Problem 315 - Intermediate Algebra

Question

In the following exercises, simplify. $$ \frac{3}{8} g+\frac{1}{12} h+\frac{7}{8} g+\frac{5}{12} h $$

Step-by-Step Solution

Answer

\(\frac{5}{4}g + \frac{1}{2}h\).

Step 1: Combine Like Terms

Identify and group the like terms in the expression. Here, \(\frac{3}{8}g\) and \(\frac{7}{8}g\) are like terms. Similarly, \(\frac{1}{12}h\) and \(\frac{5}{12}h\) are like terms.

Step 2: Add the Coefficients of Like Terms

Add the coefficients of the like terms. For \(\frac{3}{8}g + \frac{7}{8}g\), add \(\frac{3}{8} + \frac{7}{8} = \frac{10}{8} = \frac{5}{4}\). For \(\frac{1}{12}h + \frac{5}{12}h\), add \(\frac{1}{12} + \frac{5}{12} = \frac{6}{12} = \frac{1}{2}\).

Step 3: Rewrite the Simplified Expression

Combine the results from Step 2 to rewrite the simplified expression: \(\frac{5}{4}g + \frac{1}{2}h\).

Key Concepts

Like Terms

When dealing with algebraic expressions, identifying 'like terms' is crucial. Like terms are terms that have the same variables raised to the same power. For example, in the exercise given, \(\frac{3}{8}g\) and \(\frac{7}{8}g\) are like terms because they both contain the variable \(g\) raised to the same power (which is 1 here). Similarly, \(\frac{1}{12}h\) and \(\frac{5}{12}h\) are also like terms because both contain the variable \(h\).
By focusing on like terms, we can simplify expressions more easily by combining them, which we will discuss next.

Combining Coefficients

Once we have identified like terms, the next step is combining their coefficients. Coefficients are the numerical factors in terms; in \(\frac{3}{8}g\), the coefficient is \( \frac{3}{8} \). To combine coefficients of like terms, simply add them together.

For instance, in the expression \(\frac{3}{8}g + \frac{7}{8}g\):

• Add the coefficients: \( \frac{3}{8} + \frac{7}{8} = \frac{10}{8} = \frac{5}{4}\).
• So, the combined term becomes \( \frac{5}{4}g \).

Similarly, for the terms with \(h\): \

• Add the coefficients: \( \frac{1}{12} + \frac{5}{12} = \frac{6}{12} = \frac{1}{2} \).
• The simplified term is \( \frac{1}{2}h \).

Now, we can rewrite the simplified expression more clearly.

Fraction Addition

Adding fractions is a crucial skill when working with coefficients in algebraic expressions. To add fractions, follow these steps:

• Ensure that the fractions have a common denominator.
• If they do not, find the least common denominator (LCD).
• Convert each fraction to an equivalent fraction with the LCD.
• Finally, add the numerators while keeping the denominator the same.

In our exercise, the fractions were \( \frac{3}{8} \) and \( \frac{7}{8} \) for \(g\), which already have a common denominator of 8. Adding them is straightforward:
\( \frac{3}{8} + \frac{7}{8} = \frac{10}{8} \), which simplifies to \( \frac{5}{4} \).

Similarly, for the terms with \(h\), \( \frac{1}{12} \) and \( \frac{5}{12} \) also have a common denominator of 12. Adding them gives:
\( \frac{1}{12} + \frac{5}{12} = \frac{6}{12} = \frac{1}{2} \).
Combining these results helps to simplify the entire algebraic expression efficiently.