Problem 39 - Introductory Algebra for College Students

Question

Simplify each fraction by reducing it to its lowest terms. $$\frac{120}{86}$$

Step-by-Step Solution

Answer

The simplified form of \(\frac{120}{86}\) is \(\frac{60}{43}\).

Step 1: Identify the Numerator and Denominator

In the fraction \(\frac{120}{86}\), the numerator is 120 and the denominator is 86.

Step 2: Find the Greatest Common Divisor (GCD)

The GCD of 120 and 86 is 2. This is the largest number that can evenly divide both numbers.

Step 3: Simplify the Fraction

Divide both the numerator and the denominator by the GCD. \(\frac{120}{2} = 60\) and \(\frac{86}{2} = 43\). So, the fraction simplified to its lowest terms is \(\frac{60}{43}\).

Key Concepts

Numerator and Denominator

When dealing with fractions, the numbers used represent different parts of the whole. The numerator is the number on the top of the fraction. It indicates how many parts of the whole are being considered or used. On the other hand, the denominator is the number at the bottom, which shows the total number of equal parts the whole is divided into.

For example, in the fraction \(\frac{120}{86}\), 120 is the numerator, which means there are 120 parts we are focusing on. The denominator is 86, representing that the whole is divided into 86 equal parts. Understanding these terms is crucial as they form the basis of all operations involving fractions.

Greatest Common Divisor

The Greatest Common Divisor (GCD), also known as the greatest common factor, is the highest number that can exactly divide two or more integers without leaving a remainder. To simplify fractions effectively, finding the GCD of the numerator and the denominator is vital. It is the cornerstone of reducing fractions to their lowest terms because it allows you to divide both the numerator and the denominator by the same number, thereby scaling down the fraction.

To find the GCD, you can use several methods such as listing out factors, employing the Euclidean algorithm, or using prime factorization. Once the GCD is identified, both the numerator and the denominator can be divided by it to simplify the fraction.

Reducing Fractions to Lowest Terms

Reducing a fraction to its lowest terms makes it simpler and often easier to understand or work with in calculations. After identifying the numerator and the denominator, and finding the GCD, you proceed to divide both the numerator and the denominator by the GCD. The resulting fraction is in its simplest form because it cannot be reduced further by division without introducing decimals or fractions.

In the given exercise, the fraction \(\frac{120}{86}\) is simplified using the GCD of 2, which is the largest number that divides both 120 and 86. After dividing 120 and 86 by 2, we obtain \(\frac{60}{43}\), a fraction that cannot be reduced any further. This systematic approach ensures that you always achieve the most reduced version of any fraction.