Problem 89 - Elementary and Intermediate Algebra

Question

Perform the operations and, if possible, simplify. $$ \frac{2}{3}-\frac{1}{4}+\frac{1}{12} $$

Step-by-Step Solution

Answer

The simplified result of the operation is \( \frac{1}{2} \).

Step 1: Identify the Problem

We need to perform the arithmetic operations on the fractions: \( \frac{2}{3} - \frac{1}{4} + \frac{1}{12} \) and simplify if possible.

Step 2: Find the Common Denominator

The denominators of the fractions are 3, 4, and 12. The least common denominator (LCD) for these numbers is 12. We will rewrite each fraction with a denominator of 12.

Step 3: Convert Fractions to Common Denominator

Convert each fraction:- \( \frac{2}{3} = \frac{2 imes 4}{3 imes 4} = \frac{8}{12} \)- \( \frac{1}{4} = \frac{1 imes 3}{4 imes 3} = \frac{3}{12} \)- \( \frac{1}{12} \) is already with a denominator of 12.

Step 4: Perform the Addition and Subtraction

With a common denominator, combine the fractions: \( \frac{8}{12} - \frac{3}{12} + \frac{1}{12} \).Perform the operations: \( \frac{8-3+1}{12} = \frac{6}{12} \).

Step 5: Simplify the Resulting Fraction

The fraction \( \frac{6}{12} \) can be simplified:Divide both the numerator and the denominator by the greatest common divisor, which is 6:\( \frac{6 \div 6}{12 \div 6} = \frac{1}{2} \).

Key Concepts

Least Common Denominator

When dealing with multiple fractions, finding a common denominator is essential for combining them through addition or subtraction. The least common denominator (LCD) is the smallest multiple that is common to all the denominators involved.
This is crucial because it allows each fraction to be converted into an equivalent fraction with the same denominator.

• First, list out the multiples of each denominator.
• Identify the smallest multiple that appears in all lists.

For example, in the given exercise, the denominators are 3, 4, and 12. The multiples of these numbers are:

• 3: 3, 6, 9, 12, 15, ...
• 4: 4, 8, 12, 16, ...
• 12: 12, 24, 36, ...

The smallest common multiple is 12, which becomes the LCD. Once the LCD is determined, you adjust each fraction by multiplying the numerator and denominator, so they share this common base, allowing you to easily perform the required operations.

Simplifying Fractions

Simplifying fractions is the process of reducing them to their simplest form, which means expressing the fraction with the smallest possible numerator and denominator. This makes the fraction easier to understand and use in calculations.
Here’s how you simplify:

• Find the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD.

In the example provided in the solution, the result of the operations was the fraction \( \frac{6}{12} \).
The GCD of 6 and 12 is 6. So, you divide both the numerator and the denominator by 6, resulting in \( \frac{1}{2} \).
This reduced form is much easier to interpret and is the clearest way to express the answer.

Numerator and Denominator

Understanding the roles of the numerator and denominator is key to working with fractions effectively.
The numerator is the top part of a fraction. It represents the number of parts being considered. The denominator, on the other hand, is on the bottom and indicates the total number of equal parts into which the whole is divided.
To perform arithmetic operations like addition and subtraction on fractions:

• Ensure all fractions refer to the same whole, using a common denominator.
• Operate only on the numerators, as you keep the denominator constant.

For example, in the exercise given, once all fractions shared a common denominator of 12, it was straightforward:
You combined the numerators from the fractions: numerator \(8\) from \( \frac{8}{12} \), subtracting numerator \(3\) from \(\frac{3}{12}\), and adding numerator \(1\) from \(\frac{1}{12}\).
This results in the computation \(8 - 3 + 1 = 6\), so the answer is \(\frac{6}{12}\). With simplification tools, this can become \(\frac{1}{2}\), making your work easier to manage and understand.