Use the GCF to simplify each fraction.
32/128, 51/85 and 42/35
The Correct Answer and Explanation is:
To simplify fractions using the greatest common factor (GCF), we first need to determine the GCF of the numerator and the denominator of each fraction. Once we identify the GCF, we can divide both the numerator and the denominator by this value to simplify the fraction to its lowest terms. Let’s work through each of the fractions you’ve provided.
1. Simplifying ( \frac{32}{128} )
Step 1: Find the GCF
- The factors of 32 are: 1, 2, 4, 8, 16, 32.
- The factors of 128 are: 1, 2, 4, 8, 16, 32, 64, 128.
- The common factors are: 1, 2, 4, 8, 16, 32.
- The GCF is 32.
Step 2: Simplify the fraction
[
\frac{32 \div 32}{128 \div 32} = \frac{1}{4}
]
2. Simplifying ( \frac{51}{85} )
Step 1: Find the GCF
- The factors of 51 are: 1, 3, 17, 51.
- The factors of 85 are: 1, 5, 17, 85.
- The common factors are: 1, 17.
- The GCF is 17.
Step 2: Simplify the fraction
[
\frac{51 \div 17}{85 \div 17} = \frac{3}{5}
]
3. Simplifying ( \frac{42}{35} )
Step 1: Find the GCF
- The factors of 42 are: 1, 2, 3, 6, 7, 14, 21, 42.
- The factors of 35 are: 1, 5, 7, 35.
- The common factors are: 1, 7.
- The GCF is 7.
Step 2: Simplify the fraction
[
\frac{42 \div 7}{35 \div 7} = \frac{6}{5}
]
Final Results:
- ( \frac{32}{128} ) simplifies to ( \frac{1}{4} ).
- ( \frac{51}{85} ) simplifies to ( \frac{3}{5} ).
- ( \frac{42}{35} ) simplifies to ( \frac{6}{5} ).
Conclusion:
In summary, simplifying fractions using the GCF involves identifying the largest number that divides both the numerator and denominator without leaving a remainder. This process ensures the fraction is expressed in its simplest form, making it easier to understand and work with. The fractions ( \frac{32}{128} ), ( \frac{51}{85} ), and ( \frac{42}{35} ) were successfully simplified to ( \frac{1}{4} ), ( \frac{3}{5} ), and ( \frac{6}{5} ), respectively.