Fractions — Operations, Conversion & Comparison - Cheat Sheet
Your last-minute revision companion for mastering fractions operations, conversions, and comparisons before the exam
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
Proper Fraction
Example
3/8, 5/7, 2/9
Fraction where numerator < denominator (value < 1)
Improper Fraction
Example
9/4, 7/3, 12/5
Fraction where numerator ≥ denominator (value ≥ 1)
Mixed Fraction
Example
2 1/3, 5 2/7, 1 4/5
Whole number + proper fraction
Diagrams to know
- Fraction bars showing numerator/denominator relationship
- Visual representations using pie charts or rectangles
Conversion Between Fraction Types
Formulas
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
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
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
Equivalent Fractions
Example
1/2 = 2/4 = 3/6 = 4/8
Different fractions representing the same value
Reactions or equations
4 2/3 = (4×3+2)/3 = 14/3
Note
Multiply whole by denominator, add numerator
Conditions
Converting mixed to improper
17/5 = 3 2/5 (17÷5 = 3 remainder 2)
Note
Quotient becomes whole, remainder becomes numerator
Conditions
Converting improper to mixed
Addition and Subtraction of Fractions
Formulas
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
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
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
LCD (Least Common Denominator)
Example
LCD of 4 and 6 is 12
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
3/8 + 5/8 = 8/8 = 1
Note
Add numerators, keep denominator
Conditions
Same denominators
1/4 + 1/6 = 3/12 + 2/12 = 5/12
Note
Convert to equivalent fractions with LCD
Conditions
Different denominators, LCD = 12
Multiplication and Division of Fractions
Formulas
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
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
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
Reciprocal
Example
Reciprocal of 3/4 is 4/3
Fraction flipped upside down (numerator becomes denominator)
Diagrams to know
- Cross multiplication visualization
- Reciprocal method flowchart
Reactions or equations
2/3 × 4/5 = 8/15
Note
Multiply across: numerator × numerator, denominator × denominator
Conditions
Standard multiplication
3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8
Note
Flip second fraction and multiply
Conditions
Division using reciprocal method
Comparing and Ordering Fractions
Formulas
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
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
Equivalent Fractions
Example
1/2 = 2/4 = 3/6
Fractions that represent the same value
Diagrams to know
- Number line showing fraction positions
- Visual fraction bars for comparison
Reactions or equations
Compare 2/3 vs 3/4: 2×4 = 8, 3×3 = 9, so 2/3 < 3/4
Note
Larger cross product indicates larger original fraction
Conditions
Cross multiplication comparison
Fraction to Decimal Conversion
Formulas
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
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
Terminating Decimal
Example
1/4 = 0.25, 3/8 = 0.375
Decimal that ends (finite number of digits)
Repeating Decimal
Example
1/3 = 0.333..., 2/7 = 0.285714285714...
Decimal with digit pattern that repeats infinitely
Reactions or equations
3/10 = 0.3, 47/100 = 0.47, 125/1000 = 0.125
Note
Move decimal point left by number of zeros in denominator
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
Fraction Operations Comparison
| Operation | Same Denominators | Different Denominators | Key Rule |
|---|---|---|---|
| Addition | Add numerators, keep denominator | Find LCD first, then add | Never add denominators |
| Subtraction | Subtract numerators, keep denominator | Find LCD first, then subtract | Never subtract denominators |
| Multiplication | Multiply across | Multiply across | No LCD needed |
| Division | Multiply by reciprocal | Multiply by reciprocal | Flip second fraction |
Fraction Types Comparison
| Type | Numerator vs Denominator | Value | Example |
|---|---|---|---|
| Proper | Numerator < Denominator | < 1 | 3/7 |
| Improper | Numerator ≥ Denominator | ≥ 1 | 9/4 |
| Mixed | Whole + Proper | > 1 | 2 1/3 |
Diagrams
Fractions Concept Overview
Complete overview of all fraction concepts and their relationships
Fraction Operations Decision Tree
Step-by-step decision process for solving any fraction operation
Mixed to Improper Conversion Process
Visual guide for converting mixed fractions to improper fractions with example