Write the fraction 45/54 in simplest form
The Correct Answer and Explanation is :
To simplify the fraction ( \frac{45}{54} ), we first need to find the greatest common divisor (GCD) of the numerator (45) and the denominator (54). The GCD is the largest number that divides both numbers without leaving a remainder.
Step 1: Find the GCD
- Prime Factorization:
- For 45: The prime factors of 45 are ( 3 \times 3 \times 5 = 3^2 \times 5 ).
- For 54: The prime factors of 54 are ( 2 \times 3 \times 3 \times 3 = 2 \times 3^3 ).
- Identify Common Factors:
- The common prime factor is ( 3 ).
- The lowest power of ( 3 ) common to both factorizations is ( 3^2 ) (from 45), which equals 9.
Thus, the GCD of 45 and 54 is 9.
Step 2: Divide Numerator and Denominator by the GCD
Now, we divide both the numerator and the denominator of the fraction by the GCD:
[
\frac{45 \div 9}{54 \div 9} = \frac{5}{6}
]
Conclusion
The simplest form of the fraction ( \frac{45}{54} ) is ( \frac{5}{6} ).
Explanation
Simplifying fractions is a vital skill in mathematics, as it allows for easier calculations and clearer representation of numbers. By reducing a fraction to its simplest form, we express it in the lowest terms, which means both the numerator and denominator share no common factors other than 1. This simplification is not just for aesthetics; it is crucial for comparing fractions, performing arithmetic operations, and solving real-world problems efficiently.
In this case, ( \frac{45}{54} ) can be seen as a ratio, which represents a part of a whole. Simplifying to ( \frac{5}{6} ) reveals that for every 6 parts, 5 parts are taken, providing a clearer understanding of the relationship between these quantities.