Problem 31 - Introductory and Intermediate Algebra for College Students 4th

Question

Perform the indicated subtraction. $$-\frac{4}{5}-\frac{1}{5}$$

Step-by-Step Solution

Answer

The result of the subtraction \(-\frac{4}{5} - (-\frac{1}{5})\) is \(-\frac{3}{5}\).

Step 1: Identify the fractions

The fractions to be subtracted are \(-\frac{4}{5}\) (minuend) and \(-\frac{1}{5}\) (subtrahend).

Step 2: Perform the subtraction

Subtract the subtracted numerator from the minuend's numerator while keeping the same denominator. Therefore, \(-\frac{4}{5} - (-\frac{1}{5}) = -\frac{4}{5} + \frac{1}{5} = -\frac{3}{5}\).

Step 3: Simplify if possible

In this case, no further simplification is possible after subtraction. The fraction \(-\frac{3}{5}\) is the simplest form.

Key Concepts

Negative Numbers

Negative numbers are values less than zero. They are used to represent a deficit or a loss. In the context of subtraction, especially with fractions, negative numbers can initially appear complicated, but they follow the same fundamental properties as positive numbers.
When subtracting negative numbers, remember that subtracting a negative is the same as adding the absolute value. For example:

• Subtracting \(-\frac{1}{5}\) is equivalent to adding \(+\frac{1}{5}\).
• This is because two negatives make a positive (like a double flip).

Using this rule makes calculations straightforward, even when dealing with multiple negative numbers.

Fractions

Fractions consist of a numerator (top number) and a denominator (bottom number). They represent parts of a whole. For example, in the fraction \(-\frac{4}{5}\), 4 is the numerator and 5 is the denominator.
When working with fractions, it's essential to ensure they have a common denominator before performing operations like subtraction.
Some key ideas about fractions include:

• The numerator indicates how many parts we have or are considering.
• The denominator shows how many parts make up a whole.
• Fractions such as \(-\frac{4}{5}\) are negative because the negative sign is in the numerator, implying that four out of five parts are less than zero.

This understanding helps you perform operations more accurately.

Simplification

Simplification in math means reducing an expression or a fraction to its simplest form. This often involves making the fraction as concise as possible without changing its value. In the solution \(-\frac{3}{5}\), simplification is crucial to obtain the final answer in the most understandable way.
Here's how simplification works:

• Look for common factors in the numerator and the denominator that can be divided out.
• If there are no further common factors besides 1, the fraction is already in its simplest form.
• Fractions such as \(-\frac{3}{5}\) cannot be simplified further as 3 and 5 share no common factors other than 1.

Through simplification, you can ensure that mathematical expressions are as straightforward and as unambiguous as possible.