Comprehensive Guide to Fractions

What is a Fraction?

A fraction represents a part of a whole or, more generally, any number of equal parts. It consists of:

Numerator: The number above the line, representing how many parts we have.
Denominator: The number below the line, representing the total number of equal parts in the whole.

Three-fourths or three quarters (3/4)

Types of Fractions

1. Proper Fractions

When the numerator is less than the denominator (value less than 1).

Examples: 1/2, 3/4, 5/8

In a proper fraction, the part is smaller than the whole.

2. Improper Fractions

When the numerator is greater than or equal to the denominator (value greater than or equal to 1).

Examples: 5/3, 7/4, 11/8

In an improper fraction, the part is equal to or larger than the whole.

3. Mixed Numbers

A whole number and a proper fraction combined.

Examples: 1 1/2, 2 3/4, 5 2/3

A mixed number represents a quantity that is between two whole numbers.

4. Equivalent Fractions

Fractions that represent the same value.

Examples: 1/2 = 2/4 = 3/6 = 4/8

We get equivalent fractions by multiplying or dividing both the numerator and denominator by the same non-zero number.

5. Like Fractions

Fractions with the same denominator.

Examples: 3/8 and 5/8, 2/7 and 4/7

6. Unlike Fractions

Fractions with different denominators.

Examples: 1/2 and 3/5, 4/9 and 7/12

7. Unit Fractions

Fractions with a numerator of 1.

Examples: 1/2, 1/3, 1/4, 1/5

Operations with Fractions

1. Addition of Fractions

Like Fractions (Same Denominator): Add the numerators, keep the denominator the same.

Example: 3/8 + 2/8 = (3+2)/8 = 5/8

Unlike Fractions (Different Denominators): Find a common denominator, convert to equivalent fractions, then add.

Example: 1/2 + 1/3

Find the Least Common Multiple (LCM) of denominators: LCM(2,3) = 6
Convert to equivalent fractions with denominator 6:
- 1/2 = 3/6
- 1/3 = 2/6
Add numerators: 3/6 + 2/6 = 5/6

2. Subtraction of Fractions

Like Fractions: Subtract the numerators, keep the denominator the same.

Example: 7/9 - 4/9 = (7-4)/9 = 3/9 = 1/3

Unlike Fractions: Find a common denominator, convert to equivalent fractions, then subtract.

Example: 5/6 - 1/4

Find LCM of denominators: LCM(6,4) = 12
Convert to equivalent fractions:
- 5/6 = 10/12
- 1/4 = 3/12
Subtract: 10/12 - 3/12 = 7/12

3. Multiplication of Fractions

Multiply the numerators, multiply the denominators.

Example: 2/3 × 3/4

Multiply numerators: 2 × 3 = 6
Multiply denominators: 3 × 4 = 12
Result: 6/12 = 1/2

Shortcut: You can simplify before multiplying by finding common factors between numerators and denominators.

Example: 3/8 × 4/9

Identify common factors: 3 and 9 share a factor of 3
Cross-simplify: (3÷3)/(8) × (4)/(9÷3) = 1/8 × 4/3
Multiply: 1 × 4 = 4 (numerator), 8 × 3 = 24 (denominator)
Result: 4/24 = 1/6

4. Division of Fractions

Multiply by the reciprocal of the divisor (flip the second fraction).

Example: 2/3 ÷ 3/4

Take the reciprocal of the second fraction: 3/4 becomes 4/3
Change division to multiplication: 2/3 × 4/3
Multiply numerators: 2 × 4 = 8
Multiply denominators: 3 × 3 = 9
Result: 8/9

Converting Between Different Types of Fractions

1. Converting Improper Fractions to Mixed Numbers

Divide the numerator by the denominator.
The quotient is the whole number part.
The remainder is the numerator of the fraction part.
The denominator remains the same.

Example: Convert 17/5 to a mixed number

Divide 17 by 5: 17 ÷ 5 = 3 with a remainder of 2
The whole number is 3
The fraction part is 2/5
Result: 17/5 = 3 2/5

2. Converting Mixed Numbers to Improper Fractions

Multiply the whole number by the denominator.
Add the result to the numerator.
Place this sum over the original denominator.

Example: Convert 2 3/4 to an improper fraction

Multiply the whole number by the denominator: 2 × 4 = 8
Add the result to the numerator: 8 + 3 = 11
Place this sum over the original denominator: 11/4
Result: 2 3/4 = 11/4

Simplifying Fractions

Finding the Greatest Common Divisor (GCD)

To simplify a fraction to its lowest terms, divide both the numerator and denominator by their Greatest Common Divisor (GCD).

Example: Simplify 12/18

Find the GCD of 12 and 18: GCD(12,18) = 6
Divide both numbers by the GCD: 12 ÷ 6 = 2 and 18 ÷ 6 = 3
Result: 12/18 = 2/3

Methods to find GCD:

Prime Factorization: Break down both numbers into prime factors, then multiply the common prime factors.
Euclidean Algorithm: Repeatedly divide the larger number by the smaller and keep the remainder until the remainder is 0.
Common Divisors: List all divisors of both numbers and find the largest common one.

Comparing Fractions

Methods for Comparing Fractions

1. Same Denominator: Compare the numerators.

Example: Compare 5/8 and 3/8

5 > 3, so 5/8 > 3/8

2. Same Numerator: Compare the denominators (inversely).

Example: Compare 3/4 and 3/7

4 < 7, so 3/4 > 3/7 (smaller denominator means larger fraction)

3. Different Numerators and Denominators: Convert to a common denominator and compare numerators.

Example: Compare 2/3 and 3/5

Find LCD: LCM(3,5) = 15
Convert: 2/3 = 10/15 and 3/5 = 9/15
Compare: 10 > 9, so 2/3 > 3/5

4. Cross Multiplication: Multiply the numerator of each fraction by the denominator of the other.

Example: Compare 2/3 and 3/5

Cross multiply: 2 × 5 = 10 and 3 × 3 = 9
Compare: 10 > 9, so 2/3 > 3/5

5. Convert to Decimals: Divide the numerator by the denominator and compare.

Example: Compare 2/3 and 3/5

2/3 ≈ 0.667 and 3/5 = 0.6
Compare: 0.667 > 0.6, so 2/3 > 3/5

Word Problems with Fractions

Example 1: Division of Whole by Fraction

Problem: If you have 4 pizzas and each person gets 1/8 of a pizza, how many people can you serve?

Solution:

Set up division: 4 ÷ 1/8
Convert to multiplication by reciprocal: 4 × 8/1
Calculate: 4 × 8 = 32
Answer: 32 people can be served.

Example 2: Multi-Step Problem

Problem: Sarah had a chocolate bar. She ate 1/4 of it and gave 1/3 of what remained to her brother. How much of the original chocolate bar does she have left?

Solution: