Problem 90 - Algebra 1: Concepts and Skills

Question

Write the fraction in simplest form. (Skills Review p. 763) $$ \frac{30}{48} $$

Step-by-Step Solution

Answer

The fraction \( \frac{30}{48} \) in simplest form is \( \frac{5}{8} \).

Step 1: Identify common factors of numerator and denominator

Identify the common factors of 30 and 48. The common factors are \(1, 2, 3, 6\). The greatest common factor (GCF) is 6.

Step 2: Simplify the fraction by the GCF

Divide both the numerator and the denominator of the fraction by the GCF. So, \( \frac{30}{6}\) = 5 and \( \frac{48}{6}\) = 8.

Step 3: Write the simplified fraction

Write down the simplified fraction. The simplified form of \( \frac{30}{48}\) is \( \frac{5}{8}\).

Key Concepts

Greatest Common Factor

Understanding the Greatest Common Factor (GCF) is essential when simplifying fractions. The GCF is the largest number that can evenly divide both the numerator and the denominator without leaving a remainder. For example, when looking at the fraction \(\frac{30}{48}\), we determine the GCF by listing out all the factors of the numerator, 30, which are 1, 2, 3, 5, 6, 10, 15, and 30, and the factors of the denominator, 48, which are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48. The common factors between these two sets are 1, 2, 3, and 6. Out of these, the greatest common factor is 6.

Simplify Fractions Step by Step

The process of simplifying fractions is straightforward if you follow it step by step. Begin by identifying the Greatest Common Factor as we discussed earlier. Next, divide both the numerator and the denominator by the GCF to reduce the fraction to its simplest form. Using our example, dividing the numerator (30) and the denominator (48) by the GCF (6), we get \(\frac{30}{6} = 5\) and \(\frac{48}{6} = 8\), respectively. The fraction simplifies to \(\frac{5}{8}\). This step-by-step process is repetitively applicable for any fraction you want to simplify.

Common Factors

Discovering common factors between the numerator and the denominator is an integral part of simplifying fractions. Common factors are numbers that divide both components of a fraction evenly. Always start with the number 1 (since it's a universal factor for all integers) and progress towards larger numbers until you find all common factors. When working with larger numbers, using a prime factorization tool can speed up the process by listing prime multiples. Always look for the highest common factor to efficiently simplify a fraction in the least number of steps.

Arithmetic Skills Review

Simplifying fractions is an important arithmetic skill that requires a strong understanding of basic math concepts, including multiplication, division, and factorization. To reinforce these skills, it is advisable to practice prime factorization and identify multiples. These exercises strengthen mental math ability and enhance your capacity to simplify fractions quickly and correctly. Practice with different numbers to gain confidence and speed, solidifying your arithmetic foundation.