Simplification: BODMAS, Fractions, Decimals, HCF-LCM, Squares, Cubes, Divisibility, Remainders & Approximation

Simplification is the single most important building-block topic in TNPSC Aptitude and Mental Ability. Almost every other topic in Quantitative Aptitude — Ratio & Proportion, Percentage, Profit & Loss, Time & Work, Average, Data Interpretation — depends on your speed and accuracy in simplification. In TNPSC Group 1, Group 2, Group 2A, Group 4, and VAO exams, 3 to 6 questions are directly asked from Simplification every single year, and indirectly, simplification skills are used in almost 60-70% of all Aptitude questions. This chapter covers every sub-topic you need — BODMAS, Fractions, Decimals, HCF & LCM, Squares & Square Roots, Cubes & Cube Roots, Divisibility Rules, Remainder Theorem basics, Approximation, and Basic Algebraic Simplification — with deep explanations, shortcut tricks, and TNPSC-pattern solved examples.

Exam Tip: In TNPSC exams, calculators are NOT allowed. So your entire preparation strategy for this chapter must focus on mental calculation speed, not just conceptual understanding.

1. BODMAS Rule

1.1 What is BODMAS?

BODMAS is the rule that decides the correct ORDER OF OPERATIONS when a mathematical expression contains multiple operations like addition, subtraction, multiplication, division, and brackets. Without a fixed order, the same expression could give different answers to different people. BODMAS removes this confusion by fixing a universal sequence.

Letter	Stands For	Meaning
B	Brackets	Solve ( ), { }, [ ] first — in the order: small ( ), curly { }, square [ ]
O	Of	"Of" means multiplication, but it is solved before normal ×, ÷ (e.g., ½ of 20 = 10)
D	Division	Solve ÷ operations
M	Multiplication	Solve × operations
A	Addition	Solve + operations
S	Subtraction	Solve − operations

Important Rule: Division and Multiplication have EQUAL priority — solve them left to right as they appear, NOT division always before multiplication. Same rule applies to Addition and Subtraction — solve left to right.

1.2 Order of Solving Brackets

When multiple types of brackets appear together, always solve in this sequence:

1. Vinculum / Bar ( ‾‾‾ ) — the line above numbers, solved first of all
2. Small brackets ( )
3. Curly brackets { }
4. Square brackets [ ]

1.3 Worked Examples

Example 1: Simplify: 45 ÷ 9 × 3 + 6 − 2

Step 1 (D & M, left to right): 45 ÷ 9 = 5, then 5 × 3 = 15
Step 2: Expression becomes 15 + 6 − 2
Step 3 (A & S, left to right): 15 + 6 = 21, 21 − 2 = 19
Answer: 19

Example 2: Simplify: 12 + [18 − {15 − (8 − 3)}]

Step 1 (innermost bracket): 8 − 3 = 5
Step 2: {15 − 5} = 10
Step 3: [18 − 10] = 8
Step 4: 12 + 8 = 20
Answer: 20

Example 3 (TNPSC pattern with "of"): Simplify: 1/2 of 40 + 3/5 of 25 − 6

Step 1 (Of first): 1/2 × 40 = 20; 3/5 × 25 = 15
Step 2: 20 + 15 − 6 = 29
Answer: 29

1.4 Common Mistakes in BODMAS

• Solving division before multiplication automatically — WRONG. Follow left to right.
• Ignoring "of" as a separate high-priority operation.
• Solving brackets from outside to inside instead of inside to outside.
• Sign errors while removing brackets, especially with a negative sign before a bracket (all signs inside flip).

2. Fractions

2.1 Definition

A fraction represents a part of a whole and is written as p/q, where p is the numerator and q is the denominator (q ≠ 0).

Type	Definition	Example
Proper Fraction	Numerator < Denominator	3/7
Improper Fraction	Numerator ≥ Denominator	9/4
Mixed Fraction	Whole number + Proper fraction	2¾
Like Fractions	Same denominator	2/9, 5/9
Unlike Fractions	Different denominators	2/3, 3/7
Equivalent Fractions	Different form, same value	1/2 = 2/4 = 4/8

2.2 Basic Operations on Fractions

Addition/Subtraction: Convert to like fractions using LCM of denominators, then add/subtract numerators.

Multiplication: Multiply numerators together and denominators together: (a/b) × (c/d) = ac/bd

Division: Multiply by the reciprocal of the second fraction: (a/b) ÷ (c/d) = (a/b) × (d/c)

2.3 Converting Mixed Fraction to Improper Fraction

Formula: Improper fraction = (Whole number × Denominator + Numerator) / Denominator

Example: 3⅖ = (3×5 + 2)/5 = 17/5

2.4 Comparing Fractions Quickly (Cross Multiplication Trick)

To compare a/b and c/d, cross multiply: if a×d > b×c, then a/b > c/d.

Example: Compare 5/7 and 8/11 → 5×11=55, 7×8=56. Since 55<56, 5/7 < 8/11.

2.5 Worked Examples

Example 1: Simplify: 2/3 + 3/4 − 1/6

LCM of 3, 4, 6 = 12
= 8/12 + 9/12 − 2/12 = 15/12 = 5/4 = 1¼
Answer: 5/4

Example 2 (TNPSC pattern): Simplify: (2/3 of 3/4) ÷ (5/6 of 2/5)

Step 1: 2/3 of 3/4 = 1/2; 5/6 of 2/5 = 1/3
Step 2: (1/2) ÷ (1/3) = 1/2 × 3/1 = 3/2
Answer: 3/2

2.6 Common Mistakes

• Adding numerators and denominators directly without finding LCM.
• Forgetting to flip the second fraction during division.
• Not reducing the final answer to lowest terms.

3. Decimals

3.1 Definition

A decimal number is a number that uses a decimal point to represent fractional parts based on powers of 10. Example: 45.67 means 45 + 6/10 + 7/100.

3.2 Operations on Decimals

Operation	Rule
Addition/Subtraction	Align decimal points, then add/subtract as usual
Multiplication	Multiply as whole numbers, then place the decimal point by counting total decimal places from both numbers
Division	Convert divisor to a whole number by shifting decimal, shift dividend's decimal the same number of places

3.3 Converting Fraction to Decimal and Vice Versa

Fraction to Decimal: Divide numerator by denominator. Example: 3/4 = 0.75

Decimal to Fraction: Write decimal digits as numerator over appropriate power of 10, then simplify. Example: 0.375 = 375/1000 = 3/8

3.4 Recurring Decimals

Pure Recurring Decimal: All digits after decimal repeat. Formula: 0.ab repeating = ab/99

Mixed Recurring Decimal: Some digits don't repeat. Formula: (whole number formed by non-repeating+repeating digits − non-repeating part) / (9 for each repeating digit, followed by 0 for each non-repeating digit)

Example: Convert 0.333... (0.3̄) to fraction = 3/9 = 1/3

Example: Convert 0.1666... (0.16̄) to fraction = (16−1)/90 = 15/90 = 1/6

3.5 Worked Examples

Example 1: Simplify: 12.5 × 0.4 + 3.6 ÷ 0.9

12.5 × 0.4 = 5
3.6 ÷ 0.9 = 4
5 + 4 = 9
Answer: 9

Example 2: What is 0.0025 × 0.16?

25 × 16 = 400. Total decimal places = 4+2 = 6. Answer = 0.000400 = 0.0004
Answer: 0.0004

3.6 Common Mistakes

• Misplacing the decimal point after multiplication.
• Forgetting to shift both divisor and dividend equally in division.
• Wrong formula application for mixed recurring decimals.

4. HCF and LCM

4.1 Definitions

HCF (Highest Common Factor): The largest number that divides two or more numbers exactly, without leaving a remainder. Also called GCD (Greatest Common Divisor).

LCM (Least Common Multiple): The smallest number that is exactly divisible by two or more given numbers.

4.2 Key Formula (Very Important for TNPSC)

Product of two numbers = HCF × LCM

This formula is heavily tested. If you know any three of (Number 1, Number 2, HCF, LCM), you can find the fourth.

4.3 Methods to Find HCF

Method	Process
Prime Factorization	Break each number into prime factors, multiply the COMMON prime factors with lowest powers
Division Method (Euclid's Method)	Divide larger number by smaller; divide previous divisor by remainder; repeat till remainder = 0; last divisor is HCF

4.4 Methods to Find LCM

Method	Process
Prime Factorization	Take ALL prime factors (common and uncommon) with HIGHEST powers, multiply them
Division Method	Divide numbers by common prime numbers simultaneously till no common factor remains; multiply all divisors and remaining quotients

4.5 HCF and LCM of Fractions

HCF of fractions = HCF of numerators / LCM of denominators
LCM of fractions = LCM of numerators / HCF of denominators

4.6 Worked Examples

Example 1: Find HCF and LCM of 24 and 36.

24 = 2³ × 3; 36 = 2² × 3²
HCF = 2² × 3 = 12
LCM = 2³ × 3² = 72
Check: 24×36 = 864; 12×72 = 864 ✓
Answer: HCF = 12, LCM = 72

Example 2 (TNPSC pattern): The HCF of two numbers is 12 and their LCM is 144. If one number is 36, find the other.

Other number = (HCF × LCM) / Given number = (12 × 144)/36 = 1728/36 = 48
Answer: 48

Example 3 (Real-life application): Find the greatest number that divides 245 and 1029 leaving remainder 5 in each case.

Subtract remainder: 245−5=240; 1029−5=1024
Find HCF of 240 and 1024 → HCF = 16
Answer: 16

Example 4: Find the least number which when divided by 12, 16, and 24 leaves remainder 5 in each case.

LCM of 12, 16, 24 = 48
Required number = 48 + 5 = 53
Answer: 53

4.7 Shortcut Tricks

• To find the greatest number dividing x, y, z leaving same remainder r each time: HCF of (x−r), (y−r), (z−r)
• To find the greatest number dividing x, y, z leaving remainders a, b, c respectively: HCF of (x−a), (y−b), (z−c)
• To find the least number divisible by x, y, z leaving remainder r each time: LCM(x,y,z) + r
• To find the least number divisible by x, y, z which is exactly divisible (leaves no remainder): LCM(x,y,z)

5. Squares and Square Roots

5.1 Definition

Square of a number = number multiplied by itself (n²). Square root of a number is the value which, multiplied by itself, gives the original number (√n).

5.2 Important Squares to Memorize (1 to 30)

Memorizing squares up to 30 saves enormous time in TNPSC exams: 11²=121, 12²=144, 13²=169, 14²=196, 15²=225, 16²=256, 17²=289, 18²=324, 19²=361, 20²=400, 21²=441, 22²=484, 23²=529, 24²=576, 25²=625, 26²=676, 27²=729, 28²=784, 29²=841, 30²=900

5.3 Methods to Find Square Root

Method	Best Used For
Prime Factorization	Perfect squares — pair up identical prime factors, take one from each pair
Long Division Method	Large numbers, decimals, non-perfect squares
Estimation Method	Quick approximate answers in MCQs

5.4 Square Root Shortcut for Perfect Squares (Last Digit Trick)

For a perfect square ending in a particular digit, its square root's last digit follows a pattern:

Last digit of perfect square	Possible last digit of square root
1	1 or 9
4	2 or 8
9	3 or 7
6	4 or 6
5	5
0	0
2, 3, 7, 8	Not a perfect square

Example: Find √7921. Last digit 1 → root ends in 1 or 9. Since 7921 is between 80²(6400) and 90²(8100), root is between 80-90, closer to 89. Check 89² = 7921. Answer: 89

5.5 Square Root of Decimals and Fractions

√(a/b) = √a/√b. For decimals, make the number of decimal digits even by adding a zero if needed, then apply long division method.

5.6 Worked Examples

Example 1: Find the square root of 1156 using factorization.

1156 = 2×2×17×17 = 2² × 17²
√1156 = 2×17 = 34
Answer: 34

Example 2 (TNPSC pattern): If √0.0081 = ?

0.0081 = 81/10000
√81=9, √10000=100
= 9/100 = 0.09
Answer: 0.09

5.7 Common Mistakes

• Not pairing decimal digits correctly before applying long division.
• Confusing square root shortcuts with actual verification — always verify with one multiplication in MCQs.

6. Cubes and Cube Roots

6.1 Definition

Cube of a number = number multiplied by itself three times (n³). Cube root is the value which, when cubed, gives the original number (∛n).

6.2 Important Cubes to Memorize (1 to 15)

1³=1, 2³=8, 3³=27, 4³=64, 5³=125, 6³=216, 7³=343, 8³=512, 9³=729, 10³=1000, 11³=1331, 12³=1728, 13³=2197, 14³=2744, 15³=3375

6.3 Cube Root Shortcut (Last Digit Trick)

Unlike square roots, cube roots have a UNIQUE last digit for each ending digit — this makes cube root shortcuts extremely reliable:

Last digit of perfect cube	Last digit of cube root
1	1
8	2
7	3
4	4
5	5
6	6
3	7
2	8
9	9
0	0

6.4 Method to Find Cube Root of Large Perfect Cubes (Grouping Trick)

Step 1: Group digits in pairs of 3 from the right.
Step 2: The last group gives the units digit of the cube root using the table above.
Step 3: The first group tells you the tens digit — find the largest cube ≤ first group's value.

Example: Find ∛175616
Group: 175 | 616
Last group 616 ends in 6 → cube root ends in 6
First group 175: largest cube ≤175 is 5³=125 (since 6³=216 is too big) → tens digit = 5
Answer: 56 (Verify: 56³ = 175616 ✓)

6.5 Worked Examples

Example 1: Find ∛13824

Group: 13 | 824
824 ends in 4 → root ends in 4
13: largest cube ≤13 is 2³=8 → tens digit=2
Answer: 24 (Verify: 24³=13824 ✓)

Example 2: Simplify ∛(0.000343)

0.000343 = 343/1000000
∛343=7, ∛1000000=100
= 7/100 = 0.07
Answer: 0.07

6.6 Common Mistakes

• Grouping digits in pairs of 2 instead of 3 (that's for square root, not cube root).
• Forgetting cube root last-digit is unique (unlike square root which has two options).

7. Divisibility Rules

7.1 Why This Matters for TNPSC

Divisibility rules help you quickly check whether a large number is divisible by another number WITHOUT actually performing division — essential for speed in HCF/LCM, factorization, and elimination-based MCQ solving.

Divisor	Divisibility Rule
2	Last digit is 0, 2, 4, 6, or 8
3	Sum of all digits is divisible by 3
4	Last two digits form a number divisible by 4
5	Last digit is 0 or 5
6	Number is divisible by both 2 and 3
7	Double the last digit, subtract from remaining number; result divisible by 7 (repeat if needed)
8	Last three digits form a number divisible by 8
9	Sum of all digits is divisible by 9
10	Last digit is 0
11	(Sum of digits at odd places − Sum of digits at even places) is 0 or divisible by 11
12	Number is divisible by both 3 and 4
13	Multiply last digit by 4, add to remaining number; result divisible by 13

7.2 Worked Examples

Example 1: Is 4368 divisible by 11?

Digits from right: 8(odd pos 1),6(even pos2),3(odd pos3),4(even pos4)
Sum at odd places (1st,3rd from right)=8+3=11
Sum at even places (2nd,4th from right)=6+4=10
Difference=11−10=1 → not divisible by 11
Answer: Not divisible

Example 2 (TNPSC pattern): Find the smallest number to be added to 5678 to make it divisible by 9.

Sum of digits of 5678 = 5+6+7+8=26
Next multiple of 9 after 26 is 27
Number to add = 27−26=1
Answer: 1

7.3 Common Mistakes

• Applying the divisibility rule for 3 to check divisibility by 9 (different digit-sum targets).
• Confusing odd/even position counting direction in the rule for 11.

8. Remainders

8.1 Basic Concept

When a number (Dividend) is divided by another number (Divisor), we get a Quotient and a Remainder.

Dividend = (Divisor × Quotient) + Remainder

Remainder is always less than the Divisor.

8.2 Properties of Remainders (Useful for TNPSC MCQs)

• If a number N leaves remainder r when divided by d, then N = dk + r for some integer k.
• Remainder of sum = (Remainder1 + Remainder2) mod Divisor
• Remainder of product = (Remainder1 × Remainder2) mod Divisor
• To find the remainder of a large power, look for a repeating cycle (pattern) of remainders — very useful in unit-digit-of-power questions.

8.3 Worked Examples

Example 1: A number when divided by 5 gives remainder 3. What is the remainder when its square is divided by 5?

N=5k+3 → N²=25k²+30k+9
25k² and 30k are divisible by 5. Remainder from 9÷5 = 4
Answer: 4

Example 2 (TNPSC pattern): Find the remainder when 2^50 is divided by 7.

Powers of 2 mod 7 cycle: 2¹=2, 2²=4, 2³=1, then repeats every 3 terms
50 ÷ 3 = 16 remainder 2 → same as 2² = 4
Answer: 4

8.4 Common Mistakes

• Forgetting that remainder must always be smaller than the divisor — a "remainder" larger than divisor means recalculate.
• Not reducing remainder-of-sum/product results with a final mod operation.

9. Approximation

9.1 Concept

Approximation questions ask you to round off numbers to estimate an answer close to the actual value, without doing exact lengthy calculations — vital for saving time in objective-type TNPSC exams.

9.2 Rules of Rounding Off

• If the digit to be dropped is 5 or more, round UP the previous digit.
• If the digit to be dropped is less than 5, the previous digit remains unchanged.
• For "≈" (approximately equal to) type questions, round each number to the nearest convenient whole number before calculating.

9.3 Worked Examples

Example 1: Find the approximate value of 3998.9 ÷ 19.99 × 5.001

Approximate: 4000 ÷ 20 × 5 = 200×5=1000
Answer: ≈1000

Example 2: What approximate value should come in place of "?" in: 419.99 + 218.001 − 99.98 = ?

Approximate: 420 + 218 − 100 = 538
Answer: ≈538

9.4 Common Mistakes

• Rounding numbers too aggressively, causing large errors in multi-step calculations.
• Rounding in the wrong direction (down instead of up).

10. Basic Algebraic Simplification

10.1 Important Algebraic Identities (Must Memorize)

Identity	Formula
Square of sum	(a+b)² = a² + 2ab + b²
Square of difference	(a−b)² = a² − 2ab + b²
Difference of squares	a² − b² = (a+b)(a−b)
Cube of sum	(a+b)³ = a³ + 3a²b + 3ab² + b³
Cube of difference	(a−b)³ = a³ − 3a²b + 3ab² − b³
Sum of cubes	a³ + b³ = (a+b)(a² − ab + b²)
Difference of cubes	a³ − b³ = (a−b)(a² + ab + b²)
Sum & difference relation	(a+b)² − (a−b)² = 4ab
	(a+b)² + (a−b)² = 2(a² + b²)

10.2 Why These Identities Matter in TNPSC

Many "Simplification" questions look extremely lengthy but are actually direct identity applications in disguise. Recognizing the identity pattern instantly cuts your solving time from 2 minutes to 15 seconds.

10.3 Worked Examples

Example 1: Simplify (97)² using identity.

97 = 100−3
(100−3)² = 100² − 2×100×3 + 3² = 10000−600+9=9409
Answer: 9409

Example 2 (TNPSC pattern): If a+b=12 and ab=27, find a²+b²

a²+b² = (a+b)² − 2ab = 144−54=90
Answer: 90

Example 3: Simplify: (25.4×25.4 − 14.6×14.6) / (25.4−14.6)

This is a²−b² form: numerator=(a+b)(a−b), and denominator=(a−b)
So expression simplifies to (a+b) = 25.4+14.6=40
Answer: 40

Example 4: Simplify: (7.83×7.83 + 2×7.83×2.17 + 2.17×2.17)

This is (a+b)² form where a=7.83, b=2.17
= (7.83+2.17)² = 10² = 100
Answer: 100

10.4 Common Mistakes

• Not recognizing identity patterns and instead doing long multiplication (wastes precious exam time).
• Sign errors in (a−b)² and (a−b)³ expansions.

📝 Model MCQ Questions (TNPSC Exam Pattern)

Q.No	Question	Options	Answer with Solution
1	Simplify: 8 + 4 ÷ 2 × 3 − 5	a) 9 b) 15 c) 5 d) 11	b) 15 — 4÷2=2, 2×3=6, 8+6−5=9... Recheck: 8+4÷2×3−5 = 8+(4÷2)×3−5=8+2×3−5=8+6−5=9. Correct Answer: a) 9
2	The HCF of two numbers is 18 and LCM is 720. If one number is 72, find the other.	a) 150 b) 180 c) 200 d) 240	b) 180 — (18×720)/72 = 180
3	Find √3969	a) 61 b) 63 c) 67 d) 69	b) 63 — 63×63=3969
4	Find ∛35937	a) 31 b) 33 c) 35 d) 39	b) 33 — Last digit 7→root ends in 3; 35→3³=27≤35→tens digit3; 33³=35937
5	Which of the following is divisible by 11?	a) 2531 b) 4521 c) 7942 d) 9042	d) 9042 — (9+4)−(0+2)=11, divisible by 11
6	Simplify: 3/5 of 2/3 + 1/4 ÷ 1/2	a) 1.1 b) 1 c) 0.9 d) 1.5	a) 1.1 — 3/5×2/3=2/5=0.4; 1/4÷1/2=1/2=0.5; 0.4+0.5=0.9... Recheck: 2/5+1/2 = 4/10+5/10=9/10=0.9. Correct Answer: c) 0.9

🎯 TNPSC Exam Strategy for This Chapter

• Memorize squares (1-30), cubes (1-15), and basic fraction-decimal conversions BEFORE attempting practice questions — this alone saves 30-40 seconds per question.
• In the exam hall, first scan the question for identity patterns (a²−b², (a+b)², etc.) before starting calculation.
• For BODMAS questions, write down each step on rough sheet — mental-only solving increases error risk under exam pressure.
• For HCF/LCM word problems, identify keywords: "greatest number that divides" → HCF concept; "least number divisible by" → LCM concept.
• Practice divisibility rules for 7, 11, 13 repeatedly — these are less intuitive and commonly tested.
• In approximation questions, always round to the NEAREST convenient number, not just the nearest 10.
• Time target: Aim to solve each simplification MCQ within 45-50 seconds during practice, gradually reducing to 30 seconds.

✅ Final Revision Points

• BODMAS: Brackets → Of → Division/Multiplication (left to right) → Addition/Subtraction (left to right)
• Fractions: LCM for addition/subtraction; cross-multiply for division; direct multiply for multiplication
• Decimals: Align points for +/−; count total decimal places for ×; shift decimal for ÷
• HCF × LCM = Product of two numbers — the single most tested formula
• Square root last digit trick: two possibilities; Cube root last digit trick: one unique possibility
• Divisibility: Master rules for 2,3,4,5,6,8,9,10,11 thoroughly; know 7 and 13 for advanced level
• Remainder: Dividend = Divisor×Quotient + Remainder; remainder always less than divisor
• Approximation: Round to nearest convenient number before calculating
• Algebraic identities: (a+b)², (a−b)², a²−b², cube identities — recognize patterns instantly

Conclusion

Simplification is not just one chapter — it is the FOUNDATION on which your entire TNPSC Aptitude preparation is built. A strong grip on BODMAS, fractions, decimals, HCF-LCM, squares, cubes, divisibility, remainders, approximation, and algebraic identities directly boosts your speed and accuracy in every other quantitative topic. Practice daily with a timer, revise the formula tables regularly, and attempt previous year TNPSC question papers to identify the exact question patterns the Commission prefers. Consistent practice of this chapter alone can help you comfortably secure 4-6 marks in the Aptitude section of TNPSC Group 1, Group 2, Group 2A, Group 4, and VAO examinations.