Problem 26 - Prealgebra

Question

Find each product. Use an area model if necessary. $$\frac{3}{4}\left(-\frac{1}{3}\right)$$

Step-by-Step Solution

Answer

The product is \(-\frac{1}{4}\).

Step 1: Multiply the Numerators

To find the product of two fractions, multiply the numerators together. The numerators in this case are 3 and -1. Thus, multiply them: \[3 \times (-1) = -3\].

Step 2: Multiply the Denominators

Next, multiply the denominators together. The denominators in the fractions \(\frac{3}{4}\) and \(-\frac{1}{3}\) are 4 and 3, respectively. Multiply these:\[4 \times 3 = 12\].

Step 3: Form the Fraction from the Results

Now, take the results of the numerator and the denominator from the previous steps to form a new fraction:\[-\frac{3}{12}\].

Step 4: Simplify the Fraction

To simplify \(-\frac{3}{12}\), divide both the numerator and the denominator by their greatest common divisor, which is 3:\[-\frac{3 \div 3}{12 \div 3} = -\frac{1}{4}\].

Key Concepts

Multiplying Fractions

When multiplying fractions, remember that you operate directly with the numerators and denominators. To multiply two fractions:

• Multiply the numerators together to get the new numerator.
• Multiply the denominators together to get the new denominator.

For instance, to multiply \(\frac{3}{4}\) by \(-\frac{1}{3}\):

• Multiply the numerators: \(3 \times -1 = -3\).
• Multiply the denominators: \(4 \times 3 = 12\).

This process gives us the fraction \(-\frac{3}{12}\). Notice how we first deal with each part of the fraction separately, making our calculations manageable and systematic.

Area Model

The area model is a visual method to understand fraction multiplication. It represents fractions as parts of a shape, like a rectangle. Each fraction covers a part of this shape.
To use an area model for \(\frac{3}{4}\times -\frac{1}{3}\):

• Consider a rectangle divided into 4 equal parts; 3 parts are shaded because of \(\frac{3}{4}\).
• Next, divide the same rectangle into 3 equal parts in the other direction; shading 1 part negatively gives us \(-\frac{1}{3}\).

The overlapping shaded area represents the product of the two fractions, allowing us to visually verify the calculation resulting in \(-\frac{3}{12}\). The area model is especially helpful for beginners to grasp how fractions interact during multiplication.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form. This is done by dividing the numerator and the denominator by their greatest common divisor (GCD).
For the fraction \(-\frac{3}{12}\), find the GCD of 3 and 12, which is 3. Divide both parts of the fraction by the GCD:

• Numerator: \(-3/3 = -1\).
• Denominator: \(12/3 = 4\).

Thus, \(-\frac{3}{12}\) simplifies to \(-\frac{1}{4}\). Simplification makes fractions easier to understand and work with in further calculations. Always aim to simplify to maintain clarity.

Numerators and Denominators

Understanding numerators and denominators is crucial because they define parts of a fraction.

• The numerator is the fraction's top number, indicating how many parts are being considered.
• The denominator is the bottom number, showing into how many equal parts the whole is divided.

In \(\frac{3}{4}\), 3 is the numerator and 4 is the denominator. When you multiply fractions, you handle numerators and denominators separately, maintaining the logic of partitions and portions. This focused approach helps in correctly executing operations and maintaining accurate representations of quantities.