Problem 87 - Elementary Algebra

Question

Simplify each algebraic expression by combinang similar terms. $$\frac{3}{5} n-\frac{1}{4} n+\frac{3}{10} n$$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{13}{20}n \).

Step 1: Convert to Common Denominator

The first step is to convert all fractions to have a common denominator. Here, the fractions are \( \frac{3}{5}\), \(-\frac{1}{4}\), and \(\frac{3}{10}\). The least common denominator (LCD) of 5, 4, and 10 is 20.

Step 2: Rewrite Fractions

Write each fraction with the common denominator of 20:\[ \frac{3}{5}n = \frac{12}{20}n \]\[ -\frac{1}{4}n = -\frac{5}{20}n \]\[ \frac{3}{10}n = \frac{6}{20}n \]

Step 3: Combine Like Terms

Combine all the terms \( \frac{12}{20}n - \frac{5}{20}n + \frac{6}{20}n \):\[ \left(\frac{12}{20} - \frac{5}{20} + \frac{6}{20}\right)n = \frac{13}{20}n \]

Step 4: Simplified Expression

The simplified expression is found by combining the terms, resulting in the final simplified expression:\[ \frac{13}{20}n \]

Key Concepts

Like Terms

In algebra, "like terms" are terms that have the same variables raised to the same power. They can be combined or simplified into one term. This is an essential step when simplifying algebraic expressions.

For example, in the expression \(3n + 5n - n\), all terms are "like terms" because they all contain the variable \(n\).
- To combine them, simply add or subtract the coefficients (numbers in front of the variables).
- The expression simplifies to \((3+5-1)n = 7n\).

Combining like terms makes expressions simpler and easier to work with, especially in equations and algebraic fractions.

Common Denominator

When dealing with fractions, the "common denominator" is a shared multiple of the denominators of the fractions. Converting to a common denominator allows you to easily add or subtract fractions because it ensures the fractions are parts of the same whole.

To find a common denominator:

• Identify the denominators of the fractions you want to work with.
• Find a shared multiple, often by finding their least common denominator (LCD). The LCD is the smallest number that all denominators divide into without a remainder.

Once fractions share a common denominator, they can be added or subtracted by operating on their numerators while keeping the denominator the same.

Least Common Denominator

The "least common denominator" (LCD) is a specific kind of common denominator that is the smallest multiple shared by all denominators in a set of fractions. It's especially useful for simplifying algebraic fractions by making calculations easier and quicker.

How to find the LCD:

• List the multiples of each denominator.
• Identify the smallest multiple that appears in all lists.

Using the LCD means you won't have large or unwieldy numbers to deal with in calculations. This is particularly helpful in problems like the one given in the original exercise, where aligning fractions under a common denominator enables smooth combination of like terms.

Algebraic Fractions

Algebraic fractions are fractions that contain algebraic expressions in either the numerator, the denominator, or both. Simplifying these fractions can involve combining like terms, factoring, and finding common denominators.

To simplify algebraic fractions:

• Ensure all fractions share a common denominator for addition or subtraction.
• Combine like terms by adding or subtracting numerators.
• Factor expressions when possible, which can sometimes further simplify a problem.

Understanding and working with algebraic fractions is foundational for solving more complex algebraic equations and helps in developing strong skills in algebraic manipulation. This is why exercises involving these fractions often emphasize finding the least common denominator and combining like terms efficiently.