9/40 in simplest form?
The Correct Answer and Explanation is:
To simplify the fraction ( \frac{9}{40} ), we need to find the greatest common divisor (GCD) of the numerator (9) and the denominator (40). The GCD is the largest number that divides both numbers without leaving a remainder.
Step 1: Find the Prime Factorization
Prime factorization of 9:
- The number 9 can be expressed as ( 3 \times 3 ) or ( 3^2 ).
Prime factorization of 40:
- The number 40 can be expressed as ( 2 \times 20 ).
- Further breaking it down, ( 20 = 2 \times 10 ), and ( 10 = 2 \times 5 ).
- Thus, the complete prime factorization of 40 is ( 2^3 \times 5 ).
Step 2: Determine the GCD
Now, we can identify the common prime factors between 9 and 40.
- The prime factorization of 9 is ( 3^2 ).
- The prime factorization of 40 is ( 2^3 \times 5 ).
Since there are no common factors between 9 and 40, the GCD is 1.
Step 3: Simplify the Fraction
To simplify the fraction ( \frac{9}{40} ), we divide both the numerator and the denominator by their GCD (which is 1):
[
\frac{9 \div 1}{40 \div 1} = \frac{9}{40}
]
Since the GCD is 1, the fraction ( \frac{9}{40} ) is already in its simplest form.
Conclusion
In summary, the fraction ( \frac{9}{40} ) cannot be simplified further since 9 and 40 share no common factors other than 1. Therefore, the final simplified form of ( \frac{9}{40} ) remains ( \frac{9}{40} ). This process demonstrates the importance of finding the GCD when simplifying fractions, as it ensures that the fraction is expressed in the lowest terms possible.