Problem 70 - Elementary Algebra

Question

For exercises \(23-74\), evaluate. $$ \frac{3}{14}-\left(-\frac{9}{14}\right) $$

Step-by-Step Solution

Answer

The value is \(\frac{6}{7}\).

Step 1 - Understand the Problem

The goal is to evaluate the expression \(\frac{3}{14} - \big(-\frac{9}{14}\big) \). This is a subtraction problem involving two fractions.

Step 2 - Rewrite the Expression

Notice the double negative in the expression \(\frac{3}{14} - \big(-\frac{9}{14}\big)\). A double negative becomes positive, so rewrite the expression as \(\frac{3}{14} + \frac{9}{14}\).

Step 3 - Add the Fractions

Since both fractions have the same denominator, add the numerators. \[ \frac{3}{14} + \frac{9}{14} = \frac{3 + 9}{14} \]

Step 4 - Simplify the Numerator

Add the numerators: \[ \frac{3 + 9}{14} = \frac{12}{14} \]

Step 5 - Simplify the Fraction

Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 2: \[ \frac{12 \/ 2}{14 \/ 2} = \frac{6}{7} \]

Key Concepts

adding fractions

Adding fractions might seem tricky at first, but it becomes simple once you understand the steps. When fractions have the same denominator, you can add the numerators directly. For example, let's look at the expression \(\frac{3}{14} + \frac{9}{14}\). Since both fractions have the same denominator, you simply need to add the numerators (3 and 9) together: \(\frac{3+9}{14}\) which equals \(\frac{12}{14}\).

negative numbers in fractions

Using negative numbers can be confusing in fraction operations, particularly when you deal with double negatives. For instance, consider the expression \(\frac{3}{14} - \big(-\frac{9}{14}\big)\). The minus and negative signs together turn the operation into addition because negative times negative equals positive. Thus, \(\frac{3}{14} - \big(-\frac{9}{14}\big)\) converts to \(\frac{3}{14} + \frac{9}{14}\). Understanding this simple rule helps in evaluating similar expressions correctly.

simplifying fractions

Simplifying fractions means reducing them to their smallest form. After adding or subtracting fractions, you often have to simplify the result. Let's take our previous example, \(\frac{12}{14}\). To simplify, find the greatest common divisor (GCD) of 12 and 14, which is 2. Divide both the numerator and the denominator by this number: \(\frac{12}{2} \big/ \frac{14}{2} = \frac{6}{7}\). The fraction \(\frac{6}{7}\) cannot be simplified further, so it is in its simplest form. This step is crucial for getting the correct and finalized answer.