Simplification
Overview
Simplification is the backbone of quantitative aptitude in MPESB Group 1/2/3 exams. Almost every mathematics section contains 3–5 direct simplification questions, and the skill underlies nearly all other arithmetic topics—from percentage to profit-loss to data interpretation. Mastering simplification means you solve faster and with fewer careless errors.
The core challenge is executing operations in the correct order while handling fractions, decimals, and surds efficiently. Students who internalize BODMAS and develop mental math shortcuts consistently outperform those who rely on lengthy written calculations. This topic rewards practice and pattern recognition over complex theory.
For MPESB exams, expect questions that test your ability to simplify nested brackets, convert between fractions and decimals rapidly, and rationalize surds. Time pressure is real—aim to solve each simplification question in 30–45 seconds.
Key Concepts
- BODMAS Rule: The universal order of operations—Brackets, Orders (powers/roots), Division, Multiplication, Addition, Subtraction. Division and Multiplication share equal priority (left to right), as do Addition and Subtraction.
- Bracket Hierarchy: Solve innermost brackets first. Order: Vinculum (bar) → Parentheses ( ) → Curly braces { } → Square brackets [ ].
- Fractions: A fraction a/b represents a parts of b equal divisions. Proper fractions have numerator < denominator; improper fractions have numerator ≥ denominator.
- Decimals: Positional representation where digits after the decimal point represent tenths, hundredths, thousandths, etc. Every fraction can be expressed as a terminating or recurring decimal.
- Surds: Irrational roots that cannot be simplified to rational numbers (e.g., √2, ³√5). Surds follow specific rules for multiplication, division, and rationalization.
- Like and Unlike Fractions: Like fractions share the same denominator and can be added/subtracted directly. Unlike fractions require LCM-based conversion first.
- Recurring Decimals: Decimals where a digit or group of digits repeats infinitely (e.g., 0.333... = 1/3). Converting them to fractions uses algebraic methods.
Formulas / Key Facts
BODMAS Operations
- B → Brackets (vinculum, then parentheses, then curly, then square)
- O → Orders (powers and roots)
- D/M → Division and Multiplication (left to right)
- A/S → Addition and Subtraction (left to right)
Fraction Operations
- Addition: a/b + c/d = (ad + bc)/bd
- Subtraction: a/b − c/d = (ad − bc)/bd
- Multiplication: a/b × c/d = ac/bd
- Division: a/b ÷ c/d = a/b × d/c = ad/bc
Decimal-Fraction Conversions
- 0.5 = 1/2, 0.25 = 1/4, 0.75 = 3/4, 0.125 = 1/8, 0.2 = 1/5, 0.333... = 1/3
Surd Rules
- √a × √b = √(ab)
- √a ÷ √b = √(a/b)
- (√a)² = a
- √a + √b ≠ √(a+b) — surds cannot be added unless they are like surds
- Rationalization: a/(√b) = a√b/b; a/(√b + √c) = a(√b − √c)/(b − c)
Useful Squares and Cubes
- 11² = 121, 12² = 144, 13² = 169, 14² = 196, 15² = 225
- 2³ = 8, 3³ = 27, 4³ = 64, 5³ = 125, 6³ = 216
Worked Examples
Example 1: BODMAS with Mixed Operations
Simplify: 48 ÷ 12 × 3 + 8 − 2 × 3
Step 1: No brackets or orders, so start with D/M from left to right.
- 48 ÷ 12 = 4
- 4 × 3 = 12
- 2 × 3 = 6
Step 2: Expression becomes 12 + 8 − 6
Step 3: Addition and subtraction left to right.
- 12 + 8 = 20
- 20 − 6 = 14
Example 2: Fraction Simplification
Simplify: 3/4 + 2/5 − 1/2
Step 1: Find LCM of denominators (4, 5, 2) = 20
Step 2: Convert each fraction.
- 3/4 = 15/20
- 2/5 = 8/20
- 1/2 = 10/20
Step 3: Perform operations.
- 15/20 + 8/20 − 10/20 = (15 + 8 − 10)/20 = 13/20
Answer: 13/20
Example 3: Surd Rationalization
Simplify: 6/(√3 + √2)
Step 1: Multiply numerator and denominator by the conjugate (√3 − √2).
= 6(√3 − √2) / [(√3 + √2)(√3 − √2)]
Step 2: Apply difference of squares in denominator. = 6(√3 − √2) / (3 − 2) = 6(√3 − √2) / 1
Answer: 6√3 − 6√2 or equivalently 6(√3 − √2)
Example 4: Nested Brackets
Simplify: 25 − [10 − {8 − (6 − 3)}]
Step 1: Solve innermost bracket first.
- (6 − 3) = 3
Step 2: Move outward to curly braces.
- {8 − 3} = 5
Step 3: Square brackets.
- [10 − 5] = 5
Step 4: Final subtraction.
- 25 − 5 = 20
Common Mistakes
- Ignoring left-to-right rule for D/M and A/S: Students think division always comes before multiplication. Correct approach: D and M have equal priority—solve whichever appears first from left to right.
- Adding surds incorrectly: Writing √2 + √3 = √5 is wrong. Correct approach: Unlike surds cannot be combined; √2 + √3 stays as √2 + √3.
- Forgetting to simplify final fractions: Leaving answer as 12/16 instead of 3/4. Correct approach: Always reduce to lowest terms by dividing numerator and denominator by their HCF.
- Mishandling negative signs in brackets: Especially when a minus precedes a bracket. Correct approach: Distribute the negative sign to all terms inside: −(a − b) = −a + b.
- Decimal place errors: Misplacing the decimal point in multiplication/division. Correct approach: Count total decimal places in factors; the product must have that many decimal places.
Quick Reference
- BODMAS order: Brackets → Orders → Division/Multiplication (L→R) → Addition/Subtraction (L→R)
- Fraction division: Flip the second fraction and multiply.
- Surd addition: Only like surds (same radicand) can be combined.
- Rationalize: Multiply by conjugate to remove surd from denominator.
- Recurring decimal to fraction: 0.abab... = ab/99; 0.aaa... = a/9
- Speed tip: Memorize squares (1–25), cubes (1–10), and common fraction-decimal equivalents.