Problem 42 - Algebra 1

Question

Simplify. \left(-\frac{1}{2}\right)\left(\frac{2}{3}\right)

Step-by-Step Solution

Answer

The simplified expression is -2/6.

Step 1: Multiplication of Numerators

We start by multiplying the two numerators together. The equation becomes: -1 * 2, which is -2.

Step 2: Multiplication of Denominators

Next, we multiply the two denominators together. The equation becomes: 2 * 3, which is 6.

Step 3: Form the Product

Finally, combine the results from step 1 and step 2 to form the product. The final expression is -2/6

Key Concepts

Multiplication of Fractions

When multiplying fractions, it’s important to keep the process straightforward. You always deal with the numerators and denominators separately. This means you multiply the numerators of the fractions together and then the denominators together.
For instance, in multiplying \(-\frac{1}{2}\) and \(\frac{2}{3}\), start by

• Multiplying the numerators: \(-1 \times 2\), which results in \(-2\).
• Then, multiply the denominators: \(2 \times 3\), resulting in \(6\).

These two results will form the new fraction: \(\frac{-2}{6}\). This ensures that you maintain the integrity of the multiplication process, making sure both parts of the fraction are treated accurately.
Once you form the product fraction, remember to simplify further if possible to make it easier to interpret.

Numerator and Denominator

Understanding the roles of the numerator and denominator helps in simplifying and performing operations with fractions easily. In fractions:

• The numerator is the top part, representing how many parts of a whole you have.
• The denominator is the bottom part, showing how many equal parts the whole is divided into.

In our example, the first fraction, \(-\frac{1}{2}\), has:
Numerator: \(-1\), meaning it's one negative part out of two.
Denominator: \(2\), indicating the whole is divided into two parts.
The second fraction, \(\frac{2}{3}\), has:

• Numerator: \(2\), two parts of the divided whole.
• Denominator: \(3\), showing the whole is split into three equal parts.

This basic understanding allows you to properly perform operations like multiplication, addition, or simplification. A firm grasp of these terms simplifies solving fraction problems considerably.

Negative Fractions

Negative fractions might seem tricky at first, but they follow simple arithmetic rules. A negative sign in a fraction means that the value of the fraction is less than zero, indicating a direction on a number line rather than a value increase.
When dealing with negative fractions during multiplication, you simply follow these steps:

• If both fractions are negative or just one, the product depends on their signs.
• For example, \(-\frac{1}{2}\ \times\ \frac{2}{3}\), consider the negative sign affects only the numerator initially, \(-1\), while the denominator, \(2\), remains positive.

The result also carries over the negative sign, giving \(\frac{-2}{6}\). The rule is: multiplying two negatives results in a positive, while one negative gives a negative product.
Understanding these basics aids in predicting the outcome of multiplying, dividing, or simplifying negative fractions, making complex problems easier to tackle.