Civil Service Exam (Subprofessional) Numerical Ability — Fractions — Operations, Conversion & ComparisonCheat Sheet
Fractions — Operations, Conversion & Comparison cheat sheet for Civil Service Exam (Subprofessional) aspirants. If you could only take one sheet of paper into your review session, this is what it would look like. Civil Service Commission (CSC)'s most-tested concepts, all in one place.
Exam context
The Career Service Examination — Subprofessional Level is conducted by Civil Service Commission (CSC) and is scheduled for Bi-annual — March and August 2026. The Numerical Ability subtest is marked as "~25% weightage" in the official pattern, and Fractions — Operations, Conversion & Comparison appears in position 2nd of 9 in the Civil Service Exam (Subprofessional) Numerical Ability review rotation. Passing mark: 80%. Recent Civil Service Exam (Subprofessional) 2026 papers have drawn roughly 17 questions from this subject.
Fractions — Operations, Conversion & Comparison - Cheat sheet
Your last-minute revision companion for mastering fractions operations, conversions, and comparisons before the exam
Sections
Section Title
Types of Fractions
Important Facts
- Proper fractions are always less than 1
- Improper fractions are always ≥ 1
- Mixed fractions contain both whole and fractional parts
- All three types can represent the same value (equivalent fractions)
Key Definitions
Term
Proper Fraction
Example
3/8, 5/7, 2/9
Definition
Fraction where numerator < denominator (value < 1)
Term
Improper Fraction
Example
9/4, 7/3, 12/5
Definition
Fraction where numerator ≥ denominator (value ≥ 1)
Term
Mixed Fraction
Example
2 1/3, 5 2/7, 1 4/5
Definition
Whole number + proper fraction
Diagrams To Know
- Fraction bars showing numerator/denominator relationship
- Visual representations using pie charts or rectangles
Formulas
Formula
Mixed to Improper: (Whole × Denominator) + Numerator / Denominator
Meaning
Multiply whole number by denominator, add numerator, keep denominator
Watch Out
Don't forget to multiply the whole number by the denominator first
When To Use
Converting mixed fractions before operations
Formula
Improper to Mixed: Quotient + Remainder/Divisor
Meaning
Divide numerator by denominator: quotient = whole, remainder = new numerator
Watch Out
The divisor (original denominator) stays the same
When To Use
Final answer conversion or simplifying results
Section Title
Conversion Between Fraction Types
Important Facts
- Only convert between improper and mixed fractions (not to proper)
- Division gives: quotient = whole number, remainder = numerator
- Original denominator always remains the same in conversion
- Equivalent fractions multiply/divide numerator and denominator by same number
Key Definitions
Term
Equivalent Fractions
Example
1/2 = 2/4 = 3/6 = 4/8
Definition
Different fractions representing the same value
Reactions Or Equations
Note
Multiply whole by denominator, add numerator
Equation
4 2/3 = (4×3+2)/3 = 14/3
Conditions
Converting mixed to improper
Note
Quotient becomes whole, remainder becomes numerator
Equation
17/5 = 3 2/5 (17÷5 = 3 remainder 2)
Conditions
Converting improper to mixed
Formulas
Formula
Same Denominator: a/c ± b/c = (a±b)/c
Meaning
Add/subtract numerators, keep denominator
Watch Out
Only add/subtract numerators, never denominators
When To Use
When denominators are identical
Formula
Different Denominators: Get LCD, convert, then add/subtract
Meaning
Find LCD, make equivalent fractions, then compute
Watch Out
Must find LCD first - cannot add/subtract with different denominators
When To Use
When denominators are different
Section Title
Addition and Subtraction of Fractions
Important Facts
- Same denominator: add/subtract numerators only
- Different denominators: find LCD first
- Always simplify final answer to lowest terms
- For mixed fractions: convert to improper first
Key Definitions
Term
LCD (Least Common Denominator)
Example
LCD of 4 and 6 is 12
Definition
Smallest number that is a multiple of all denominators
Diagrams To Know
- LCD finding using multiples method
- Step-by-step conversion to equivalent fractions
Reactions Or Equations
Note
Add numerators, keep denominator
Equation
3/8 + 5/8 = 8/8 = 1
Conditions
Same denominators
Note
Convert to equivalent fractions with LCD
Equation
1/4 + 1/6 = 3/12 + 2/12 = 5/12
Conditions
Different denominators, LCD = 12
Formulas
Formula
Multiplication: a/b × c/d = (a×c)/(b×d)
Meaning
Multiply numerators together, multiply denominators together
Watch Out
Don't need common denominators - multiply straight across
When To Use
All fraction multiplication problems
Formula
Division: a/b ÷ c/d = a/b × d/c
Meaning
Multiply by reciprocal of divisor
Watch Out
Flip the second fraction (divisor) then multiply
When To Use
All fraction division problems
Section Title
Multiplication and Division of Fractions
Important Facts
- Multiplication is easier than addition - no LCD needed
- Division = multiplication by reciprocal
- Always simplify before multiplying when possible
- Convert mixed fractions to improper before operations
Key Definitions
Term
Reciprocal
Example
Reciprocal of 3/4 is 4/3
Definition
Fraction flipped upside down (numerator becomes denominator)
Diagrams To Know
- Cross multiplication visualization
- Reciprocal method flowchart
Reactions Or Equations
Note
Multiply across: numerator × numerator, denominator × denominator
Equation
2/3 × 4/5 = 8/15
Conditions
Standard multiplication
Note
Flip second fraction and multiply
Equation
3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8
Conditions
Division using reciprocal method
Formulas
Formula
Cross Multiplication: If a/b vs c/d, compare a×d with b×c
Meaning
Multiply diagonally to compare fractions
Watch Out
Larger cross product indicates larger fraction
When To Use
Comparing two fractions with different denominators
Common Values
Value
1/2
Symbol
0.5
Quantity
One Half
Value
1/4
Symbol
0.25
Quantity
One Quarter
Value
3/4
Symbol
0.75
Quantity
Three Quarters
Section Title
Comparing and Ordering Fractions
Important Facts
- Same denominator: compare numerators directly
- Same numerator: smaller denominator is larger fraction
- Different denominators: convert to LCD or use cross multiplication
- Mixed fractions: compare whole numbers first
Key Definitions
Term
Equivalent Fractions
Example
1/2 = 2/4 = 3/6
Definition
Fractions that represent the same value
Diagrams To Know
- Number line showing fraction positions
- Visual fraction bars for comparison
Reactions Or Equations
Note
Larger cross product indicates larger original fraction
Equation
Compare 2/3 vs 3/4: 2×4 = 8, 3×3 = 9, so 2/3 < 3/4
Conditions
Cross multiplication comparison
Formulas
Formula
Fraction to Decimal: Divide numerator by denominator
Meaning
Use long division: numerator ÷ denominator
Watch Out
May result in terminating or repeating decimal
When To Use
Converting any fraction to decimal form
Common Values
Value
0.1
Symbol
one tenth
Quantity
1/10
Value
0.01
Symbol
one hundredth
Quantity
1/100
Value
0.001
Symbol
one thousandth
Quantity
1/1000
Section Title
Fraction to Decimal Conversion
Important Facts
- Denominators of 10, 100, 1000 give easy decimal conversion
- Place decimal point based on denominator: 1 place (÷10), 2 places (÷100), etc.
- Some fractions give repeating decimals
- Add zeros as needed when placing decimal point
Key Definitions
Term
Terminating Decimal
Example
1/4 = 0.25, 3/8 = 0.375
Definition
Decimal that ends (finite number of digits)
Term
Repeating Decimal
Example
1/3 = 0.333..., 2/7 = 0.285714285714...
Definition
Decimal with digit pattern that repeats infinitely
Reactions Or Equations
Note
Move decimal point left by number of zeros in denominator
Equation
3/10 = 0.3, 47/100 = 0.47, 125/1000 = 0.125
Conditions
Denominators are powers of 10
Must Remember
- Same denominator: operate on numerators only, keep denominator
- Different denominators: find LCD for addition/subtraction
- Division = multiply by reciprocal (flip second fraction)
- Mixed to improper: (whole × denominator) + numerator
- Improper to mixed: divide numerator by denominator
- Multiplication: multiply straight across (no LCD needed)
- Always simplify final answers to lowest terms
- Cross multiplication for comparing fractions
- Equivalent fractions: multiply/divide top and bottom by same number
- Decimal conversion: divide numerator by denominator
Last Minute Tips
- For word problems with 'of', use multiplication (3/4 of 200 = 3/4 × 200)
- When comparing fractions, convert to same denominator or use cross multiplication
- In multiple choice, eliminate obviously wrong answers first
- If answer choices are mixed fractions, convert your improper fraction result
- Check if your answer can be simplified further before selecting final answer
Comparison Tables
Rows
Values
- Add numerators, keep denominator
- Find LCD first, then add
- Never add denominators
Property
Addition
Values
- Subtract numerators, keep denominator
- Find LCD first, then subtract
- Never subtract denominators
Property
Subtraction
Values
- Multiply across
- Multiply across
- No LCD needed
Property
Multiplication
Values
- Multiply by reciprocal
- Multiply by reciprocal
- Flip second fraction
Property
Division
Columns
- Operation
- Same Denominators
- Different Denominators
- Key Rule
Table Title
Fraction Operations Comparison
Rows
Values
- Numerator < Denominator
- < 1
- 3/7
Property
Proper
Values
- Numerator ≥ Denominator
- ≥ 1
- 9/4
Property
Improper
Values
- Whole + Proper
- > 1
- 2 1/3
Property
Mixed
Columns
- Type
- Numerator vs Denominator
- Value
- Example
Table Title
Fraction Types Comparison
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