Fractions and Mixed Numbers | Fifth Grade
Complete Notes & Formulas
1. Fractions Review
Definition: A fraction represents a part of a whole or a part of a group.
📐 Parts of a Fraction:
3
4
Numerator (top) = Number of parts we have
Denominator (bottom) = Total number of equal parts
🔑 Types of Fractions:
- Proper Fraction: Numerator < Denominator (Example: 3/4, 2/5)
- Improper Fraction: Numerator ≥ Denominator (Example: 7/4, 9/5)
- Mixed Number: Whole number + Proper fraction (Example: 2 1/3, 4 2/5)
- Unit Fraction: Numerator is 1 (Example: 1/2, 1/8)
2. Equivalent Fractions
Definition: Equivalent fractions are fractions that have different numerators and denominators but represent the same value.
📐 How to Find Equivalent Fractions:
Method 1: Multiply
a/b = (a × n)/(b × n)
Multiply numerator and denominator by the same number
Method 2: Divide
a/b = (a ÷ n)/(b ÷ n)
Divide numerator and denominator by the same number
✏️ Example:
Find equivalent fractions for 2/3
Multiply by 2: (2 × 2)/(3 × 2) = 4/6
Multiply by 3: (2 × 3)/(3 × 3) = 6/9
Multiply by 4: (2 × 4)/(3 × 4) = 8/12
2/3 = 4/6 = 6/9 = 8/12
3. Write Fractions in Lowest Terms (Simplifying)
Definition: A fraction is in lowest terms (simplest form) when the numerator and denominator have no common factors other than 1.
📝 Steps to Simplify:
- Step 1: Find the GCF (Greatest Common Factor) of numerator and denominator
- Step 2: Divide both numerator and denominator by the GCF
- Step 3: Write the simplified fraction
✏️ Example: Simplify 18/24
Step 1: Find GCF of 18 and 24
Factors of 18: 1, 2, 3, 6, 9, 18
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
GCF = 6
Step 2: Divide: (18 ÷ 6)/(24 ÷ 6) = 3/4
Answer: 3/4 (lowest terms)
4. Convert Between Improper Fractions and Mixed Numbers
Definition: Converting between these two forms helps in different types of calculations.
📐 Conversion A: Improper Fraction → Mixed Number
- Divide numerator by denominator
- Quotient = Whole number
- Remainder = New numerator
- Denominator stays the same
a/b = Quotient (Remainder/b)
✏️ Example 1: Convert 17/5 to a mixed number
Step 1: Divide 17 ÷ 5 = 3 R2
Step 2: Quotient = 3 (whole number)
Step 3: Remainder = 2 (new numerator)
Step 4: Denominator stays 5
Answer: 3 2/5
📐 Conversion B: Mixed Number → Improper Fraction
- Multiply: denominator × whole number
- Add the numerator to the result
- This becomes the new numerator
- Denominator stays the same
W a/b = (W × b + a)/b
✏️ Example 2: Convert 3 2/5 to an improper fraction
Step 1: Multiply: 5 × 3 = 15
Step 2: Add numerator: 15 + 2 = 17
Step 3: New numerator = 17
Step 4: Denominator stays 5
Answer: 17/5
5. Least Common Denominator (LCD)
Definition: The Least Common Denominator (LCD) is the smallest number that is a common denominator for a set of fractions. It's the LCM (Least Common Multiple) of the denominators.
📝 How to Find LCD:
- Method 1: List multiples of each denominator until you find the smallest common one
- Method 2: Use prime factorization and take highest powers
- Method 3: Multiply denominators and simplify (if needed)
✏️ Example: Find LCD of 1/4 and 1/6
Method 1: Listing Multiples
Multiples of 4: 4, 8, 12, 16, 20, 24...
Multiples of 6: 6, 12, 18, 24, 30...
Smallest common multiple = 12
LCD = 12
Convert to equivalent fractions:
1/4 = (1 × 3)/(4 × 3) = 3/12
1/6 = (1 × 2)/(6 × 2) = 2/12
💡 Why Use LCD?
LCD is essential for:
- Adding fractions with different denominators
- Subtracting fractions with different denominators
- Comparing fractions
6. Round Mixed Numbers
Definition: Rounding mixed numbers to the nearest whole number helps estimate answers quickly.
🔑 Rounding Rules:
If the fraction is less than 1/2:
Round DOWN to the whole number
Example: 5 1/4 rounds to 5
If the fraction is 1/2 or greater:
Round UP to the next whole number
Example: 5 3/4 rounds to 6
✏️ Examples:
| Mixed Number | Fraction Part | Rounds To |
|---|---|---|
| 3 1/8 | < 1/2 | 3 |
| 7 1/2 | = 1/2 | 8 |
| 9 5/6 | > 1/2 | 10 |
7. Reciprocals
Definition: The reciprocal of a fraction is obtained by flipping the numerator and denominator. When you multiply a fraction by its reciprocal, the result is always 1.
📐 Reciprocal Formula:
Reciprocal of a/b = b/a
a/b × b/a = 1
✏️ Examples:
| Number | Reciprocal | Check (multiply) |
|---|---|---|
| 3/4 | 4/3 | 3/4 × 4/3 = 12/12 = 1 ✓ |
| 5/8 | 8/5 | 5/8 × 8/5 = 40/40 = 1 ✓ |
| 7 (or 7/1) | 1/7 | 7 × 1/7 = 7/7 = 1 ✓ |
| 2 1/3 (or 7/3) | 3/7 | 7/3 × 3/7 = 21/21 = 1 ✓ |
💡 Special Cases:
- Reciprocal of a whole number n: 1/n
- Reciprocal of 1: 1 (because 1/1 flipped is 1/1)
- Reciprocal of 0: Undefined (cannot divide by 0)
- For mixed numbers: Convert to improper fraction first, then flip
🔑 Why Use Reciprocals?
Reciprocals are used for:
- Dividing fractions (multiply by reciprocal instead)
- Solving equations
- Finding unit rates
Quick Reference Chart
| Concept | Key Formula/Rule |
|---|---|
| Equivalent Fractions | a/b = (a × n)/(b × n) or (a ÷ n)/(b ÷ n) |
| Simplify (Lowest Terms) | Divide numerator and denominator by GCF |
| Improper → Mixed | Divide: quotient (remainder/denominator) |
| Mixed → Improper | (whole × denominator + numerator)/denominator |
| LCD | LCM of all denominators |
| Rounding Mixed Numbers | If fraction < 1/2, round down; if ≥ 1/2, round up |
| Reciprocal | Flip numerator and denominator: a/b → b/a |
💡 Important Fraction Properties:
Multiply by 1
n/n = 1
Reciprocal Product
a/b × b/a = 1
GCF = 1
Fraction in lowest terms
Zero
0/n = 0; n/0 = undefined
🔑 Key Tips:
- Always simplify fractions to lowest terms in final answers
- To compare fractions, find LCD first
- Convert mixed numbers to improper fractions for multiplication/division
- To divide by a fraction, multiply by its reciprocal
- Remember: Equivalent fractions have the same value but different forms
📚 Fifth Grade Fractions and Mixed Numbers - Complete Study Guide
Master these concepts for math excellence! ✨