks above ÷ and × in BODMAS (solve immediately after brackets). |
| Subtracting before adding when both appear together. | Addition and Subtraction share the same priority — solve strictly left to right. |
| Ignoring the bar (vinculum) over part of an expression. | The bar acts as a hidden bracket; simplify whatever is under the bar first. |
| Rounding too early in multi-step calculations. | Approximate only in the final step or when options are clearly separated by large gaps. |
Quick Reference
- BODMAS sequence: Brackets → Of → Division → Multiplication → Addition → Subtraction
- Brackets order: Vinculum (¯) → ( ) → { } → [ ]
- Key fractions: ½ = 0.5, ¼ = 0.25, ⅓ ≈ 0.333, ⅔ ≈ 0.667, ¾ = 0.75, ⅕ = 0.2
- Squaring shortcut (ending in 5): 45² → 4×5 = 20, append 25 → 2025
- Multiplying by 11: For two-digit ab → a, (a+b), b. Example: 34 × 11 = 3, 7, 4 = 374
- Check your answer: Substitute back or use rough estimation to verify plausibility.
Total ≈ 750 words. Revise BODMAS daily with 10 timed problems to build speed and accuracy.