Problem 36 - Prealgebra

Question

Add or subtract as indicated. $$4+\frac{5}{3 x}$$

Step-by-Step Solution

Answer

The expression is \( \frac{12x + 5}{3x} \).

Step 1: Identify the Terms

In the expression, we have two terms: a whole number, 4, and a fractional term \( \frac{5}{3x} \). Since they have different forms, we'll need to combine them into a single expression.

Step 2: Find a Common Denominator

To combine these terms, they need a common denominator. Since 4 is equivalent to \( \frac{4}{1} \), the least common denominator between 1 and \(3x\) is \(3x\).

Step 3: Rewrite Each Term with the Common Denominator

Convert the whole number to a fraction using the common denominator: \[ 4 = \frac{4 imes 3x}{3x} = \frac{12x}{3x} \]Now the first term is expressed in terms of the common denominator. The second term remains \( \frac{5}{3x} \).

Step 4: Perform the Addition

Now that both terms have the denominator \(3x\), you can add them: \[ \frac{12x}{3x} + \frac{5}{3x} = \frac{12x + 5}{3x} \]

Step 5: Simplify the Expression

The expression can be left as \( \frac{12x + 5}{3x} \) since both the numerator and denominator have no common factors to further simplify.

Key Concepts

Common Denominator

When performing operations with fractions, such as adding or subtracting, it is essential that they have a common denominator. The common denominator is a shared multiple between the denominators of the fractions involved. This makes it easier to combine the fractions, as they are essentially being brought into the same 'format'.
To find a common denominator:

• Identify the denominators of each fraction you are working with.
• Determine the least common multiple (LCM) of these denominators.

In the given exercise, we have a whole number, 4, which can be converted into a fraction with a denominator of 1, and another fraction, \( \frac{5}{3x} \). The LCM of 1 and \(3x\) is \(3x\), making it the common denominator.
By transforming all terms to this common denominator, you create compatibility between the differing forms of the numbers, allowing for straightforward addition or subtraction. This step is crucial in ensuring you accurately manage and solve fractional expressions.

Numerator and Denominator

Every fraction consists of two main parts: the numerator and the denominator. Understanding these elements is important for operations involving fractions, such as addition, subtraction, and even more complex operations.
The numerator is the top number that represents how many parts of the whole are being considered. For example, in \( \frac{5}{3x} \), the 5 is the numerator. The denominator, on the other hand, is the number below the line. It tells us into how many equal parts the whole is divided. In \( \frac{5}{3x} \), \(3x\) is the denominator.
When you add fractions, you add the numerators together while keeping the common denominator unchanged. This is possible because the denominators ensure both parts are in the same fractional terms, allowing appending of these numerators.
Understanding and manipulating these components will help you to rewrite expressions correctly and achieve accurate results when solving fraction problems.

Simplifying Expressions

After combining fraction terms by finding a common denominator and adding the numerators, the next step is simplifying expressions. Simplification is about making the expression as straightforward as possible by reducing terms.
Follow these steps for simplification:

• Look for common factors in the numerator and the denominator.
• If there are any, divide both by the greatest common factor (GCF) to reduce the fraction.
• If there are no common factors, then the expression is already in its simplest form.

In the exercise, after performing the addition, we have \( \frac{12x + 5}{3x} \). Checking both the numerator and denominator reveals no common factors, which means this expression is already simplified.
Simplification helps not only in cleaning up expressions but also ensures they are easier to interpret, analyze, and use in further mathematical operations. This step is a best practice in algebra to ensure clarity and precision.