Problem 22 - Basic College Mathematics

Question

Simplify. $$ \frac{42}{48} $$

Step-by-Step Solution

Answer

The simplified fraction is \(\frac{7}{8}\).

Step 1 - Find the Greatest Common Divisor (GCD)

Identify the largest number that can evenly divide both the numerator (42) and the denominator (48). The GCD of 42 and 48 is 6.

Step 2 - Divide Both Numerator and Denominator

Divide both the numerator and the denominator by the GCD. So, divide 42 by 6 and 48 by 6.

Step 3 - Simplify the Fraction

Perform the division: \frac{42}{6} = 7 \frac{48}{6} = 8 Thus, the simplified fraction is \(\frac{7}{8}\).

Key Concepts

Greatest Common Divisor

When simplifying fractions, the first and most important step is identifying the Greatest Common Divisor (GCD). The GCD is the largest number that can evenly divide both the numerator and the denominator of the fraction. For example, to simplify the fraction \(\frac{42}{48}\), we need to find the GCD of 42 and 48. We can do this by listing the factors of each number:

• Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42
• Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

The common factors are 1, 2, 3, and 6, so the greatest common factor among them is 6. Therefore, 6 is the GCD of 42 and 48.

Numerator and Denominator

In any fraction, the top number is called the numerator and the bottom number is called the denominator. The numerator represents how many parts we have, while the denominator represents how many parts make up a whole. For example, in the fraction \(\frac{42}{48}\), 42 is the numerator and 48 is the denominator. When simplifying a fraction, our goal is to reduce both the numerator and the denominator by their GCD. This makes the fraction as simple as possible. In this example, once we find that the GCD is 6, we can divide both the numerator (42) and the denominator (48) by 6 to simplify the fraction.

Fraction Simplification

After finding the GCD and identifying the numerator and denominator, the next step is to simplify the fraction. Simplification involves dividing both the numerator and the denominator by their GCD. For our fraction \(\frac{42}{48}\), we divide 42 by 6, which equals 7, and we divide 48 by 6, which equals 8. Thus, \(\frac{42}{48}\) simplifies to \(\frac{7}{8}\). Simplifying fractions makes them easier to work with and understand, whether you are adding, subtracting, multiplying, or comparing them. It's a fundamental skill in math that helps you solve problems more efficiently.