Problem 51 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

The simplified form of \(\left(-\frac{1}{2}\right)\left(\frac{2}{3}\right\) is \(-\frac{1}{3}\).

Step 1: Multiply the Numerators

First, multiply the numerators of the two fractions together. This means multiplying \(-1\) and \(2\), which results in \(-2\).

Step 2: Multiply the Denominators

Next, multiply the denominators of the two fractions together. This involves multiplying \(2\) and \(3\) which gives \(6\).

Step 3: Write down the answer

Your result is the fraction formed by the product of the numerators above the product of the denominators. So you have \(-2/6\).

Key Concepts

Numerators

When it comes to fraction multiplication, the role of the numerator is quite straightforward but pivotal.

• The numerator is the top number in a fraction.
• It tells us how many parts of a whole we have.
• For example, in the fraction \(-\frac{1}{2}\), \(-1\) is the numerator, indicating that we have \-1\ parts of a whole that has been equally divided into 2 parts.

During fraction multiplication, the numerators are multiplied directly. So, for the two fractions \(-\frac{1}{2}\) and \(\frac{2}{3}\), you multiply the numerators \(-1\) and \(2\). This multiplication gives us \(-2\), which will be the new numerator of our product.
Understanding numerators as the 'count of parts' is fundamental to managing fractions effectively.

Denominators

Next, let's talk about denominators. They play a critical role in understanding how many total parts the whole has been divided into.

• The denominator is located at the bottom of the fraction.
• It indicates the number of equal parts in the whole.
• In the fraction \(-\frac{1}{2}\), the denominator is \(2\), signifying that the whole is divided into 2 parts.

When multiplying fractions, you also multiply the denominators. So with our fractions, \(-\frac{1}{2}\) and \(\frac{2}{3}\), the denominators \(2\) and \(3\) are multiplied to result in \(6\).
This means our product will be spread over 6 parts of a new whole. Remember, the denominators offer insights into the scale on which we operate.

Simplification of Fractions

Simplifying fractions is like tidying up your math homework; it makes the results cleaner and easier to understand.

• To simplify a fraction, you are trying to find an equivalent fraction with the smallest possible numerator and denominator.
• This involves dividing both the numerator and the denominator by their greatest common divisor (GCD).

In our multiplication example, we ended up with \(-\frac{2}{6}\). Noticing that both 2 and 6 share a common factor of 2, we can simplify the fraction, dividing both numerator and denominator by 2, which gives us \(-\frac{1}{3}\).
Practicing simplification not only ensures that your answer is in its simplest form but also enhances understanding by reducing complexity.