Problem 70 - Prealgebra

Question

Multiply. $$\left(-\frac{4}{3}\right)\left(-\frac{3}{4}\right)$$

Step-by-Step Solution

Answer

The product is 1.

Step 1: Identify the Operation

We need to multiply two fractions: \(-\frac{4}{3}\) and \(-\frac{3}{4}\). Since both numbers are negative, multiplying them will result in a positive product due to the rule of signs.

Step 2: Multiply the Numerators

Multiply the numerators of both fractions. The numerators are \(-4\) and \(-3\). \(-4 \times -3 = 12\).

Step 3: Multiply the Denominators

Multiply the denominators of both fractions. The denominators are \(3\) and \(4\). \(3 \times 4 = 12\).

Step 4: Simplify the Fraction

Now that we have the fraction \(\frac{12}{12}\), simplify it by dividing both the numerator and the denominator by their greatest common divisor, which is 12.\(\frac{12}{12} = 1\).

Key Concepts

Understanding Fractions

Fractions represent a part of a whole. They are essential in mathematics and seen in everyday life. A fraction has two components:

• The numerator, which is the top number, shows how many equal parts we have.
• The denominator, which is the bottom number, tells how many parts make up a whole.

For example, in the fraction \( \frac{3}{4} \), the number 3 is the numerator, and 4 is the denominator. This indicates we have three out of four parts of something. It's crucial to express fractions clearly because they show relationships and proportions. Knowing this distinction will help in understanding operations involving fractions.

Numerator and Denominator: Key Roles

The numerator and denominator in fractions serve different roles but are both crucial to the fraction's meaning.

• The numerator counts how many parts we have.
• The denominator defines the total number of equal divisions needed to make a complete whole.

When multiplying fractions, remember:

• The numerators are multiplied with the numerators.
• The denominators are multiplied with the denominators.

For example, multiplying the numerators of \( -\frac{4}{3} \) and \( -\frac{3}{4} \), gives us \( -4 \times -3 = 12 \). Similarly, multiplying the denominators \( 3 \times 4 = 12 \), shows how the process follows specific steps based on these definitions. Understanding the distinct functions of numerators and denominators allows smoother operations and contributes to better simplification outcomes.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form. This involves making the numerator and denominator as small as possible but still maintaining their ratio.Here’s how you can simplify fractions:

• Find the greatest common divisor (GCD) of the numerator and denominator.
• Divide both the numerator and the denominator by their GCD.

In solving \( \frac{12}{12} \), the number 12 is both the numerator and the denominator. Dividing both by their GCD, which is 12, results in 1, thus \( \frac{12}{12} = 1 \). This simplification reflects that the fraction represents a whole, showing the significance of simplification in revealing true quantity or relationship. By always reducing fractions, you ensure they are in the most comprehensible format.