Problem 74 - Prealgebra

Question

Apply the distributive property, then find the LCD and simplify. $$\frac{3 x}{4}-\frac{2 x}{3}$$

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{x}{12} \).

Step 1: Distribute the Terms

The given expression consists of two fractions: \( \frac{3x}{4} \) and \( \frac{2x}{3} \). There is no immediate need to distribute any factors since each part is already distributed.

Step 2: Identify the Least Common Denominator (LCD)

To combine the fractions, we need a common denominator. The denominators are 4 and 3. Find the least common multiple of these denominators. The least common multiple (LCM) of 4 and 3 is 12, so the least common denominator (LCD) is 12.

Step 3: Adjust Fractions to Have the LCD

Convert each fraction to have the same denominator, 12.- For \( \frac{3x}{4} \), multiply the numerator and the denominator by 3: \[ \frac{3x}{4} \times \frac{3}{3} = \frac{9x}{12} \]- For \( \frac{2x}{3} \), multiply the numerator and the denominator by 4:\[ \frac{2x}{3} \times \frac{4}{4} = \frac{8x}{12} \]

Step 4: Subtract the Fractions

Now that both fractions have the same denominator, subtract the numerators:\[ \frac{9x}{12} - \frac{8x}{12} = \frac{9x - 8x}{12} \]

Step 5: Simplify the Result

Simplify the expression obtained after the subtraction:\[ \frac{9x - 8x}{12} = \frac{x}{12} \]

Step 6: Final Step: Review

The original expression \( \frac{3 x}{4}-\frac{2 x}{3} \) simplifies to \( \frac{x}{12} \).

Key Concepts

Distributive Property

The distributive property is a fundamental principle in algebra that involves multiplying a single term across a sum or difference. It allows us to simplify expressions and solve equations more easily. In this specific exercise, the distributive property was not explicitly used because each term already stands alone with its own fraction. However, understanding this property is crucial in more complex problems where you might encounter expressions like \( a(b + c) \). Here, you would distribute \( a \) across both \( b \) and \( c \), resulting in \( ab + ac \). The distributive property helps us re-structure expressions in a way that makes them more manageable. This exercise sets the foundation to recognize when or when not to apply it, ensuring that you only use it when truly necessary.

Least Common Multiple

The least common multiple (LCM) is essential when dealing with fractions having different denominators. In order to perform operations like addition or subtraction on fractions, they must share a common denominator. Finding the LCM:

• **List the multiples**: For each denominator, identify their multiples. For example, for 4, you have 4, 8, 12, 16, etc. For 3, the multiples are 3, 6, 9, 12, etc.
• **Find the smallest common multiple**: Look for the smallest number that appears in both lists. In this exercise, the smallest common multiple of 4 and 3 is 12.

Once you determine the LCM, you can adjust each fraction to have this shared denominator, making addition or subtraction possible. Applying these steps ensures that you can handle even more complex fractional equations with ease.

Simplifying Fractions

Simplifying fractions is all about making a fraction as simple as possible. It involves reducing the fraction to its simplest form. This concept is pivotal after combining fractions, like in this exercise.To simplify a fraction:

• **Look for common factors**: Identify any common factors in the numerator and the denominator.
• **Divide by the largest common factor**: Divide both parts of the fraction by their greatest common factor. In the solution, \( \frac{x}{12} \) is already in its simplest form since there are no common factors between \( x \) and 12.

Simplifying helps in making calculation results cleaner and more understandable, eliminating any unnecessary complexity. It is a basic yet powerful tool in algebra that ensures effective problem-solving and clear communication of your results.