Problem 13 - Beginning and Intermediate Algebra

Question

Multiply. $$ -\frac{1}{2}\left(-\frac{3}{5}\right) $$

Step-by-Step Solution

Answer

The product is \(\frac{3}{10}\).

Step 1: Identify the multiplication of fractions

The expression we need to solve is \(-\frac{1}{2}\times -\frac{3}{5}\). Here, we are asked to multiply two fractions: \(-\frac{1}{2}\) and \(-\frac{3}{5}\).

Step 2: Multiply numerators and denominators

To multiply fractions, multiply the numerators together and the denominators together. Here, multiply the numerators: \(-1\) and \(-3\), which gives us: \[-1 \times -3 = 3\]. Next, multiply the denominators: \[2\times 5 = 10\].

Step 3: Combine the results into a fraction

Now that we have the products of the numerators and denominators, combine them to form a single fraction: \(\frac{3}{10}\).

Step 4: Simplify the fraction

The fraction \(\frac{3}{10}\) is already in its simplest form because 3 and 10 have no common factors other than 1.

Key Concepts

Fractions

Fractions are a fundamental concept in mathematics, representing parts of a whole. A fraction is composed of two numbers, one placed above the other, separated by a line. This line is known as the fraction bar. The number on top is the "numerator," and the one on the bottom is the "denominator." Fractions can be used to describe things like one-half of a chocolate bar or three-fifths of an hour.
Fractions are helpful because they allow us to express numbers that are not whole, enabling us to work with divisions and parts that do not result in whole numbers.
Understanding how to manipulate fractions, such as multiplying them, is an essential skill. When we multiply fractions, we essentially find what fraction of a fraction is represented, leading us to concepts such as ratios and proportions.

Numerator

The numerator is the top part of a fraction. It tells us how many parts of the entire thing we are dealing with. For example, in the fraction \(\frac{3}{5}\), the number 3 is the numerator.
When multiplying fractions, we multiply the numerators together to find the numerator of the resulting fraction. This means that when we multiplied \(-\frac{1}{2}\) by \(-\frac{3}{5}\), we multiplied \(-1\) by \(-3\) to get the numerator \(3\).
The numerator is like the "counter" of our operation, showing how many parts we have from the division of our whole.

Denominator

The denominator is the bottom part of a fraction. It indicates the number of equal parts that make up the whole. Continuing with our example, in \(\frac{3}{5}\), the 5 serves as the denominator, showing that the whole is divided into five parts.
When multiplying fractions, the rule is to multiply their denominators together to get the denominator of the resulting product. For example, in our multiplication process of \(-\frac{1}{2}\) and \(-\frac{3}{5}\), we take the denominators \(2\) and \(5\) to get the denominator \(10\).
Denominators are crucial because they tell the size of each part, helping us understand how big or small each piece of the whole is.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form where the numerator and denominator have no common factors other than 1. This process ensures that the fraction is presented in the most concise manner.
After multiplying fractions, it's essential to check if the resulting fraction can be simplified. In our example, after multiplying, we got \(\frac{3}{10}\). Both 3 and 10 have no common factors besides 1, indicating that \(\frac{3}{10}\) is already simplified.
Simplifying makes fractions easier to understand and compare. It involves dividing the numerator and the denominator by their greatest common divisor (GCD) until no further reduction is possible. Thus, always consider whether a fraction can be simplified to ensure clarity and simplicity.