Problem 37 - Elementary Algebra

Question

Add or subtract as indicated and express your answers in simplest form. (Objective 3) $$\frac{7 n}{12}-\frac{4 n}{3}$$

Step-by-Step Solution

Answer

The simplest form of the expression is \( \frac{-3n}{4} \).

Step 1: Identify the denominators

The denominators in the fractions \( \frac{7n}{12} \) and \( \frac{4n}{3} \) are 12 and 3, respectively. To perform subtraction, we need a common denominator.

Step 2: Find the least common denominator (LCD)

The least common multiple of 12 and 3 is 12. Thus, 12 is our least common denominator.

Step 3: Rewrite fractions with the LCD

The fraction \( \frac{7n}{12} \) already has the common denominator. For \( \frac{4n}{3} \), convert it by multiplying both the numerator and denominator by 4 to get \( \frac{16n}{12} \).

Step 4: Subtract the fractions

Now, subtract \( \frac{16n}{12} \) from \( \frac{7n}{12} \) as follows: \( \frac{7n}{12} - \frac{16n}{12} = \frac{7n - 16n}{12} = \frac{-9n}{12} \).

Step 5: Simplify the result

The fraction \( \frac{-9n}{12} \) can be simplified by dividing the numerator and the denominator by their greatest common divisor (GCD), which is 3. This gives us \( \frac{-3n}{4} \).

Key Concepts

Least Common Denominator

When working with fractions, especially when adding or subtracting them, it's important to find the least common denominator (LCD). The LCD is the smallest number that is a multiple of each of the denominators involved in the fractions. By using the LCD, we ensure that we are comparing or combining parts of fractions that are of the same size.
This simplifies the process of combining the fractions because it transforms them into equivalent fractions that have denominators in common. For example, if you have fractions like \( \frac{1}{3} \) and \( \frac{1}{4} \), their denominators are 3 and 4. The least common multiple of these numbers, and thus their least common denominator, is 12. Converting each fraction into having a denominator of 12 allows straightforward comparisons and operations to be conducted.

Subtracting Fractions

Subtracting fractions involves a few clear steps once you have found a common denominator. Here's how it works:

• Ensure that each fraction has the same denominator. This ensures that you are working with like terms and is necessary for accurately performing the subtraction.
• Subtract the numerators, which represent the parts of each fraction, while keeping the common denominator unchanged.
• Write the result as a single fraction. Be sure to check accuracy by ensuring that the numerator reflects the correct result of the subtraction.

In terms of our example with \( \frac{7n}{12} \) and \( \frac{4n}{3} \), we first convert \( \frac{4n}{3} \) to \( \frac{16n}{12} \) so each fraction has a common denominator (12). Then, we subtract: \( \frac{7n}{12} - \frac{16n}{12} = \frac{7n - 16n}{12} = \frac{-9n}{12} \). Always ensure you have correctly aligned the fractions before subtracting.

Simplifying Fractions

Simplifying fractions is all about making them easier to read and understand by reducing them to their simplest form. This involves finding the greatest common divisor (GCD) of both the numerator and the denominator.
To simplify a fraction:\
1. Determine the GCD of the numerator and the denominator.
2. Divide both the numerator and the denominator by this GCD.
This results in a fraction whose numerator and denominator have no common factors (other than 1), thus representing the same value as the original fraction but in its simplest form.
In the example, we found the fraction \( \frac{-9n}{12} \), and the GCD of 9 and 12 is 3. Dividing both by the GCD, we simplify this to \( \frac{-3n}{4} \). Simplification is crucial as it provides the cleanest representation of the solution.