You cross out common factors by identifying numbers that divide evenly into both the numerator and the denominator of a fraction and then dividing by those numbers. This process is known as simplifying or reducing fractions.
Understanding Common Factors
A common factor is a number that divides two or more numbers without leaving a remainder. When working with fractions, identifying common factors in the numerator and the denominator is essential for simplification.
The Process of Crossing Out Common Factors
The video, "Multiplying Fractions using Cancellation," from the reference, provides a practical example of how to cancel common factors. Here’s how it’s done:
- Identify: Find common factors between the numerators and denominators. For instance, in a fraction multiplication problem, you might see a 3 in a numerator and a 9 in a denominator. Both are divisible by 3.
- Divide: Divide both the numerator and the denominator by the common factor. Using the previous example, divide the 3 by 3 (resulting in 1) and divide the 9 by 3 (resulting in 3).
- Rewrite: Replace the original numbers with the results of the division. This simplifies the fractions before you multiply them.
- Repeat: If there are multiple pairs of numbers with common factors, repeat steps 1-3 until all possible reductions have been made. According to the reference, you can cancel out numerators and denominators by identifying common factors.
Example
Let’s illustrate this with a specific problem, inspired by the video example: Imagine you are multiplying the fraction 3/5 by 5/9.
- Initial Problem: (3/5) * (5/9)
- Identify Common Factors:
- 3 and 9 share a common factor of 3.
- 5 and 5 share a common factor of 5.
- Divide:
- Divide 3 by 3 = 1.
- Divide 9 by 3 = 3.
- Divide 5 by 5 = 1.
- Divide 5 by 5 = 1.
- Rewrite: (1/1) * (1/3)
- Multiply: The simplified multiplication is (1/1) * (1/3) = 1/3
In this problem, all the numerators and denominators were reduced using the common factors.
Benefits of Crossing Out Common Factors
- Simplified Calculations: By reducing fractions before multiplying, you work with smaller, easier to manage numbers.
- Fewer Errors: Simplifying early reduces the likelihood of making mistakes when multiplying large numbers.
- Final Fraction in Lowest Terms: The result is automatically in its simplest form.
By identifying common factors, dividing by them, and rewriting the fractions, you make multiplying and working with fractions much more manageable and efficient.
