Verified Answer :

  • Identify the prime factorization of the numerator and denominator to find the greatest common divisor.
  • Divide both 30 and 72 by their greatest common divisor, which is 6.
  • The resulting fraction \(\frac{5}{12}\) is the simplified form of the original fraction.
  • Final answer is \(\boxed{\frac{5}{12}}\).

Explanation

  1. Analyzing the Fraction

To simplify the fraction \(\frac{30}{72}\), we need to find the greatest common divisor (GCD) of the numerator (30) and the denominator (72). A fraction is in its lowest terms when the only common factor between its numerator and denominator is 1.

  1. Finding the Greatest Common Divisor

First, find the prime factorization of both numbers:

  • The prime factorization of 30 is: \(30 = 2 \times 3 \times 5\).
  • The prime factorization of 72 is: \(72 = 2 \times 36 = 2 \times 2 \times 18 = 2 \times 2 \times 2 \times 9 = 2^3 \times 3^2\).

Now, identify the common factors: both numbers share one factor of 2 and one factor of 3. Thus, the greatest common divisor is \(2 \times 3 = 6\).

  1. Simplifying the Fraction

Divide both the numerator and the denominator by their greatest common divisor (6):

  • Numerator: \(\frac{30}{6} = 5\)
  • Denominator: \(\frac{72}{6} = 12\)

Thus, the simplified fraction is \(\frac{5}{12}\). Since 5 and 12 have no common factors other than 1, the fraction is in its lowest terms.

Examples

Simplifying fractions is essential in daily tasks such as cooking, where you might need to adjust recipe portions. For instance, if a recipe calls for \(\frac{30}{72}\) of a cup of flour, simplifying it to \(\frac{5}{12}\) makes it much easier to measure out using standard kitchen measuring tools. Similarly, in construction or design, reducing ratios helps maintain accurate proportions while using smaller, more manageable numbers.

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