Problem 9 - Elementary Algebra

Question

For the following problems, reduce, if possible, each fraction lowest terms. \(\frac{10}{12}\)

Step-by-Step Solution

Answer

Question: Reduce the fraction 10/12 to its lowest terms. Answer: The reduced fraction is 5/6.

Step 1: Find the Greatest Common Divisor (GCD) of the Numerator and Denominator

The numerator is 10 and the denominator is 12. First, we will list the factors of both numbers. Factors of 10: 1, 2, 5, 10 Factors of 12: 1, 2, 3, 4, 6, 12 The greatest common divisor of 10 and 12 is 2.

Step 2: Divide the Numerator and Denominator by the GCD

Now, we will divide both the numerator (10) and the denominator (12) by the GCD (2). \(\frac{10}{2} = 5\) \(\frac{12}{2} = 6\)

Step 3: Rewrite the Fraction in Lowest Terms

After dividing the numerator and denominator by the GCD, we rewrite the fraction with the new values. \(\frac{10}{12} = \frac{5}{6}\) The fraction in the lowest terms is \(\frac{5}{6}\).

Key Concepts

Greatest Common Divisor

Understanding the greatest common divisor (GCD) is essential when simplifying fractions. It is the largest number that can exactly divide two or more numbers without leaving a remainder.
When simplifying fractions, determining the GCD helps in reducing the fraction to its lowest terms. To find the GCD of two numbers, like 10 and 12 from our problem, list the factors of each number:

• Factors of 10: 1, 2, 5, 10
• Factors of 12: 1, 2, 3, 4, 6, 12

The GCD is the largest common factor found in both lists. In this case, it is 2.
Using this GCD, you can adjust both the numerator and the denominator of a fraction to make it simpler, which makes calculations easier and more efficient.

Numerator and Denominator

Every fraction consists of two parts: the numerator and the denominator.
These play a crucial role in defining the value of a fraction. The numerator is the top number in a fraction. It indicates how many parts of the whole are being considered. For example, in the fraction \(\frac{10}{12}\), 10 is the numerator, representing 10 parts of a certain whole.
The denominator, on the other hand, is the bottom number. It shows into how many equal parts the whole is divided. In our example, 12 is the denominator, meaning the whole is divided into 12 equal parts.
Understanding these two components is key to performing operations on fractions, such as addition, subtraction, and particularly simplification.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form. This means changing a fraction so that the numerator and the denominator are as small as possible while still having the same value.
To simplify a fraction like \(\frac{10}{12}\), once you have the GCD (which we've found to be 2), divide both the numerator and the denominator by this number.

• \(\frac{10}{12} \rightarrow\) Divide both by 2
• New numerator: \(\frac{10}{2} = 5\)
• New denominator: \(\frac{12}{2} = 6\)

Now, you re-write the fraction with these new numbers, resulting in \(\frac{5}{6}\).
The simplified or lowest terms of the fraction \(\frac{10}{12}\) is \(\frac{5}{6}\). This not only makes the fraction easier to work with but also maintains its equality to the original expression.