Whole Numbers, Decimals and Fractions

Whole Numbers, Decimals and Fractions is the foundation chapter of Quantitative Aptitude. In SSC CGL, 1 to 2 questions directly come from this chapter and indirectly every other chapter uses these concepts. This chapter covers properties of whole numbers, decimal operations, types of fractions and their calculations in complete depth. Students who master this chapter find the rest of the Quantitative Aptitude section significantly easier. Every concept in this chapter is important from the SSC examination perspective and nothing should be skipped.

1. Number System - Complete Overview

1.1 Types of Numbers

Understanding the number system is the first and most important step. Each type of number has its own specific properties.

Natural Numbers (N):

• Definition: Counting numbers starting from 1
• Set: {1, 2, 3, 4, 5, 6, 7, 8, ...}
• Smallest natural number: 1
• There is no largest natural number (infinite set)
• Every natural number except 1 is either prime or composite
• Zero is NOT a natural number

Whole Numbers (W):

• Definition: Natural numbers together with zero
• Set: {0, 1, 2, 3, 4, 5, ...}
• Smallest whole number: 0
• Difference from natural numbers: Only zero is additionally included
• Every natural number is a whole number but every whole number is not a natural number

Integers (Z):

• Definition: Whole numbers together with negative numbers
• Set: {..., -4, -3, -2, -1, 0, 1, 2, 3, 4, ...}
• Positive integers: {1, 2, 3, 4, ...} which are the same as natural numbers
• Negative integers: {-1, -2, -3, -4, ...}
• Zero is neither positive nor negative
• Every whole number is an integer but every integer is not a whole number

Rational Numbers (Q):

• Definition: Numbers that can be written in p/q form where p and q are integers and q ≠ 0
• Examples: 3/4, -5/2, 7 (= 7/1), 0 (= 0/1), 0.5 (= 1/2), 0.333... (= 1/3)
• Every integer is a rational number
• Both terminating decimals and recurring decimals are rational numbers
• The decimal expansion of a rational number is either terminating or recurring

Irrational Numbers:

• Definition: Numbers that CANNOT be written in p/q form
• Examples: √2, √3, √5, π, e, √7
• Their decimal expansion is non-terminating and non-recurring
• √2 = 1.41421356... (never ends, never repeats)
• π = 3.14159265... (never ends, never repeats)
• Important: √4 = 2 (rational), √9 = 3 (rational) - square roots of perfect squares are rational

Real Numbers (R):

• Definition: Rational numbers together with Irrational numbers
• Every number on the number line is a real number
• Real numbers cover the complete number line

Complex Numbers:

• Definition: Numbers in a + bi form where i = √(-1)
• For SSC CGL, knowing the name and basic form is sufficient

Relationship (subset notation): N ⊂ W ⊂ Z ⊂ Q ⊂ R

1.2 Even and Odd Numbers - Complete Rules

Even Numbers:

• Definition: Numbers exactly divisible by 2
• General form: 2n where n is any integer
• Set: {..., -4, -2, 0, 2, 4, 6, 8, ...}
• 0 is an even number
• 2 is the only even prime number

Odd Numbers:

• Definition: Numbers not divisible by 2
• General form: 2n + 1 where n is any integer
• Set: {..., -3, -1, 1, 3, 5, 7, 9, ...}

Operations on Even and Odd Numbers:

Important SSC Trick:

• Sum of n consecutive integers starting from 1:
  • If n is odd: sum = n × middle number
  • If n is even: sum = (n/2) × (first + last)

1.3 Prime and Composite Numbers - Deep Coverage

Prime Numbers:

• Definition: Numbers with exactly 2 factors - 1 and itself
• Smallest prime number: 2
• Only even prime number: 2
• All primes other than 2 are odd
• 1 is neither prime nor composite
• 0 is neither prime nor composite

List of prime numbers up to 100 (25 prime numbers): 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

How to check whether a number is prime:

• Calculate √n
• Check divisibility by all primes up to √n
• If not divisible by any prime up to √n, the number is prime
• Example: Is 97 prime? √97 ≈ 9.8. Check divisibility by 2, 3, 5, 7. None divides 97. So 97 is prime.

Composite Numbers:

• Definition: Numbers with more than 2 factors
• Smallest composite number: 4
• Every composite number has at least one prime factor

Fundamental Theorem of Arithmetic: Every composite number can be uniquely expressed as a product of prime numbers.

• Example: 360 = 2³ × 3² × 5

Number of factors of a number: If N = p^a × q^b × r^c then: Total number of factors = (a+1)(b+1)(c+1)

• Example: 360 = 2³ × 3² × 5¹
• Total factors = (3+1)(2+1)(1+1) = 4 × 3 × 2 = 24

Sum of all factors: = [(p^(a+1) - 1)/(p-1)] × [(q^(b+1) - 1)/(q-1)] × ...

Co-prime Numbers:

• Two numbers whose HCF = 1
• Examples: (8, 9), (4, 9), (15, 16)
• Note: Neither number needs to itself be prime
• Consecutive integers are always co-prime
• If (a, b) are co-prime and c divides ab, then c divides a or c divides b

Twin Primes: Pairs of prime numbers with a difference of 2

• (3,5), (5,7), (11,13), (17,19), (29,31), (41,43), (59,61), (71,73)

Prime Triplet: (3, 5, 7) is the only prime triplet

1.4 Divisibility Rules - All Rules with Examples

Divisibility by 2:

• Rule: Last digit is 0, 2, 4, 6, or 8
• Example: 47,832 - last digit 2, so divisible by 2 ✓

Divisibility by 3:

• Rule: Sum of all digits is divisible by 3
• Example: 52,416 → 5+2+4+1+6 = 18 → 18 ÷ 3 = 6 ✓

Divisibility by 4:

• Rule: The number formed by the last two digits is divisible by 4
• Example: 7,316 → 16 ÷ 4 = 4 ✓
• Example: 7,318 → 18 ÷ 4 = 4.5 ✗

Divisibility by 5:

• Rule: Last digit is 0 or 5
• Example: 3,475 - last digit 5 ✓

Divisibility by 6:

• Rule: Divisible by both 2 AND 3
• Example: 5,412 → even (divisible by 2 ✓) and 5+4+1+2 = 12 (divisible by 3 ✓) → divisible by 6 ✓

Divisibility by 7:

• Rule: Double the last digit, subtract from the remaining number. Result must be divisible by 7. Repeat the process if needed.
• Example: 343 → remaining part: 34, last digit doubled: 3×2 = 6 → 34 - 6 = 28 → 28 ÷ 7 = 4 ✓
• Example: 2,401 → 240 - 2 = 238 → 23 - 16 = 7 ✓

Divisibility by 8:

• Rule: The number formed by the last three digits is divisible by 8
• Example: 9,128 → 128 ÷ 8 = 16 ✓
• Example: 1,234 → 234 ÷ 8 = 29.25 ✗

Divisibility by 9:

• Rule: Sum of all digits is divisible by 9
• Example: 9,801 → 9+8+0+1 = 18 → 18 ÷ 9 = 2 ✓

Divisibility by 10:

• Rule: Last digit is 0
• Example: 4,570 ✓

Divisibility by 11:

• Rule: (Sum of digits at odd positions from right) - (Sum of digits at even positions from right) = 0 or divisible by 11
• Positions are counted from the RIGHT
• Example: 85,426 → Odd positions (from right): 6, 4, 8 = 18 → Even positions: 2, 5 = 7 → 18 - 7 = 11 ✓
• Example: 1,331 → Odd positions: 1+3 = 4 → Even positions: 3+1 = 4 → 4 - 4 = 0 ✓

Divisibility by 12:

• Rule: Divisible by both 3 and 4
• Example: 1,644 → 1+6+4+4 = 15 (divisible by 3 ✓) and last two digits 44 ÷ 4 = 11 ✓

Divisibility by 13:

• Rule: Multiply last digit by 4, add to remaining number. Result must be divisible by 13.
• Example: 286 → 28 + (6×4) = 28 + 24 = 52 → 52 ÷ 13 = 4 ✓

Divisibility by 14:

• Rule: Divisible by both 2 and 7

Divisibility by 15:

• Rule: Divisible by both 3 and 5

Divisibility by 16:

• Rule: Last four digits form a number divisible by 16

Divisibility by 25:

• Rule: Last two digits are 00, 25, 50, or 75

Divisibility by 99:

• Rule: Split number into groups of two digits from the right, sum of all groups must be divisible by 99
• Example: 99,297 → 9 + 92 + 97 = 198 = 2 × 99 ✓

Divisibility by 101:

• Rule: Split into pairs from right, alternating sum = 0 or divisible by 101

1.5 Properties of Whole Numbers - All Properties

Closure Property:

• Addition: a + b ∈ W for all a, b ∈ W (CLOSED)
  • Example: 3 + 5 = 8 ∈ W ✓
• Multiplication: a × b ∈ W for all a, b ∈ W (CLOSED)
  • Example: 3 × 5 = 15 ∈ W ✓
• Subtraction: NOT closed. 3 - 5 = -2 ∉ W ✗
• Division: NOT closed. 3 ÷ 5 = 0.6 ∉ W ✗

Commutative Property:

• Addition: a + b = b + a (COMMUTATIVE)
  • Example: 7 + 3 = 3 + 7 = 10 ✓
• Multiplication: a × b = b × a (COMMUTATIVE)
  • Example: 4 × 6 = 6 × 4 = 24 ✓
• Subtraction: NOT commutative. 7 - 3 ≠ 3 - 7 ✗
• Division: NOT commutative. 12 ÷ 4 ≠ 4 ÷ 12 ✗

Associative Property:

• Addition: (a + b) + c = a + (b + c) (ASSOCIATIVE)
  • Example: (2+3)+4 = 2+(3+4) = 9 ✓
• Multiplication: (a × b) × c = a × (b × c) (ASSOCIATIVE)
  • Example: (2×3)×4 = 2×(3×4) = 24 ✓
• Subtraction: NOT associative. (9-4)-2 = 3 ≠ 9-(4-2) = 7 ✗
• Division: NOT associative ✗

Distributive Property:

• Left distributive: a × (b + c) = a×b + a×c
  • Example: 5 × (3+4) = 5×3 + 5×4 = 35 ✓
• Right distributive: (b + c) × a = b×a + c×a
  • Also works with subtraction: a × (b - c) = a×b - a×c

Identity Elements:

• Additive identity: 0 (a + 0 = a = 0 + a)
  • Example: 7 + 0 = 7
• Multiplicative identity: 1 (a × 1 = a = 1 × a)
  • Example: 7 × 1 = 7

Additive Inverse:

• For integers: Additive inverse of a is -a (a + (-a) = 0)
• Additive inverse does not exist in whole numbers (except for 0)

Multiplicative Inverse (Reciprocal):

• Multiplicative inverse of a is 1/a (a × 1/a = 1)
• Among whole numbers, only 1 has a multiplicative inverse that is also a whole number (1 × 1 = 1)
• 0 has no multiplicative inverse

Zero Properties:

• a + 0 = a (additive identity)
• a × 0 = 0 (zero multiplication property)
• 0 × a = 0
• a - 0 = a
• 0 - a = -a
• 0 ÷ a = 0 (where a ≠ 0)
• a ÷ 0 = undefined
• 0⁰ = undefined (or sometimes taken as 1 in specific mathematical contexts)

One Properties:

• a × 1 = a (multiplicative identity)
• a ÷ 1 = a
• 1^n = 1 for all values of n
• a^1 = a for all values of a

1.6 BODMAS - Complete with All Cases

BODMAS determines the order in which mathematical operations must be performed.

B - Brackets (solve from innermost to outermost)

• Types of brackets in order: ( ) then { } then [ ]
• ( ) = parentheses or simple brackets
• { } = curly brackets or braces
• [ ] = square brackets

O - Of / Orders (powers and roots)

• Exponents (2³), Square roots (√9), Cube roots (∛8) etc.
• "Of" means multiplication: 1/2 of 10 = 5

D - Division (left to right)

M - Multiplication (left to right)

A - Addition (left to right)

S - Subtraction (left to right)

Important Note: Division and Multiplication have equal priority - solve left to right. Addition and Subtraction have equal priority - solve left to right.

Example 1 (Basic): 15 + 3 × (8 - 2)² ÷ 9 = 15 + 3 × 6² ÷ 9 (brackets solved) = 15 + 3 × 36 ÷ 9 (orders solved) = 15 + 108 ÷ 9 (multiplication) = 15 + 12 (division) = 27 (addition)

Example 2 (Multiple brackets): [{(5 + 3) × 2} - 4] ÷ 3 = [{8 × 2} - 4] ÷ 3 = [16 - 4] ÷ 3 = 12 ÷ 3 = 4

Example 3 (With fractions): 3/4 of 48 + 6² - (9 × 2) = 3/4 × 48 + 36 - 18 = 36 + 36 - 18 = 54

Common Mistakes in BODMAS:

• -3² = -(3²) = -9 (NOT (-3)² = 9)
• When no operation sign appears between a number and a bracket (like 2(3)), it means multiplication

1.7 Important Number Computation Formulas

Sum of first n natural numbers: = n(n+1)/2

• Example: Sum of 1 to 100 = 100 × 101/2 = 5050

Sum of first n odd numbers: = n²

• Example: 1+3+5+7+9 = 5² = 25 (first 5 odd numbers)

Sum of first n even numbers: = n(n+1)

• Example: 2+4+6+8+10 = 5×6 = 30 (first 5 even numbers)

Sum of squares of first n natural numbers: = n(n+1)(2n+1)/6

• Example: 1²+2²+3²+...+10² = 10×11×21/6 = 385

Sum of cubes of first n natural numbers: = [n(n+1)/2]²

• Example: 1³+2³+3³+...+10³ = [10×11/2]² = 55² = 3025

Important: Sum of cubes of first n natural numbers = (Sum of first n natural numbers)²

Counting numbers in a range:

• Count of integers from a to b (inclusive) = b - a + 1
• Count of even numbers from a to b = (b - a)/2 + 1 (when both a and b are even)
• Count of odd numbers from 1 to n (when n is odd) = (n+1)/2

2. Decimals - Complete Coverage

2.1 Place Value System

Understanding place value thoroughly is critical for working with decimals.

Places to the left of the decimal point (whole number part):

• Units place: 10⁰ = 1
• Tens place: 10¹ = 10
• Hundreds place: 10² = 100
• Thousands place: 10³ = 1,000
• Ten thousands place: 10⁴ = 10,000
• Lakhs place: 10⁵ = 1,00,000
• Ten lakhs place: 10⁶ = 10,00,000
• Crores place: 10⁷ = 1,00,00,000

Places to the right of the decimal point (fractional part):

• Tenths place: 10⁻¹ = 0.1 = 1/10
• Hundredths place: 10⁻² = 0.01 = 1/100
• Thousandths place: 10⁻³ = 0.001 = 1/1000
• Ten thousandths place: 10⁻⁴ = 0.0001
• Hundred thousandths place: 10⁻⁵ = 0.00001

Example: In the number 4,635.278:

• 4 is in the thousands place (value = 4000)
• 6 is in the hundreds place (value = 600)
• 3 is in the tens place (value = 30)
• 5 is in the units place (value = 5)
• 2 is in the tenths place (value = 0.2)
• 7 is in the hundredths place (value = 0.07)
• 8 is in the thousandths place (value = 0.008)

2.2 Types of Decimals - Complete Classification

Terminating Decimals:

• Definition: Decimal expansion ends after a finite number of digits
• Examples: 0.5, 0.25, 0.125, 3.75, 0.3125
• These are always rational numbers
• A fraction p/q in its lowest terms gives a terminating decimal if and only if q has no prime factors other than 2 and 5
• Example: 7/20 = 7/(4×5) → only prime factors 2 and 5 → terminating → 0.35
• Example: 1/3 → has prime factor 3 → non-terminating

Non-terminating Recurring Decimals:

• Definition: Decimal expansion never ends but a fixed block of digits repeats continuously
• Also called repeating decimals
• Examples:
  • 1/3 = 0.333... = 0.3̄
  • 1/6 = 0.16666... = 0.16̄
  • 1/7 = 0.142857142857... = 0.1̄4̄2̄8̄5̄7̄
  • 1/11 = 0.090909... = 0.0̄9̄
• The repeating block is called the "repetend" or "period"
• These are rational numbers

Non-terminating Non-recurring Decimals:

• Definition: Never end and never repeat any pattern
• These are irrational numbers
• Examples:
  • √2 = 1.41421356237...
  • √3 = 1.73205080757...
  • π = 3.14159265358...
  • e = 2.71828182845...

Pure Recurring Decimals:

• All digits after the decimal point form the repeating block
• Examples: 0.3̄ = 0.333..., 0.1̄4̄ = 0.141414...

Mixed Recurring Decimals:

• Some digits after the decimal are non-recurring, after which the repeating block begins
• Examples: 0.16̄ = 0.1666..., 0.13̄6̄ = 0.13636...

2.3 Converting Recurring Decimals to Fractions

Type 1: Pure Recurring (0.aaa... type)

Formula: Value = (Recurring part) / (Same number of nines)

• 0.3̄ = 3/9 = 1/3
• 0.7̄ = 7/9
• 0.2̄7̄ = 27/99 = 3/11
• 0.1̄4̄2̄8̄5̄7̄ = 142857/999999 = 1/7
• 0.1̄ = 1/9
• 0.9̄ = 9/9 = 1 (very important - 0.999... = 1 exactly)

Type 2: Mixed Recurring (0.ab̄ type)

Formula: Value = (Complete number after decimal - Non-recurring part) / (Nines for recurring digits followed by zeros for non-recurring digits)

• 0.16̄ = (16-1)/90 = 15/90 = 1/6
• 0.13̄6̄ = (136-1)/990 = 135/990 = 3/22
• 0.83̄ = (83-8)/90 = 75/90 = 5/6
• 0.46̄3̄ = (463-4)/990 = 459/990 = 51/110

Verification method: Convert the result back to decimal to verify correctness. 5/6 = 0.8333... = 0.83̄ ✓

General Formula: If decimal = 0.(non-recurring part)(recurring part) Let x = number of non-recurring digits after decimal Let y = number of recurring digits Then denominator = (10^y - 1) × 10^x = (99...9)(00...0) Numerator = Full number after decimal point - Non-recurring part

2.4 Converting Fractions to Decimals

Method: Long Division

Step 1: Divide the numerator by the denominator 

Step 2: If remainder becomes 0 at any point, it is a terminating decimal 

Step 3: If the same remainder repeats, the decimal is recurring

Predicting the type of decimal: For a fraction p/q in its simplest form:

• If q = 2^a × 5^b (factors only 2 and 5), decimal terminates
• If q has any prime factor other than 2 and 5, decimal recurs
• Length of repeating block divides (q-1) when q is prime

Important conversions to memorize for SSC:

1/2 = 0.5 

1/3 = 0.333... 

1/4 = 0.25 

1/5 = 0.2 

1/6 = 0.1666... 

1/7 = 0.142857142857... 

1/8 = 0.125 

1/9 = 0.111... 

1/11 = 0.0909... 

1/12 = 0.08333... 

1/13 = 0.076923076923... 

1/14 = 0.0714285... 

1/15 = 0.0666... 

1/16 = 0.0625 

1/18 = 0.0555... 

1/20 = 0.05 

1/25 = 0.04 

1/50 = 0.02 

1/125 = 0.008

2.5 Operations on Decimals - Full Detail

Addition:

• Align all decimal points (add trailing zeros if needed to make lengths equal)
• Add as you would whole numbers
• Place the decimal point in the result aligned with the others

Example: 

23.456 + 7.8 + 0.09 = 23.456

• 7.800
• 0.090 = 31.346

Subtraction:

• Same procedure as addition - align decimal points first
• Subtract as whole numbers after alignment

Example: 

45.6 - 18.37 = 45.60

• 18.37 = 27.23

Multiplication: 

Step 1: Ignore decimal points completely and multiply as whole numbers 

Step 2: Count the total decimal places in both numbers being multiplied 

Step 3: Place the decimal point in the result counting from the right equal to the total decimal places

Example: 

3.24 × 1.5 

= 324 × 15 = 4860 

Total decimal places = 2 + 1 = 3 

Answer = 4.860 = 4.86

Example: 

0.003 × 0.07 

= 3 × 7 = 21 

Total decimal places = 3 + 2 = 5 

Answer = 0.00021

Division: Method 1: Convert to whole numbers by multiplying both dividend and divisor by the same power of 10 to eliminate the decimal from the divisor.

Example: 

4.86 ÷ 0.6 

= 48.6 ÷ 6 (multiply both by 10) 

= 8.1

Example: 

0.0072 ÷ 0.08 

= 0.72 ÷ 8 (multiply both by 100) 

= 0.09

Method 2: Convert to fraction form 

4.86 ÷ 0.6 = 486/100 ÷ 6/10 = 486/100 × 10/6 = 4860/600 = 8.1

Effect of multiplying by powers of 10:

• × 10: Move decimal point 1 place to the right
• × 100: Move decimal point 2 places to the right
• × 1000: Move decimal point 3 places to the right
• × 0.1: Move decimal point 1 place to the left
• × 0.01: Move decimal point 2 places to the left

Effect of dividing by powers of 10:

• ÷ 10: Move decimal point 1 place to the left
• ÷ 100: Move decimal point 2 places to the left
• ÷ 0.1: Move decimal point 1 place to the right
• ÷ 0.01: Move decimal point 2 places to the right

2.6 Comparison of Decimals

Method: Compare digit by digit from left to right, beginning with the whole number part.

Example: Compare 3.4567 and 3.4589

• Whole number part: 3 = 3
• Tenths digit: 4 = 4
• Hundredths digit: 5 = 5
• Thousandths digit: 6 vs 8 → 8 > 6 Therefore 3.4589 > 3.4567

Example: Arrange in descending order: 0.7, 0.77, 0.707, 0.770 

Writing with equal decimal places: 0.700, 0.770, 0.707, 0.770 

Note: 0.77 and 0.770 are identical 

Descending order: 0.77 = 0.770 > 0.707 > 0.7

2.7 Rounding Off Decimals

Rules:

• If the digit to be dropped is less than 5, round down (keep the previous digit unchanged)
• If the digit to be dropped is 5 or greater, round up (increase the previous digit by 1)

Examples:

• 3.456 rounded to 2 decimal places = 3.46 (third decimal place is 6, which is ≥ 5, so round up)
• 3.454 rounded to 2 decimal places = 3.45 (third decimal place is 4, which is < 5, so round down)
• 3.445 rounded to 2 decimal places = 3.45 (third decimal place is 5, round up)
• 47.836 rounded to nearest whole number = 48 (first decimal place is 8, which is ≥ 5)
• 47.436 rounded to nearest whole number = 47 (first decimal place is 4, which is < 5)

2.8 Significant Figures

Rules for counting significant figures:

1. All non-zero digits are significant
2. Zeros between non-zero digits are significant
3. Leading zeros (before the first non-zero digit) are NOT significant
4. Trailing zeros after the decimal point ARE significant
5. Trailing zeros in whole numbers without a decimal point are ambiguous

Examples:

• 3.456 has 4 significant figures
• 0.00456 has 3 significant figures (leading zeros are not significant)
• 3.450 has 4 significant figures (the trailing zero after the decimal is significant)
• 3400 has 2, 3, or 4 significant figures (ambiguous)
•  
  1. has 4 significant figures (decimal point confirms trailing zeros are significant)

2.9 Important Decimal Shortcuts for SSC Examinations

• 0.5 = 1/2 → multiplying by 0.5 is the same as dividing by 2
• 0.25 = 1/4 → multiplying by 0.25 is the same as dividing by 4
• 0.125 = 1/8 → multiplying by 0.125 is the same as dividing by 8
• 0.0625 = 1/16
• 0.2 = 1/5 → multiplying by 0.2 is the same as dividing by 5
• 0.1̄ = 1/9
• 1.5 = 3/2
• 2.5 = 5/2
• 1.25 = 5/4
• 0.75 = 3/4

Quick comparison of decimals with fractions:

• Is 0.68 > 2/3? Since 2/3 = 0.666... < 0.68 → Yes, 0.68 > 2/3
• Is 3/7 > 0.43? Since 3/7 = 0.4285... < 0.43 → No, 3/7 < 0.43

3. Fractions - Complete Coverage

3.1 Fraction Fundamentals

A fraction p/q represents:

• p parts taken out of q equal parts
• p = numerator
• q = denominator
• q ≠ 0 always (denominator can never be zero)

Position of fractions on the number line:

• Proper fractions lie between 0 and 1
• Improper fractions lie at 1 or to the right of 1
• Negative fractions lie to the left of 0

3.2 Types of Fractions - Complete List

Proper Fraction:

• Numerator < Denominator
• Value is always between 0 and 1 (for positive fractions)
• Examples: 1/2, 3/4, 7/9, 5/8, 11/17
• The complement of a proper fraction: complement of 3/7 is 4/7, and their sum = 1

Improper Fraction:

• Numerator ≥ Denominator
• Value is always greater than or equal to 1
• Examples: 7/4, 9/2, 11/3, 5/5, 15/7
• Note: 5/5 = 1 exactly

Mixed Fraction:

• A whole number combined with a proper fraction
• Examples: 2(3/4), 5(1/2), 3(2/7)
• Converting mixed to improper fraction: a(b/c) = (a×c + b)/c
  • 2(3/4) = (2×4 + 3)/4 = 11/4
  • 5(1/2) = (5×2 + 1)/2 = 11/2
• Converting improper to mixed fraction: divide numerator by denominator
  • 11/4 = 2 remainder 3 = 2(3/4)

Like Fractions:

• Fractions with the same denominator
• Examples: 1/7, 2/7, 5/7, 9/7
• Easy to add and subtract (simply add or subtract numerators)

Unlike Fractions:

• Fractions with different denominators
• Examples: 1/2, 2/3, 3/5

Equivalent Fractions:

• Fractions with the same value but different appearance
• Obtained by multiplying or dividing both numerator and denominator by the same non-zero number
• Examples: 1/2 = 2/4 = 3/6 = 4/8 = 5/10 = 50/100
• To verify: cross multiply (a/b = c/d if and only if ad = bc)

Unit Fraction:

• Numerator equals 1
• Examples: 1/2, 1/3, 1/7, 1/100
• As the denominator increases, the value of the unit fraction decreases

Complex Fraction:

• Numerator or denominator (or both) is itself a fraction
• Example: (3/4)/(5/7) = 3/4 × 7/5 = 21/20
• Example: (2 + 1/3)/(4 - 1/2) = (7/3)/(7/2) = 7/3 × 2/7 = 2/3

Continued Fraction:

• A fraction in the form a + 1/(b + 1/(c + 1/...))
• Example: 1 + 1/(2 + 1/3) = 1 + 1/(7/3) = 1 + 3/7 = 10/7

3.3 Simplification of Fractions

Reducing to simplest form (lowest terms): Step 1: Find the HCF of numerator and denominator Step 2: Divide both numerator and denominator by the HCF

Example: Simplify 48/72

• HCF(48, 72) = 24
• 48/24 = 2, 72/24 = 3
• Simplest form = 2/3

Example: Simplify 195/385

• 195 = 3 × 5 × 13
• 385 = 5 × 7 × 11
• HCF = 5
• 195/5 = 39, 385/5 = 77
• Result = 39/77 (verify: HCF(39, 77) = HCF(3×13, 7×11) = 1 ✓)

3.4 Comparison of Fractions

Method 1 - Cross Multiplication: To compare a/b and c/d:

• Calculate a×d and b×c
• If ad > bc, then a/b > c/d
• If ad < bc, then a/b < c/d
• If ad = bc, then a/b = c/d

Example: Compare 5/8 and 7/11

• 5×11 = 55
• 7×8 = 56
• Since 55 < 56, we have 5/8 < 7/11

Method 2 - Same Denominator using LCM: Convert all fractions to the same denominator using LCM.

Example: Compare 3/4, 5/6, 7/9

• LCM(4, 6, 9) = 36
• 3/4 = 27/36
• 5/6 = 30/36
• 7/9 = 28/36
• Order: 27/36 < 28/36 < 30/36
• Therefore: 3/4 < 7/9 < 5/6

Method 3 - Decimal Conversion: Best method for quick comparison in SSC examinations. 

3/4 = 0.75, 5/6 = 0.833, 7/9 = 0.777 Result: 3/4 < 7/9 < 5/6

Method 4 - Quick Rules:

Rule 1: Same numerator → larger denominator = smaller fraction

• 3/7 vs 3/11: same numerator 3, denominator 7 < 11, therefore 3/7 > 3/11

Rule 2: Same denominator → larger numerator = larger fraction

• 5/9 vs 7/9: same denominator 9, numerator 7 > 5, therefore 7/9 > 5/9

Rule 3: Adding same positive number to both numerator and denominator:

• If a/b < 1: value increases → a/b < (a+k)/(b+k)
• If a/b > 1: value decreases → a/b > (a+k)/(b+k)
• Example: 3/5 vs 4/6: both increased by 1. Since 3/5 < 1, we have 3/5 < 4/6 ✓

3.5 Operations on Fractions - Complete

Addition of Like Fractions: a/c + b/c = (a+b)/c Example: 3/11 + 5/11 = 8/11

Addition of Unlike Fractions: Step 1: Find the LCM of all denominators Step 2: Convert each fraction to the same denominator Step 3: Add the numerators

Example: 1/3 + 1/4 + 1/6

• LCM(3, 4, 6) = 12
• = 4/12 + 3/12 + 2/12
• = 9/12 = 3/4

Example: 2(1/3) + 3(1/4) 

= 7/3 + 13/4 

= 28/12 + 39/12 

= 67/12 = 5(7/12)

Subtraction: Same process as addition - find LCM, convert, then subtract numerators.

Example: 5/6 - 3/8

• LCM(6, 8) = 24
• = 20/24 - 9/24
• = 11/24

Multiplication of Fractions: (a/b) × (c/d) = ac/bd

Where possible, simplify before multiplying by cancelling common factors.

Example: (14/15) × (25/21) 

Cancel: 14 and 21 share factor 7 → 2 and 3; 25 and 15 share factor 5 → 5 and 3 

= (2/3) × (5/3) 

= 10/9

Division of Fractions: (a/b) ÷ (c/d) = (a/b) × (d/c) = ad/bc

The "Keep, Change, Flip" method:

• KEEP the first fraction unchanged
• CHANGE the division sign to multiplication
• FLIP (take reciprocal of) the second fraction

Example: (3/5) ÷ (9/10) 

= (3/5) × (10/9) 

= 30/45 = 2/3

Example: 2(1/4) ÷ 1(1/2) 

= (9/4) ÷ (3/2) 

= (9/4) × (2/3) 

= 18/12 = 3/2 = 1(1/2)

Mixed Operations: Always follow the BODMAS rule.

Example: (1/2 + 1/3) ÷ (1/4 - 1/6) 

= (3/6 + 2/6) ÷ (3/12 - 2/12) 

= (5/6) ÷ (1/12) 

= (5/6) × 12 

= 60/6 = 10

3.6 Important Fraction Identities and Properties

Reciprocal:

• Reciprocal of a/b is b/a
• Reciprocal of a whole number n is 1/n
• Reciprocal of mixed number 2(3/4) = reciprocal of 11/4 = 4/11
• Product of any number and its reciprocal = 1
• Reciprocal of the reciprocal = original number

Special fraction values:

• a/a = 1 for all a ≠ 0
• 0/a = 0 for all a ≠ 0
• a/1 = a
• a/0 = undefined

Fraction and its complement:

• Complement of a/b = 1 - a/b = (b-a)/b
• A fraction and its complement always add up to 1

Fraction properties with inequalities: If a/b < 1 (proper fraction):

• a/b < (a+k)/(b+k) for any positive k (adding same positive number to both numerator and denominator increases the value)
• a/b > (a-k)/(b-k) for k < a (subtracting decreases the value)

If a/b > 1 (improper fraction):

• a/b > (a+k)/(b+k) for any positive k (adding same positive number to both numerator and denominator decreases the value towards 1)

3.7 Simplification Using Fractions - SSC Type Problems

Type 1: Value of nested expression

Example: 1 - 1/(1 + 1/(1 - 1/3)) 

= 1 - 1/(1 + 1/(2/3)) 

= 1 - 1/(1 + 3/2) 

= 1 - 1/(5/2) 

= 1 - 2/5 

= 3/5

Type 2: Telescoping Products

(1 - 1/2)(1 - 1/3)(1 - 1/4)...(1 - 1/n) 

= (1/2)(2/3)(3/4)...(n-1)/n 

= 1/n

Example: (1-1/2)(1-1/3)(1-1/4)...(1-1/10) = 1/10

Type 3: Telescoping Sums

1/(1×2) + 1/(2×3) + 1/(3×4) + ... + 1/(n×(n+1)) 

= (1 - 1/2) + (1/2 - 1/3) + (1/3 - 1/4) + ... + (1/n - 1/(n+1)) 

= 1 - 1/(n+1) 

= n/(n+1)

Example: 1/(1×2) + 1/(2×3) + ... + 1/(9×10) = 1 - 1/10 = 9/10

Type 4: Finding the Original Fraction

A fraction becomes p/q when a is added to both numerator and denominator. Find the original fraction. 

Let fraction = x/y 

(x+a)/(y+a) = p/q → solve for x/y

Example: A fraction becomes 4/5 when 1 is added to both numerator and denominator. Find the original fraction. 

(x+1)/(y+1) = 4/5 

5x + 5 = 4y + 4 

5x - 4y = -1 

Try x = 3, y = 4: 5(3) - 4(4) = 15 - 16 = -1 ✓ 

Original fraction = 3/4

3.8 Relationship Between Fractions and Other Concepts

Fraction and Percentage: To convert a fraction to percentage: multiply by 100

• 3/4 = 3/4 × 100 = 75%
• 7/8 = 7/8 × 100 = 87.5%

To convert percentage to fraction: divide by 100 and simplify

• 35% = 35/100 = 7/20
• 12.5% = 12.5/100 = 125/1000 = 1/8

Fraction and Ratio:

• Ratio a : b can be written as the fraction a/b
• 3 : 4 is equivalent to the fraction 3/4
• Ratio of 3/5 to 4/7 = (3/5) ÷ (4/7) = (3/5) × (7/4) = 21/20 = 21 : 20

Fraction and Division:

• a/b literally means a ÷ b
• Any division always produces a fraction unless the numbers divide exactly

3.9 Complete Table: Fractions, Decimals and Percentages

4. Number Relationships - Complete Coverage

4.1 Absolute Value (Modulus)

• |a| = a if a ≥ 0
• |a| = -a if a < 0
• |3| = 3, |-5| = 5, |0| = 0
• |a| ≥ 0 always (absolute value is never negative)
• |a × b| = |a| × |b|
• |a + b| ≤ |a| + |b| (Triangle Inequality)
• |a - b| ≥ ||a| - |b||

4.2 Factors and Multiples

Factors:

• a is a factor of b if b ÷ a gives a whole number with no remainder
• Factors of 12: 1, 2, 3, 4, 6, 12
• Every number except 1 has at least 2 factors (1 and itself)
• Number of factors of N = p^a × q^b × r^c equals (a+1)(b+1)(c+1)

Multiples:

• Multiples of 3: 3, 6, 9, 12, 15, 18...
• Every number is a multiple of 1
• Every number is a multiple of itself

Perfect Numbers:

• A number whose sum of proper divisors (all factors except the number itself) equals the number
• Example: 6 → proper factors: 1, 2, 3 → 1+2+3 = 6 ✓
• Example: 28 → proper factors: 1, 2, 4, 7, 14 → 1+2+4+7+14 = 28 ✓
• The next perfect numbers are 496 and 8128

Perfect Squares:

• Numbers whose square root is a whole number
• 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225...
• A perfect square always ends in: 0, 1, 4, 5, 6, or 9
• A perfect square NEVER ends in: 2, 3, 7, or 8
• Sum of first n odd numbers = n²

Perfect Cubes:

• 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000
• A perfect cube can end in any digit from 0 to 9

4.3 Square Roots - Complete

Definition: √a = b means b² = a (where b ≥ 0)

Perfect square roots to memorize: √1=1, √4=2, √9=3, √16=4, √25=5, √36=6, √49=7, √64=8, √81=9, √100=10, √121=11, √144=12, √169=13, √196=14, √225=15, √256=16, √289=17, √324=18, √361=19, √400=20

Properties of Square Roots:

• √(a×b) = √a × √b
• √(a/b) = √a / √b
• (√a)² = a (for a ≥ 0)
• √(a²) = |a| = a (for a ≥ 0)
• √a × √a = a
• √a + √b ≠ √(a+b) (this is a very common mistake to avoid)

Simplifying square roots:

• √72 = √(36×2) = 6√2
• √50 = √(25×2) = 5√2
• √108 = √(36×3) = 6√3
• √200 = √(100×2) = 10√2

Square roots of fractions:

• √(9/16) = √9/√16 = 3/4
• √(25/49) = 5/7

Finding square root by prime factorisation: Example: √7056 7056 = 2⁴ × 3² × 7² √7056 = 2² × 3 × 7 = 4 × 3 × 7 = 84

Approximation of irrational square roots:

• √2 ≈ 1.414
• √3 ≈ 1.732
• √5 ≈ 2.236
• √6 ≈ 2.449
• √7 ≈ 2.646
• √8 = 2√2 ≈ 2.828
• √10 ≈ 3.162
• √11 ≈ 3.317
• √13 ≈ 3.606
• √15 ≈ 3.873

SSC tip for approximating irrational square roots: √5 lies between √4 = 2 and √9 = 3, and is closer to 2, so √5 ≈ 2.2

4.4 Cube Roots

Definition: ∛a = b means b³ = a

Perfect cube roots to memorize: ∛1=1, ∛8=2, ∛27=3, ∛64=4, ∛125=5, ∛216=6, ∛343=7, ∛512=8, ∛729=9, ∛1000=10

Properties:

• ∛(a×b) = ∛a × ∛b
• ∛(a/b) = ∛a / ∛b
• (∛a)³ = a

4.5 Surds - Complete Coverage

Definition: A surd is an irrational number expressed as the n-th root of a rational number that cannot be simplified to a rational number.

Examples: √2, √3, ∛5, ⁴√7 are surds. 

But √4 = 2 is NOT a surd (it simplifies to a rational number).

Types of Surds:

Simple Surd: Contains a single term

• Examples: √3, ∛5, ⁴√2

Pure Surd: No rational number is multiplied with the surd

• Examples: √2, √5, ∛7

Mixed Surd: A rational number multiplied with a surd

• Examples: 2√3, 5√2, 3∛4

Compound Surd (Binomial Surd): Sum or difference of two surds

• Examples: √2 + √3, 3 + √5, √7 - √3

Similar Surds (Like Surds): Surds with the same irrational part

• 2√3 and 5√3 are similar surds (both have √3 as the irrational part)
• Like surds can be added: 2√3 + 5√3 = 7√3

Dissimilar Surds (Unlike Surds): Surds with different irrational parts

• √2 and √3 are dissimilar surds
• Cannot be combined into a single term: √2 + √3 ≠ √5

Order of a Surd:

• √2 is a surd of order 2 (square root)
• ∛3 is a surd of order 3 (cube root)
• ⁴√5 is a surd of order 4 (fourth root)

4.6 Laws of Surds

Multiplication (same order): √a × √b = √(ab) Example: 2√3 × 5√7 = 10√21

Division (same order): √a ÷ √b = √(a/b) Example: √12 ÷ √3 = √4 = 2

Addition and Subtraction (only like surds can be combined): a√n + b√n = (a+b)√n Example: 3√5 + 2√5 = 5√5

Powers: (√a)^n = a^(n/2) Example: (√3)⁴ = 3² = 9 Example: (√5)³ = 5^(3/2) = 5√5

4.7 Rationalisation of Surds

Definition: The process of converting an irrational denominator to a rational number by multiplying both numerator and denominator by an appropriate rationalising factor.

Type 1: Single surd in the denominator Rationalising factor = the same surd

1/√3 = (1/√3) × (√3/√3) = √3/3

3/√5 = (3/√5) × (√5/√5) = 3√5/5

Type 2: Binomial surd denominator of the form (a + √b) Rationalising factor = conjugate = (a - √b)

1/(3 + √2) 

= [1/(3 + √2)] × [(3 - √2)/(3 - √2)] 

= (3 - √2)/(9 - 2) 

= (3 - √2)/7

1/(√5 + √3) = [1/(√5 + √3)] × [(√5 - √3)/(√5 - √3)] 

= (√5 - √3)/(5 - 3) 

= (√5 - √3)/2

Type 3: Double surd in the denominator of the form (√a - √b) Use the conjugate (√a + √b)

1/(√7 - √5) = [1/(√7 - √5)] × [(√7 + √5)/(√7 + √5)] 

= (√7 + √5)/(7 - 5) 

= (√7 + √5)/2

4.8 Important Surd Identities for SSC

(a + b)² = a² + 2ab + b² Example: (√3 + √2)² = 3 + 2√6 + 2 = 5 + 2√6

(a - b)² = a² - 2ab + b² Example: (√5 - √3)² = 5 - 2√15 + 3 = 8 - 2√15

(a + b)(a - b) = a² - b² Example: (√7 + √3)(√7 - √3) = 7 - 3 = 4 Example: (2 + √3)(2 - √3) = 4 - 3 = 1

Finding the value of expressions involving surds: If x = 2 + √3, find x + 1/x: 1/x = 1/(2+√3) = (2-√3)/[(2+√3)(2-√3)] = (2-√3)/(4-3) = 2-√3 x + 1/x = (2+√3) + (2-√3) = 4

If a = 3+2√2 and b = 3-2√2, find a × b: 

a × b = (3+2√2)(3-2√2) = 9 - 8 = 1 

Note: Since ab = 1, this means b = 1/a (they are reciprocals of each other)

Comparing surds of different orders: To compare √3 and ∛4: Convert both to 6th root (LCM of orders 2 and 3): √3 = ⁶√(3³) = ⁶√27 ∛4 = ⁶√(4²) = ⁶√16 Since 27 > 16, we conclude √3 > ∛4

4.9 SSC-Style Problems on Number Relationships

Type: Finding values when sum or product is given

If x + 1/x = 5, find x² + 1/x²: 

(x + 1/x)² = x² + 2 + 1/x² = 25 

Therefore x² + 1/x² = 23

If x + 1/x = 5, find x³ + 1/x³: 

x³ + 1/x³ = (x + 1/x)³ - 3(x + 1/x) = 125 - 15 = 110

If x - 1/x = 4, find x² + 1/x²: 

(x - 1/x)² = x² - 2 + 1/x² = 16 

Therefore x² + 1/x² = 18

If x² + 1/x² = 7, find x + 1/x: 

(x + 1/x)² = x² + 2 + 1/x² = 9 

Therefore x + 1/x = 3

Essential formulas that must be memorised:

• (x + 1/x)² = x² + 2 + 1/x²
• (x - 1/x)² = x² - 2 + 1/x²
• x² + 1/x² = (x + 1/x)² - 2 = (x - 1/x)² + 2
• x³ + 1/x³ = (x + 1/x)³ - 3(x + 1/x)
• x³ - 1/x³ = (x - 1/x)³ + 3(x - 1/x)

5. Solved Examples - SSC Level

Example 1: Find the value of 0.1 + 0.1² + 0.1³ = 0.1 + 0.01 + 0.001 = 0.111

Example 2: What is the value of 3.6 × 0.48 + 0.02 × 1.44? = 1.728 + 0.0288 = 1.7568

Example 3: Simplify: [(0.6)³ - (0.2)³] / [(0.6)² + 0.6×0.2 + (0.2)²] Applying a³ - b³ = (a-b)(a²+ab+b²): = (0.6 - 0.2) × [(0.6)² + 0.6×0.2 + (0.2)²] / [(0.6)² + 0.6×0.2 + (0.2)²] = 0.6 - 0.2 = 0.4

Example 4: Arrange in descending order: 7/8, 13/16, 31/40, 63/80 Converting to a common denominator of 80: 7/8 = 70/80, 13/16 = 65/80, 31/40 = 62/80, 63/80 Descending order: 70/80 > 65/80 > 63/80 > 62/80 Therefore: 7/8 > 13/16 > 63/80 > 31/40

Example 5: Find √(6 + √(6 + √(6 + ...∞))) Let x = √(6 + √(6 + ...)) Then x = √(6 + x) x² = 6 + x x² - x - 6 = 0 (x - 3)(x + 2) = 0 x = 3 (taking the positive value since square root is non-negative)

Example 6: If √(0.0169 × x) = 1.3, find x. Squaring both sides: 0.0169 × x = 1.69 x = 1.69 / 0.0169 = 100

Example 7: The product of two fractions is 14/15. If one fraction is 7/6, find the other. Other fraction = (14/15) ÷ (7/6) = (14/15) × (6/7) = 84/105 = 4/5

Example 8: If 1.5x = 0.04y, find (y - x)/(y + x). y/x = 1.5/0.04 = 75/2 Let y = 75k and x = 2k (y - x)/(y + x) = (75k - 2k)/(75k + 2k) = 73k/77k = 73/77

Example 9: What decimal fraction of a minute is 18 seconds? 18/60 = 3/10 = 0.3

Example 10: If x = 3 + 2√2, find x + 1/x. 1/x = 1/(3+2√2) × (3-2√2)/(3-2√2) = (3-2√2)/(9-8) = 3 - 2√2 x + 1/x = (3+2√2) + (3-2√2) = 6

6. Key Formulas - Quick Reference

Whole Numbers:

• Sum of 1 to n = n(n+1)/2
• Sum of first n odd numbers = n²
• Sum of first n even numbers = n(n+1)
• Sum of squares of 1 to n = n(n+1)(2n+1)/6
• Sum of cubes of 1 to n = [n(n+1)/2]²
• Number of factors of p^a × q^b × r^c = (a+1)(b+1)(c+1)

Decimals:

• 0.aaa... = a/9
• 0.abab... = ab/99
• 0.abcabc... = abc/999
• 0.ab̄ (b repeating) = (ab - a)/90
• 0.ab̄c̄ (bc repeating) = (abc - a)/990

Fractions:

• (a/b) ÷ (c/d) = ad/bc
• (a/b) × (c/d) = ac/bd
• a/b + c/d = (ad + bc)/bd (then simplify)
• If a/b < 1: adding same positive k to both numerator and denominator increases the value
• Telescoping product: (1-1/2)(1-1/3)...(1-1/n) = 1/n
• Telescoping sum: 1/(1×2) + 1/(2×3) + ... + 1/(n(n+1)) = n/(n+1)

Surds:

• (a + √b)(a - √b) = a² - b
• (√a + √b)(√a - √b) = a - b
• If x + 1/x = k: x² + 1/x² = k² - 2, x³ + 1/x³ = k³ - 3k
• If x - 1/x = k: x² + 1/x² = k² + 2, x³ - 1/x³ = k³ + 3k