Reduce the fraction 26/48 to the simplest form
The Correct Answer and Explanation is:
To reduce the fraction ( \frac{26}{48} ) to its simplest form, we need to find the greatest common divisor (GCD) of the numerator (26) and the denominator (48). The GCD is the largest number that divides both 26 and 48 evenly.
Step 1: Find the prime factors of 26 and 48
- Prime factorization of 26:
- 26 is an even number, so it’s divisible by 2. Dividing 26 by 2 gives us 13.
- 13 is a prime number, so the prime factorization of 26 is ( 2 \times 13 ).
- Prime factorization of 48:
- 48 is an even number, so divide by 2: ( 48 \div 2 = 24 ).
- 24 is also even, so divide by 2 again: ( 24 \div 2 = 12 ).
- 12 is still even, so divide by 2 once more: ( 12 \div 2 = 6 ).
- 6 is also even, so divide by 2 again: ( 6 \div 2 = 3 ).
- Finally, 3 is a prime number. So, the prime factorization of 48 is ( 2^4 \times 3 ).
Step 2: Find the greatest common divisor (GCD)
To find the GCD, we look for the highest common factors between the prime factorizations of 26 and 48.
- Prime factorization of 26: ( 2 \times 13 )
- Prime factorization of 48: ( 2^4 \times 3 )
The only common prime factor is 2, and the smallest power of 2 that appears in both factorizations is ( 2^1 ), so the GCD of 26 and 48 is 2.
Step 3: Divide both the numerator and denominator by the GCD
Now, divide both the numerator (26) and the denominator (48) by their GCD (2):
[
\frac{26}{48} = \frac{26 \div 2}{48 \div 2} = \frac{13}{24}
]
Final Answer:
The fraction ( \frac{26}{48} ) simplifies to ( \frac{13}{24} ).
Explanation:
Simplifying fractions involves dividing both the numerator and denominator by their greatest common divisor (GCD). In this case, the GCD of 26 and 48 is 2. By dividing both the numerator and denominator by 2, we obtain the simplified fraction ( \frac{13}{24} ). This process ensures that the fraction is in its simplest form, where the numerator and denominator have no common factors other than 1.