Problem 33 - Introductory and Intermediate Algebra for College Students 4th

Question

Perform the indicated subtraction. $$-\frac{4}{5}-\left(-\frac{1}{5}\right)$$

Step-by-Step Solution

Answer

-3/5

Step 1: Identify the fractions

The subtraction involves two fractions with the same denominators: -4/5 and -1/5.

Step 2: Apply the subtraction rule

A minus sign before a parenthesis in mathematics means to take the opposite or to multiply by -1. So -(-1/5) becomes 1/5.

Step 3: Perform the subtraction

Now, we perform the subtraction -4/5 + 1/5.

Step 4: Subtract fractions

To subtract fractions with like denominators, simply subtract the numerators: -4 + 1 which gives -3.

Step 5: Write the answer

Therefore, the answer is -3/5 since the denominator remains the same when subtracting fractions with like denominators.

Key Concepts

Negative Fractions

Negative fractions are simply fractions that have a negative sign either before the numerator or in front of the fraction itself. Understanding negative fractions is crucial as it follows similar rules to negative whole numbers. For example, \(-\frac{4}{5}\) is a negative fraction where \(-4\) is the numerator and 5 is the denominator.

When working with negative fractions, keep these points in mind:

• The negative sign indicates a value less than zero.
• Negative fractions occur when the numerator is negative, the denominator is negative, or the negative sign is placed in front of the fraction.
• The value of a negative fraction can be thought of as moving left on a number line, just as positive values would move right.

It's important to remember that when you see a negative sign outside of a parenthesis, such as in \(-(-\frac{1}{5})\), it means you should take the opposite value, or essentially just change the sign to positive.

Fraction Operations

Fraction operations involve the mathematical processes of addition, subtraction, multiplication, and division applied to fractions. These operations often follow unique rules due to the nature of fractions. For subtraction, there's a key rule:

When subtracting fractions, particularly important is:

• Always ensure both fractions have the same denominator, known as a common or like denominator.
• Apply basic arithmetic operations to only the numerators, leaving denominators unchanged in subtraction and addition.

In our example, you start with \(-\frac{4}{5}\) and need to subtract \(-\frac{1}{5}\). Here, subtracting a negative fraction is the same as adding the positive of that fraction. Thus, we turn \(-(-\frac{1}{5})\) into \(+\frac{1}{5}\) and proceed by adding the numerators, keeping the denominator the same.

Like Denominators

When performing operations such as subtraction on fractions, having like denominators simplifies the process considerably. Like denominators mean that the fractions have the same bottom number or denominator, which makes subtracting or adding them straightforward.

Here's why like denominators are essential:

• They allow for direct subtraction or addition of the numerators, avoiding the need for additional steps.
• The simplest way to manage fractions, ensuring calculations are more straightforward.

In our given problem, both fractions \(-\frac{4}{5}\) and \(-\frac{1}{5}\) share the common denominator of 5. This means we can directly subtract \(-4\) and \(+1\), resulting in \(-3\), while the denominator remains 5. The overall result is \(-\frac{3}{5}\). Consistently confirming like denominators is a vital step to ensure efficient and accurate fraction arithmetic.