Problem 5 - Prealgebra

Question

Find each of the following products. (Multiply.) $$-\frac{5}{3} \cdot \frac{3}{5}$$

Step-by-Step Solution

Answer

The product is \(-1\).

Step 1: Identify the Numerators and Denominators

The fractions given are \(-\frac{5}{3}\) and \(\frac{3}{5}\). Identify that \(-5\) and \(3\) are the numerators, and \(3\) and \(5\) are the denominators of the first and second fractions, respectively.

Step 2: Multiply the Numerators

Multiply the numerators of the fractions together: \(-5 \times 3 = -15\).

Step 3: Multiply the Denominators

Multiply the denominators of the fractions together: \(3 \times 5 = 15\).

Step 4: Form the New Fraction

Combine the products of the numerators and denominators to form a new fraction: \(-\frac{15}{15}\).

Step 5: Simplify the Fraction

Simplify the fraction \(-\frac{15}{15}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 15. This simplifies to \(-1\).

Key Concepts

Numerator and Denominator

When dealing with fractions, understanding the terms *numerator* and *denominator* is crucial. These terms define the fundamental parts of a fraction.- **Numerator**: This is the top number in a fraction. It represents the number of parts you have. For instance, in the fraction \(-\frac{5}{3}\), \(-5\) is the numerator. This means you have five units, but notice the negative sign, which will be addressed in the negative fractions section.- **Denominator**: This is the bottom number in a fraction. It tells you into how many equal parts the whole is divided. In the same fraction \(-\frac{5}{3}\), the denominator is \(3\). This indicates that the whole is divided into three equal parts.For the fraction \(\frac{3}{5}\), the numerator is \(3\) and the denominator is \(5\). Understanding these terms helps when you move on to operations like addition, subtraction, or multiplication of fractions.

Simplifying Fractions

Simplifying a fraction means reducing it to its simplest form, where the numerator and denominator are as small as possible. This involves dividing both the top and bottom numbers by the greatest common divisor (GCD).Let's look at the fraction \(-\frac{15}{15}\). Here:- Both the numerator and denominator are divisible by \(15\).To simplify:1. Divide both \(-15\) and \(15\) by the GCD, which is \(15\).2. The result is \(-1\), because \(-15 \div 15 = -1\).Simplifying fractions make them easier to understand and use in calculations. It is a useful step in mathematical problem-solving.

Negative Fractions

Negative fractions can initially be confusing, but they're quite manageable once you understand the basics. Essentially, a fraction is negative if its numerator or its denominator is negative, but not both.- In our exercise, the fraction \(-\frac{5}{3}\) is negative because the numerator, \(-5\), is negative. This indicates that the overall value of the fraction is less than zero.When you multiply two fractions and one of them is negative, the resulting fraction will be negative. Here's the specific rule:- *Positive \(\times\) Negative = Negative*- Thus, \(-5 \times 3 = -15\) confirms that our resulting fraction \(-\frac{15}{15}\) is indeed negative.Understanding how negative fractions operate ensures you handle calculations accurately, keeping track of whether results should be positive or negative.