Problem 68 - Elementary Algebra

Question

For exercises \(23-74\), evaluate. $$ -\frac{1}{10}-\frac{1}{10} $$

Step-by-Step Solution

Answer

-\(\frac{1}{5}\)

Step 1: Identify the fractions

The fractions to be added are \(-\frac{1}{10}\) and \(-\frac{1}{10}\).

Step 2: Add the fractions

Since the denominators are the same, add the numerators: \(-1 - 1 = -2\).

Step 3: Simplify the fraction

The numerator is -2 and the denominator remains 10, resulting in \(-\frac{2}{10}\).

Step 4: Reduce the fraction

Simplify \(-\frac{2}{10}\): both the numerator and denominator can be divided by 2. The simplified fraction is \(-\frac{1}{5}\).

Key Concepts

negative numbers

In mathematics, negative numbers are important because they represent values less than zero. We use them to describe points on a number line, temperatures below freezing, debts, and much more.

When working with fractions that have negative signs, it's crucial to understand how the negative sign affects the operation. For instance, consider the fractions \(-\frac{1}{10}\) and \(-\frac{1}{10}\). The negative signs indicate that both fractions are less than zero.

Here’s a simple rule: the sum of two negative numbers will always be a negative number. When you add \(-\frac{1}{10} - \frac{1}{10}\), you must add their absolute values first, then apply the negative sign to the result. So, \-\frac{1}{10} - \frac{1}{10} = -\left(\frac{1}{10} + \frac{1}{10}\right) = -\frac{2}{10}\. Understanding this principle makes the operation straightforward.

fraction simplification

Fraction simplification is the process of making a fraction as simple as possible. It's often called reducing the fraction. In this process, you divide both the numerator and the denominator by their greatest common divisor (GCD).

Let's take our fraction from the problem: \(-\frac{2}{10}\). To simplify, we find the GCD of 2 and 10, which is 2. Divide both the numerator and the denominator by this value:

• Numerator: \-2 \div 2 = -1\
• Denominator: \10 \div 2 = 5\

The simplified fraction is \(-\frac{1}{5}\). Simplifying fractions not only makes them easier to work with, but it’s also often required for final answers in math problems.

common denominators

Adding and subtracting fractions requires you to have common denominators. The denominator is the bottom part of a fraction, and it signifies how many equal parts the whole is divided into. For instance, in the problem we've been working on \(-\frac{1}{10} - \frac{1}{10}\), the fractions already have a common denominator of 10.

If fractions have different denominators, you need to find a common denominator before performing the addition or subtraction. The lowest common denominator (LCD) is the smallest number that both denominators can divide into evenly.

Here’s an example with different denominators:

• Fractions: \frac{1}{4}\ and \frac{1}{6}\
• Find LCD: 12 (since 4 and 6 both divide evenly into 12)
• Convert fractions to have the same denominator: \frac{1}{4} = \frac{3}{12}\ and \frac{1}{6} = \frac{2}{12}\

Now, you can add these fractions: \frac{3}{12} + \frac{2}{12} = \frac{5}{12}\. Understanding how to find and use common denominators is fundamental to working with fractions.