Problem 83 - Beginning and Intermediate Algebra

Question

Divide. $$ -\frac{4}{9} \div \frac{4}{9} $$

Step-by-Step Solution

Answer

The result of the division is -1.

Step 1: Identify the Division Operation

The problem requires us to divide \( -\frac{4}{9} \) by \( \frac{4}{9} \). In mathematics, dividing by a fraction is equivalent to multiplying by its reciprocal.

Step 2: Find the Reciprocal

The reciprocal of a fraction \( \frac{a}{b} \) is \( \frac{b}{a} \). Therefore, the reciprocal of \( \frac{4}{9} \) is \( \frac{9}{4} \).

Step 3: Multiply by the Reciprocal

Rewrite the division as a multiplication: \(-\frac{4}{9} \times \frac{9}{4} \).

Step 4: Multiply the Fractions

To multiply two fractions, multiply the numerators and multiply the denominators: \(-\frac{4 \times 9}{9 \times 4} = -\frac{36}{36}\).

Step 5: Simplify the Fraction

Simplify the fraction \(-\frac{36}{36}\). Since the numerator and denominator are the same, the fraction is equal to -1.

Key Concepts

Reciprocal of a Fraction

Understanding the reciprocal of a fraction is crucial, especially when dividing fractions. In simple terms, the reciprocal of any fraction flips the numerator and the denominator. Imagine you have a fraction \( \frac{a}{b} \). The reciprocal of this fraction would be \( \frac{b}{a} \). This concept is essential because dividing by a fraction is the same as multiplying by its reciprocal. Reciprocals help transform division problems into multiplication ones, making calculations more straightforward. Remember, every non-zero fraction has a reciprocal, even negative fractions! For example, the reciprocal of \(-\frac{4}{9}\) is \(\frac{9}{4}\). This concept shows that all fractions, whether positive or negative, can be divided by flipping them to make multiplication easier.

Multiplying Fractions

Once you understand how to find the reciprocal, multiplying fractions becomes pretty easy. When multiplying fractions, you perform this operation across the numerators and the denominators. Let's take two fractions \( \frac{a}{b} \) and \( \frac{c}{d} \). To multiply them, multiply the numerators \( a \times c \), and do the same with the denominators \( b \times d \).
This results in a new fraction. For example, multiplying \( -\frac{4}{9} \times \frac{9}{4} \) is straightforward. Multiply the numerators: \( -4 \times 9 = -36 \), and the denominators: \( 9 \times 4 = 36 \).
Thus, the product is \( -\frac{36}{36} \). Note how keeping track of negative signs is essential throughout the multiplication process. This simple step converts our problem from complex division to a more manageable multiplication.

Simplifying Fractions

Simplifying fractions is the process of making them as simple as possible. A fraction is considered simplified when the numerator and the denominator share no common factors other than 1.
To simplify, you can divide both the top and bottom numbers by any common factor. In our example, \( -\frac{36}{36} \), the numerator and the denominator are identical. So, dividing both by 36 gives \(-1\).
In general, a fraction \( \frac{n}{n} \) where both the numerator and denominator are the same, simplifies to 1, or -1 if there is a negative sign involved. Simplifying makes fractions more understandable and easier to work with in further calculations, emphasizing clarity and neatness in mathematical expressions.