Convert the following fractions into their lowest terms (simplest form): (a) 30/42
The Correct Answer and Explanation is :
To convert the fraction ( \frac{30}{42} ) into its lowest terms, we need to simplify it by finding the greatest common divisor (GCD) of the numerator (30) and the denominator (42) and then dividing both by that GCD.
Step 1: Find the GCD of 30 and 42
To find the GCD, we can use the prime factorization method:
- Prime factorization of 30:
- ( 30 = 2 \times 3 \times 5 )
- Prime factorization of 42:
- ( 42 = 2 \times 3 \times 7 )
Next, we identify the common factors from both factorizations. The common prime factors are:
- ( 2 )
- ( 3 )
Step 2: Multiply the common factors to find the GCD
Now we multiply the common prime factors:
[
GCD = 2 \times 3 = 6
]
Step 3: Simplify the fraction
Now that we have the GCD, we can simplify the fraction ( \frac{30}{42} ) by dividing both the numerator and the denominator by the GCD (6):
[
\frac{30 \div 6}{42 \div 6} = \frac{5}{7}
]
Conclusion
Thus, the fraction ( \frac{30}{42} ) in its simplest form is ( \frac{5}{7} ).
Why Simplifying Fractions is Important
Simplifying fractions to their lowest terms is essential for clarity and ease of understanding. It helps to make calculations easier, especially in operations such as addition, subtraction, or when comparing fractions. When fractions are in their simplest form, it’s easier to see the relationship between the numbers and to communicate them effectively.
Moreover, working with simplest forms is critical in fields such as science, engineering, and mathematics, where precision and clarity are paramount. In conclusion, the fraction ( \frac{30}{42} ) simplifies to ( \frac{5}{7} ), illustrating a fundamental concept in mathematics that enhances both accuracy and communication.