Simplifying Fractions

Introduction to Simplifying Fractions

  • Simplifying a fraction means rewriting it with the smallest numbers possible without changing its value.

Understanding Simplification with Examples

The Simplest Fraction

  • Example: 1 over 2 (already in simplest form).

  • By complicating the visual representation, we can create 3 over 6, which is equivalent to 1 over 2, demonstrating the concept of equivalent fractions.

Using Factoring to Simplify

  • Given fraction 3 over 6, we can factor:

    • Bottom (6): 2 x 3

    • Top (3): 1 x 3

  • This represents the original fraction as (1/2) * (3/3).

  • Cancel out 3/3 (which equals 1) to result in 1/2, indicating 3 over 6 simplifies to 1 over 2.

  • Conceptual Idea: Identifying whole fractions hidden within the original fraction allows for simplification.

Procedure for Simplifying Fractions

  1. Factor the top and bottom numbers into their prime factors.

  2. Identify common factors on the top and bottom.

    • Cancel them out by drawing a line through them.

  3. Re-multiply any factors left over.

    • Ensure only one number remains on top and bottom of the simplified fraction.

Important Note

  • If all factors on the top or bottom cancel out, write a ‘1’ instead of ‘0’ because 1 is always a factor of any number.

Example Problems

Simple Example: 5 over 15

  1. Factor the numbers:

    • Top: 5 is prime.

    • Bottom: 15 factors into 5 x 3.

  2. Cancel common factors:

    • Cancel out the 5 from top and bottom.

  3. Reorganize:

    • Remaining on bottom: 3.

    • Result is 1/3.

More Complex Example: 30 over 36

  1. Factor the numbers:

    • Top: 30 = 5 x 2 x 3.

    • Bottom: 36 = 2 x 3 x 2 x 3.

  2. Cancel common factors:

    • Cancel one 2 from top and bottom and one 3 from top and bottom.

  3. Reorganize:

    • Remaining factors: 5 (top) and 2 x 3 = 6 (bottom).

    • Result is 5/6; simplified version of 30/36.