Problem 39 - Beginning and Intermediate Algebra

Question

Add See Examples \(\ell\) through 7 . $$ -\frac{7}{10}+\left(-\frac{3}{5}\right) $$

Step-by-Step Solution

Answer

The solution is \\( -\frac{13}{10} \\\) or \\( -1.3 \\\).

Step 1: Prepare the Fractions

Both fractions have different denominators, so the first step is to find a common denominator. The denominators are 10 and 5.

Step 2: Find the Least Common Denominator

The least common denominator (LCD) of 10 and 5 is 10.

Step 3: Convert the Fractions

Convert \( -\frac{3}{5} \) to a fraction with a denominator of 10. Multiply both the numerator and the denominator by 2: \( -\frac{3 \times 2}{5 \times 2} = -\frac{6}{10} \).

Step 4: Add the Fractions

Now that both fractions have a common denominator, add them: \( -\frac{7}{10} + (-\frac{6}{10}) = -\frac{7 + 6}{10} = -\frac{13}{10} \).

Step 5: Simplify the Result

Since \( -\frac{13}{10} \) is already in simplest form, it can be left as \( -1.3 \) or expressed as the improper fraction \( -\frac{13}{10} \).

Key Concepts

Understanding the Least Common Denominator

When you're adding fractions, having a common denominator simplifies the process. The least common denominator (LCD) is the smallest shared multiple of the denominators involved. Here’s why finding it is essential:

• First, the LCD enables a direct comparison between fractions by allowing you to line them up under one denominator.
• Second, it helps avoid errors in calculation, as all operations then occur smoothly on the numerators.

To find the LCD, you'll want to compare the denominators of the fractions. In the case of the exercise, we had 10 and 5 as denominators. The smallest number that both 10 and 5 divide into evenly is 10, thus making it our least common denominator.
After establishing the LCD, adjust each fraction to have this as its new denominator. This helps in straightforward arithmetic operations, such as addition or subtraction. Remember, multiplying the numerator and the denominator by the same number does not change the value of the fraction.

Exploring Improper Fractions

An improper fraction is simply a fraction where the numerator is larger than the denominator. This type of fraction expresses a value greater than one whole. Let's explore what to do with improper fractions:

• Improper fractions can be practical for computation, especially when adding or subtracting fractions.
• They can sometimes be rewritten as mixed numbers to interpret the solution more intuitively.
• Consider our solution of \(-\frac{13}{10}\), which is indeed an improper fraction, indicating a mixed number of \(-1 \frac{3}{10}\).

It's key to recognize the practical usability of improper fractions in your mathematical calculations. Though they represent quantities greater than one, they are equally valid as their mixed number counterparts.

Mastering Simplifying Fractions

Simplifying fractions is about reducing them to their most basic form without changing their value. A fraction is considered simplified when its numerator and denominator have no common factor other than 1. Here’s how you can simplify like a pro:

• First, identify any common factors shared between the numerator and the denominator.
• Then divide both parts of the fraction by this Greatest Common Divisor (GCD).
• Continue the process until no further simplification is possible.

In the context of our exercise, \(-\frac{13}{10}\) is already simplified. This means the fraction’s numerator and denominator share no factors other than 1, hence nothing more can be reduced. By recognizing when a fraction is in its simplest form, you make calculations easier and clearer.