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How are fractions added subtracted?

Published in Fraction Arithmetic 3 mins read

To add or subtract fractions, they must first have a common denominator. Let's explore the process:

Understanding the Basics

  • Fraction Components: A fraction consists of a numerator (top number) and a denominator (bottom number).
  • The Denominator's Role: The denominator indicates the number of equal parts the whole is divided into.
  • The Numerator's Role: The numerator indicates how many of those parts are being considered.

The Golden Rule: Common Denominators

According to the reference, to add or subtract fractions, they must have the same denominator (the bottom value). If the denominators are already the same then it is just a matter of either adding or subtracting the numerators (the top value).

Steps for Addition and Subtraction

Here's a step-by-step guide:

  1. Find a Common Denominator: If the fractions don't have the same denominator, find the least common multiple (LCM) of the denominators. This will be your new common denominator.

    • Example: For 1/2 and 1/3, the LCM of 2 and 3 is 6.

  2. Convert the Fractions: Adjust the numerators of each fraction to match the new common denominator. Remember to multiply both the numerator and the denominator by the same number to maintain the fraction's value.

    • Example:
      • 1/2 becomes (1 3) / (2 3) = 3/6
      • 1/3 becomes (1 2) / (3 2) = 2/6

  3. Add or Subtract the Numerators: Once the fractions have a common denominator, simply add or subtract the numerators. Keep the denominator the same.

    • Example (Addition): 3/6 + 2/6 = (3 + 2) / 6 = 5/6
    • Example (Subtraction): 3/6 - 2/6 = (3 - 2) / 6 = 1/6

  4. Simplify (if possible): Reduce the resulting fraction to its simplest form by dividing both the numerator and denominator by their greatest common factor (GCF).

    • Example: If the result was 4/6, both 4 and 6 are divisible by 2, simplifying to 2/3.

Examples

  • Example 1: Adding Fractions with Common Denominators

    • 2/5 + 1/5 = (2 + 1) / 5 = 3/5

  • Example 2: Subtracting Fractions with Uncommon Denominators

    • 5/8 - 1/4
    • Find the common denominator: The LCM of 8 and 4 is 8.
    • Convert 1/4: (1 2) / (4 2) = 2/8
    • Subtract: 5/8 - 2/8 = 3/8

Table Summary

Step Description Example (Adding 1/4 + 2/5)
1. Common Denominator Find the Least Common Multiple (LCM) of the denominators. LCM of 4 and 5 is 20
2. Convert Fractions Multiply the numerator and denominator of each fraction to get the common denominator. 1/4 becomes (15)/(45) = 5/20; 2/5 becomes (24)/(54) = 8/20
3. Add/Subtract Numerators Add or subtract the numerators, keeping the common denominator. 5/20 + 8/20 = (5+8)/20 = 13/20
4. Simplify Reduce the fraction to its simplest form (if possible). 13/20 is already in simplest form

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