Adding Fractions: Same and Different Denominators

Fraction Basics

• A fraction contains a numerator (top number) and a denominator (bottom number).
• The denominator indicates how many equal parts an integer is divided into.
• The numerator indicates how many of those parts are taken.

Adding Fractions with the Same Denominator

• When adding fractions sharing a denominator, that denominator remains the same in the result.
• Only the numerators are added together.
• For example: 2/4 + 1/4 = (2+1)/4 = 3/4

Examples of Adding Fractions with the Same Denominator

• 3/7 + 2/7 = (3+2)/7 = 5/7
• 4/6 + 2/6 = (4+2)/6 = 6/6 = 1
• 1/8 + 5/8 = (1+5)/8 = 6/8, simplified to 3/4 by dividing the numerator and denominator by 2.

Adding Fractions with Different Denominators

• When adding fractions with different denominators, a common denominator must be found.
• One method involves multiplying the original denominators.

Examples of Adding Fractions with Different Denominators

• 1/2 + 1/4: Multiply denominators: 2 x 4 = 8, 1/2 = 4/8 and 1/4 = 2/8, 4/8 + 2/8 = 6/8, simplifies to 3/4.
• 4/5 + 3/4: Multiply denominators: 5 x 4 = 20, 4/5 = 16/20 and 3/4 = 15/20, 16/20 + 15/20 = 31/20 = 1 and 11/20.

Subtracting Fractions with the Same Denominator

• When subtracting fractions that share a denominator, that denominator remains the same in the result.
• Only the numerators are subtracted
• For example: 3/4 - 1/4 = (3-1)/4 = 2/4, which simplifies to 1/2.

Examples of Subtracting Fractions with the Same Denominator

• 6/8 - 3/8 = (6-3)/8 = 3/8
• 8/3 - 6/3 = (8-6)/3 = 2/3
• 1/2 - 1/2 = (1-1)/2 = 0/2 = 0
• 8/5 - 3/5 = (8-3)/5 = 5/5 = 1

Subtracting Fractions with Different Denominators

• When subtracting fractions with different denominators, a common denominator must be found.
• To find a common denominator, multiply the original denominators.

Examples of Subtracting Fractions with Different Denominators

• 3/4 - 1/3: Multiply denominators: 4 x 3 = 12, 3/4 = 9/12 and 1/3 = 4/12, 9/12 - 4/12 = 5/12.
• 6/7 - 1/5: Multiply denominators: 7 x 5 = 35, 6/7 = 30/35 and 1/5 = 7/35, 30/35 - 7/35 = 23/35.

Multiplying Fractions

• To multiply fractions, multiply the numerators together and multiply the denominators together.
• (numerator1 x numerator2) / (denominator1 x denominator2)

Examples of Multiplying Fractions

• 1/2 x 1/2 = (1x1) / (2x2) = 1/4
• 2/3 x 1/5 = (2x1) / (3x5) = 2/15
• 2/5 x 1/4 = (2x1) / (5x4) = 2/20 = 1/10 (simplified)
• 2/5 x 3/3 = (2x3) / (5x3) = 6/15 = 2/5 (simplified)
• 2/6 x 2/3 = (2x2) / (6x3) = 4/18 = 2/9 (simplified)

Dividing Fractions

• To divide fractions, multiply the first fraction by the reciprocal of the second fraction (also known as cross multiplication).
• a/b ÷ c/d = a/b * d/c = (ad) / (bc)

Examples of Dividing Fractions

• 3/10 ÷ 2/7 = (3x7) / (10x2) = 21/20
• 2/3 ÷ 3/3 = (2x3) / (3x3) = 6/9 = 2/3 (simplified)
• 1/5 ÷ 1/3 = (1x3) / (5x1) = 3/5
• 4/8 ÷ 1/3 = (4x3) / (8x1) = 12/8 = 6/4 = 3/2 (simplified)

Learn the basics of fractions, including numerators and denominators. Review adding fractions with both the same and different denominators, including finding a common denominator. Examples are provided to illustrate the concepts.