By Faizah Imani – Updated Aug 30, 2022

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Fractions are the building blocks of many mathematical concepts, from algebra to trigonometry. Understanding how to manipulate them accurately sets the stage for advanced learning. Below is a concise, step‑by‑step guide that covers adding, subtracting, multiplying, and dividing fractions – all while keeping the math simple and clear.

Common Denominators

1. Add fractions with the same denominator by adding their numerators: 1/4 + 2/4 = 3/4.
2. Subtract fractions with the same denominator by subtracting the numerators: 15/8 – 4/8 = 11/8.
3. Convert improper fractions to mixed numbers when desired: 11/8 = 1 3/8.

Different Denominators

1. Determine a common denominator by multiplying the denominators: 2/6 + 4/18 → 6 × 18 = 108.
2. Convert each fraction to have the common denominator:
   • First fraction: 2/6 = 2 × 18 / 108 = 36/108.
   • Second fraction: 4/18 = 4 × 6 / 108 = 24/108.
3. Add the converted fractions: 36/108 + 24/108 = 60/108.
4. Simplify the result by dividing numerator and denominator by their greatest common divisor (12): 60/108 = 5/9.

Multiplying and Dividing Fractions

Multiplying

1. Multiply the numerators: 2 × 1 = 2.
2. Multiply the denominators: 5 × 2 = 10.
3. Place the product over the product: 2/10.
4. Simplify by dividing by the greatest common divisor (2): 2/10 = 1/5.

Dividing

1. Multiply the first numerator by the second denominator: 2 × 5 = 10.
2. Multiply the first denominator by the second numerator: 3 × 1 = 3.
3. Form the new fraction: 10/3.
4. Convert to a mixed number if the numerator exceeds the denominator: 10/3 = 3 1/3.

TL;DR

Always simplify fractions to their lowest terms after each operation. This ensures clarity and accuracy in every calculation.