Problem 41 - A Graphical Approach to College Algebra

Question

Find each sum or difference. $$\frac{3}{2 k}+\frac{5}{3 k}$$

Step-by-Step Solution

Answer

\(\frac{19}{6k}\)

Step 1: Identify the Least Common Denominator

To add fractions, we need to find a common denominator. Here, the denominators are \(2k\) and \(3k\). The least common multiple of these denominators is \(6k\).

Step 2: Rewrite Each Fraction with the Common Denominator

Convert each fraction to have the common denominator \(6k\). The first fraction \(\frac{3}{2k}\) can be rewritten as \(\frac{3 \times 3}{2k \times 3} = \frac{9}{6k}\). The second fraction \(\frac{5}{3k}\) can be rewritten as \(\frac{5 \times 2}{3k \times 2} = \frac{10}{6k}\).

Step 3: Add the Fractions

Now that both fractions have the same denominator, we can add them:\[\frac{9}{6k} + \frac{10}{6k} = \frac{9 + 10}{6k} = \frac{19}{6k}\]

Step 4: Simplify if Possible

The fraction \(\frac{19}{6k}\) is already in its simplest form since 19 is a prime number and shares no common factors with 6 or \(k\).

Key Concepts

Least Common Denominator

When working with fractions that have different denominators, the first step is to find the Least Common Denominator (LCD). The LCD is the smallest number that both denominators can divide into evenly. For example, with the denominators \(2k\) and \(3k\), you want to think about what number both of these can multiply to. You do this by finding the Least Common Multiple (LCM) of the numbers involved. Here’s how:

• List the multiples of each number. For \(2k\): \(2k, 4k, 6k, \ldots\)
• List the multiples of \(3k\): \(3k, 6k, 9k, \ldots\)
• Find the smallest multiple both lists have in common. In this case, it's \(6k\).

Once the LCD is found, both fractions can be rewritten to have this common denominator which makes it possible to add or subtract them.

Simplifying Fractions

Simplifying fractions is an important part of working with them. After performing operations like addition or subtraction, simplify your results if possible. Simplification involves reducing the fraction to its lowest terms where the numerator and denominator are as small as possible.To simplify, look for the greatest common factor (GCF) of the numerator and the denominator. Here's how:

• Find factors of both the numerator and the denominator.
• Determine the highest number that appears in both lists.
• Divide both by that number to get the fraction in its simplest form.

For instance, if \(\frac{28}{35}\), both numbers are divisible by \(7\). Divide each by \(7\) to simplify to \(\frac{4}{5}\). Simplifying ensures that fractions are as streamlined and easy to work with as possible.

Common Denominator

A common denominator is necessary when adding or subtracting fractions. It allows you to combine fractions by ensuring they both have the same base in the denominator. Finding a common denominator involves a few basic steps:

• Identify the denominators you are working with.
• Find a number into which both denominators can divide without leaving a remainder. Ideally, this is the LCD, but any common multiple works.
• Adjust the fractions by multiplying both the numerator and denominator by the necessary values so they have this common denominator.

Using a common denominator simplifies the process of adding or subtracting fractions, making it manageable. For the fractions \(\frac{3}{2k}\) and \(\frac{5}{3k}\), converting them to have \(6k\) as the denominator solves the problem effectively. It allows the numerators to be combined directly while maintaining the fraction's value.