48/52 in its simplest form
The Correct Answer and Explanation is :
To simplify the fraction ( \frac{48}{52} ), we need to find the greatest common divisor (GCD) of the numerator (48) and the denominator (52).
Step 1: Find the GCD
- Prime Factorization:
- For 48:
- ( 48 = 2 \times 24 )
- ( 24 = 2 \times 12 )
- ( 12 = 2 \times 6 )
- ( 6 = 2 \times 3 )
- Therefore, the prime factorization of 48 is ( 2^4 \times 3^1 ).
- For 52:
- ( 52 = 2 \times 26 )
- ( 26 = 2 \times 13 )
- Therefore, the prime factorization of 52 is ( 2^2 \times 13^1 ).
- Identify Common Factors:
- The common factor in both numbers is ( 2 ).
- The lowest power of 2 in the prime factorizations is ( 2^2 ).
Thus, the GCD of 48 and 52 is ( 4 ).
Step 2: Simplify the Fraction
Next, we divide both the numerator and the denominator by the GCD (4):
[
\frac{48 \div 4}{52 \div 4} = \frac{12}{13}
]
Conclusion
Therefore, the simplest form of the fraction ( \frac{48}{52} ) is ( \frac{12}{13} ).
Explanation of Simplification
Simplifying fractions involves reducing them to their most basic form, where the numerator and denominator share no common factors other than 1. This process is important in mathematics because it allows for clearer communication and easier calculations. By expressing numbers in their simplest form, you can more easily compare fractions, perform arithmetic operations, or understand ratios.
Understanding how to factor numbers and find the GCD is a key skill in mathematics. It not only aids in simplifying fractions but also enhances problem-solving abilities in various mathematical concepts, including algebra and number theory. Reducing fractions is a fundamental step in many real-world applications, such as cooking measurements, construction, and financial calculations, where precision is necessary.