Problem 35 - Beginning and Intermediate Algebra

Question

Add $$\left(\frac{3}{4} c+\frac{1}{8}\right)+\left(\frac{3}{2} c-\frac{5}{6}\right)$$

Step-by-Step Solution

Answer

The short answer is: $$\frac{9}{4}c - \frac{17}{24}$$

Step 1: Identify the Terms

In the given expression, there are two terms with the common variable 'c' and two constants. Identify these terms: Terms with 'c': \(\frac{3}{4}c\) and \(\frac{3}{2}c\) Constants: \(\frac{1}{8}\) and \(-\frac{5}{6}\)

Step 2: Adding Coefficients of 'c'

Now, add the coefficients of the common variable 'c': \(\frac{3}{4}c + \frac{3}{2}c\) To add fractions, we need a common denominator. In this case, the common denominator is 4. So, we can rewrite the expression as: \(\frac{3}{4}c + \frac{6}{4}c\) Now, add the fractions: \(\frac{3 + 6}{4}c = \frac{9}{4}c\)

Step 3: Adding Constants

Next, let's add the constants: \(\frac{1}{8} - \frac{5}{6}\) Find the common denominator for the constants, which is 24. Rewrite the fractions with the common denominator: \(\frac{3}{24} - \frac{20}{24}\) Now subtract the fractions: \(\frac{3 - 20}{24} = -\frac{17}{24}\)

Step 4: Combine Simplified Terms

Finally, combine the simplified terms: \(\frac{9}{4}c - \frac{17}{24}\) The simplified expression after adding is: $$\frac{9}{4}c - \frac{17}{24}$$

Key Concepts

Fractions

Fractions are a way to express parts of a whole. They consist of two numbers: a numerator, which sits above the line, and a denominator below it. When working with algebraic expressions, especially when adding them, understanding fractions becomes crucial.

In algebra, fractions often appear as parts of coefficients attached to variables or as constant values. For example, in our exercise, terms like \(\frac{3}{4}c\) and \(-\frac{5}{6}\) utilize fractions. Here, \(\frac{3}{4}\) and \(-\frac{5}{6}\) are fractions that need to be added by finding a common denominator.

• Numerator: The number above the fraction line.
• Denominator: The number below the fraction line, indicating how many parts a whole is divided into.

Properly handling fractions involves converting them to have the same denominator, then performing the addition or subtraction of the numerators. This concept is fundamental for correctly solving algebraic equations that involve fractions.

Coefficients

In algebra, coefficients are numbers or fractions that are multiplied by variables in an expression. Understanding coefficients is crucial for tasks like adding algebraic expressions since you often need to combine like terms.

A coefficient tells you how many times you are multiplying the variable it attaches to. Let’s use \(\frac{3}{4}c + \frac{3}{2}c\) as an example. Here, \(\frac{3}{4}\) and \(\frac{3}{2}\) act as coefficients of the variable \(c\).

• Combine Coefficients: To add the coefficients of like terms, they must first be aligned to a common denominator so they can be added as fractions.
• Resulting Coefficient: After adding, the resulting coefficient will still be multiplied by the variable.

Understanding what coefficients represent and how to manipulate them ensures that you manage variables accurately, leading to correct solutions.

Common Denominators

Finding a common denominator is a vital step when dealing with fractions, especially in algebra when adding or subtracting them. The common denominator is a shared multiple of the original denominators that allows you to equally compare and manipulate the fractions.

In our example, we needed to find a common denominator to add \(\frac{3}{4}c + \frac{3}{2}c\). The common denominator for 4 and 2 is 4, so we adjusted \(\frac{3}{2}c\) to \(\frac{6}{4}c\).

• Steps to Find Common Denominator: Identify the least common multiple (LCM) of the original denominators. Use this LCM to rewrite each fraction.
• Convert: Adjust each fraction so that the denominators match. Only then can you accurately add the numerators.

Using a common denominator allows you to work with fractions seamlessly, ensuring that every fraction is proportional to another, making addition or subtraction straightforward.

Constants

Constants in algebra are the numbers in expressions that do not multiply a variable; they remain unchanged as you work through equations.

In the solving process, you often need to add or subtract these constants. In our exercise, \(\frac{1}{8}\) and \(-\frac{5}{6}\) are constants that needed combining. Like terms involving coefficients, constants expressed as fractions require a common denominator for addition or subtraction.

• Step-by-Step Addition: Find the common denominator just like you would for any fraction to add constants correctly.
• Subtracting Constants: Treat subtraction similarly to addition but subtract the numerators once a common denominator is found.

Handling constants correctly ensures that the expressions you build during algebraic manipulation remain valid and correct throughout the process.