Problem 88 - Elementary Algebra

Question

Simplify each algebraic expression by combinang similar terms. $$\frac{2}{5} n-\frac{7}{10} n+\frac{8}{15} n$$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{7}{30}n\).

Step 1: Identify the Common Variable

We have three terms: \( \frac{2}{5}n\), \(-\frac{7}{10}n\), and \(\frac{8}{15}n\). Notice that all terms involve the variable \(n\). Our goal is to combine these terms by finding a common denominator.

Step 2: Find the Least Common Denominator (LCD)

The denominators of the fractions are 5, 10, and 15. To add or subtract fractions, we need to convert these to a common denominator. The least common multiple of these denominators is 30.

Step 3: Convert Each Term to Have a Common Denominator

Convert each fraction so they all have the denominator of 30:- \(\frac{2}{5}n = \frac{(2 \times 6)}{(5 \times 6)}n = \frac{12}{30}n\)- \(-\frac{7}{10}n = \frac{(-7 \times 3)}{(10 \times 3)}n = -\frac{21}{30}n\)- \(\frac{8}{15}n = \frac{(8 \times 2)}{(15 \times 2)}n = \frac{16}{30}n\)

Step 4: Combine Like Terms

Now that all terms have a common denominator of 30, combine them:\(\frac{12}{30}n + (-\frac{21}{30}n) + \frac{16}{30}n = \frac{12 - 21 + 16}{30}n = \frac{7}{30}n\).

Step 5: Simplify the Expression (if needed)

The resulting fraction, \(\frac{7}{30}\), is already in its simplest form, so there is no need for further simplification.

Key Concepts

Combining Like Terms

When dealing with algebraic expressions, the concept of combining like terms is fundamental. Like terms are terms that share the same variable and exponent. Here, all terms in the exercise contain the variable \( n \), making them like terms. To simplify an expression:

• Identify terms with the same variables and exponents.
• Combine them by adding or subtracting their coefficients.

In our problem, the terms \( \frac{2}{5}n \), \(-\frac{7}{10}n \), and \( \frac{8}{15}n \) can be combined because they all involve the same variable, \( n \), even though they initially have different denominators. This leads to the necessity of finding a common denominator before combining.

Least Common Denominator

Finding the least common denominator (LCD) is crucial when adding or subtracting fractions with unlike denominators. The LCD is the smallest number that can be evenly divided by each of the fraction's denominators.

To find the LCD for \( \frac{2}{5} \), \( \frac{7}{10} \), and \( \frac{8}{15} \):

• List the multiples of each denominator: 5, 10, and 15.
• Select the smallest multiple common to all: 30.

Using the LCD, you can convert each fraction so they all share a common denominator, setting the stage for combining the terms as shown in the provided solution.

Fraction Addition

Adding fractions, especially in algebraic expressions, demands a shared denominator. This ensures that the fractions are parts of a consistent whole. To add fractions like \( \frac{2}{5}n \), \(-\frac{7}{10}n \), and \( \frac{8}{15}n \):

• Convert each term to have the common denominator found before: 30.
• Rewrite fractions: \( \frac{2}{5}n = \frac{12}{30}n \), \(-\frac{7}{10}n = -\frac{21}{30}n \), \( \frac{8}{15}n = \frac{16}{30}n \).
• Combine the terms: \( \frac{12}{30}n + (-\frac{21}{30}n) + \frac{16}{30}n \).

Be sure to add or subtract the numerators, keeping the denominator constant, to simplify the expression. In the example, we achieve the simplified form \( \frac{7}{30}n \).