Problem 107 - Elementary Algebra

Question

Give a step-by-step description of how to add the fractions \(\frac{5}{4 x}\) and \(\frac{7}{6 x}\)

Step-by-Step Solution

Answer

The sum of \( \frac{5}{4x} \) and \( \frac{7}{6x} \) is \( \frac{29}{12x} \).

Step 1: Identify the Problem

We need to add two fractions: \( \frac{5}{4x} \) and \( \frac{7}{6x} \). To add them, they must have a common denominator.

Step 2: Find the Least Common Denominator (LCD)

The denominators of the fractions are \(4x\) and \(6x\). The least common denominator (LCD) will be the least common multiple of these denominators. Both denominators share \(x\), so consider the numbers 4 and 6. The least common multiple of 4 and 6 is 12, therefore, the LCD is \(12x\).

Step 3: Rewrite Fractions with LCD

Rewrite each fraction so that they both have the denominator \(12x\). - For \( \frac{5}{4x} \): Multiply both numerator and denominator by 3 to get \( \frac{15}{12x} \).- For \( \frac{7}{6x} \): Multiply both numerator and denominator by 2 to get \( \frac{14}{12x} \).

Step 4: Add the Fractions

Since both fractions now have the same denominator, add the numerators directly, keeping the denominator \(12x\):\[ \frac{15}{12x} + \frac{14}{12x} = \frac{15 + 14}{12x} = \frac{29}{12x} \]

Step 5: Verify and Simplify (if possible)

Check if the fraction \( \frac{29}{12x} \) can be simplified further. Since 29 is a prime number and does not divide evenly into 12 or \(x\), the fraction is already in its simplest form.

Key Concepts

Least Common Denominator

Adding fractions requires us to work with the concept of a common denominator. The Least Common Denominator (LCD) is the smallest multiple that two or more denominators share. In our exercise with fractions \( \frac{5}{4x} \) and \( \frac{7}{6x} \), we need to find this shared base for the denominators.

The denominators here are \(4x\) and \(6x\). Notice that both fractions have a variable part, \(x\), which we can factor out. Focus on the numerical components (4 and 6) for now. To find the least common multiple of these numbers:

• List the multiples of 4 (4, 8, 12, 16...)
• List the multiples of 6 (6, 12, 18, 24...)

You'll see that 12 is the first common multiple, making it the least. Combine this with \(x\) to make the LCD \(12x\). Achieving a shared denominator allows for straightforward fraction addition.

Simplifying Fractions

When dealing with fractions in math, simplifying is a key step to ensure that our results are in their simplest, most understandable form. Simplification involves reducing a fraction to its smallest possible numerator and denominator without changing its value.

In our problem, after adding the two fractions, we arrive at \( \frac{29}{12x} \). To simplify:

• Check the numerator and the denominator for any common factors.
• 29 is a prime number, so it cannot be divided any further except by 1 or 29 itself.
• Since 29 does not divide evenly into 12 or \(x\), \( \frac{29}{12x} \) is already in its simplest form.

Often, determining the prime nature of numbers helps establish if a fraction can be simplified. Always strive for the simplest form as it benefits computation and understanding.

Step-by-Step Math Process

Understanding the step-by-step process in solving math problems is essential for grasping fundamental concepts and being able to apply them independently. Breaking down problems into smaller, manageable steps helps in clarity and avoids errors.

For our task of adding the fractions \( \frac{5}{4x} \) and \( \frac{7}{6x} \), here is how it unfolds:

• Identify the Problem: Clearly define what needs to be done. We need a common denominator to add the fractions.
• Find the LCD: Analyze the denominators, find their least common multiple, and incorporate any variables present (in this case, \(x\)).
• Rewrite Fractions: Transform each fraction to have this shared denominator by multiplying.
• Add the Fractions: Once denominators match, simply add the numerators, keeping the shared denominator.
• Verify and Simplify: After adding, check if the result can be simplified.

Following detailed steps encourages understanding and skill in solving similar problems independently. Practicing step-by-step processes builds strong mathematical foundations.