Problem 34 - Introductory and Intermediate Algebra for College Students 4th

Question

Perform the indicated subtraction. $$-\frac{4}{9}-\left(-\frac{1}{9}\right)$$

Step-by-Step Solution

Answer

-1/3

Step 1: Simplify the expression

The expression given is \( -\frac{4}{9}-(-\frac{1}{9}) \). As the rule of operations, a minus followed by another minus makes a plus, thus the conversion of the expression will be \( -\frac{4}{9}+\frac{1}{9} \).

Step 2: Perform the operation

Now we are to perform the addition between \( -\frac{4}{9} \) and \( \frac{1}{9} \), which have the same denominator. The rule of fraction addition is that we directly add or subtract the numerators and place them over the common denominator, so we get \( -\frac{4+1}{9} = -\frac{3}{9} \).

Step 3: Simplify the result

The fraction we get, \( -\frac{3}{9} \) can be simplified. Both the numerator and the denominator are divisible by 3, so after simplifying we get \( -\frac{1}{3} \).

Key Concepts

Simplifying Fractions

When working with fractions, simplifying is a vital step to presenting your answer in the most basic form. Simplifying, in essence, involves reducing the numerator and denominator to their smallest possible values while keeping the fraction equivalent to the original. Here's how it's done:

To simplify a fraction, you need to find the greatest common divisor (GCD) of both the numerator and the denominator. The GCD is the largest number that divides both numbers without leaving a remainder. For instance, if we consider the fraction \( -\frac{3}{9} \), the GCD of 3 and 9 is 3. Dividing both the numerator and the denominator by 3, the fraction reduces to \( -\frac{1}{3} \), which is its simplest form.

Why Simplify Fractions?

It's not just about making fractions 'look nice.' Simplified fractions are easier to understand and work with, especially when performing further arithmetic operations. Keeping fractions in their simplest form can also help prevent confusion or errors in more complex mathematical processes.

Negative Fractions

Negative fractions are just like any other fractions, except that they represent quantities less than zero. The negative sign can be placed either before the fraction, numerator, or denominator, but the value remains unchanged. In mathematical operations, negative fractions follow the same rules as positive fractions.

Understanding how negative signs interact is crucial, particularly when adding or subtracting negative fractions. For example, if we subtract a negative fraction, such as in the expression \( -\frac{4}{9} - ( -\frac{1}{9} ) \), it is equivalent to adding its positive counterpart because subtracting a negative value is the same as adding a positive one. This principle is vital for correctly solving operations that involve negative fractions.

Arithmetic Operations with Fractions

Arithmetic operations with fractions, including addition, subtraction, multiplication, and division, are skills that students must master. When adding or subtracting fractions with the same denominator, you simply add or subtract the numerators and keep the common denominator. For the operation \( -\frac{4}{9} + \frac{1}{9} \), you combine the numerators to get \( -\frac{4+1}{9} \), which simplifies to \( -\frac{3}{9} \), and eventually to \( -\frac{1}{3} \) after simplification.

Homework Tips for Fractions:

• Always look for common denominators when adding or subtracting fractions.
• Remember the rules for negative numbers; two negatives make a positive.
• Simplify your fractions to their lowest terms to make further calculations or comparisons easier.

By keeping these tips in mind, handling fractions in arithmetic operations becomes more straightforward and less prone to mistakes.