Problem 15 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(1-34,\) perform the indicated multiplication. $$-\frac{3}{5} \cdot\left(-\frac{4}{7}\right)$$

Step-by-Step Solution

Answer

The multiplication of \(-\frac{3}{5}\) by \(-\frac{4}{7}\) yields \(\frac{12}{35}\).

Step 1: Identify the numerators and denominators

The numerators are -3 and -4, while the denominators are 5 and 7. It should be noted that -3 and -4 are negative numbers.

Step 2: Multiply the numerators

Multiply the numerators -3 and -4. This equals to \((-3)\times(-4) = 12\). We know that when we multiply two negative numbers it results in a positive number.

Step 3: Multiply the denominators

Multiply the denominators 5 and 7 to get 35.

Step 4: Write the result as a fraction

The result of the multiplication is given by writing the result in Step 2 above the result in Step 3. Therefore the answer is \(\frac{12}{35}\).

Key Concepts

Numerators and Denominators

Fractions have two key components: the numerator and the denominator. The numerator is the number above the fraction line, while the denominator sits below it.

• The numerator shows how many parts of a whole we are considering.
• The denominator tells us the total number of equal parts the whole is divided into.

The role of numerators and denominators is essential in fraction arithmetic. When multiplying, you multiply the numerators together and the denominators together.
In the original exercise, the number -3 is the numerator of the first fraction, and 5 is the denominator. For the second fraction, -4 is the numerator and 7 is the denominator. Understanding these elements allows us to correctly execute the multiplication process.

Negative Numbers

Negative numbers play an interesting role in multiplication, especially in fractions. They are numbers that are less than zero and are often used to represent a deficit or decrease.
When you multiply two negative numbers together, the result is a positive number. This occurs because the negatives cancel each other out. Think of it as two opposite directions merging into one forward movement.
In the given problem, both numerators, -3 and -4, are negative. Multiplying them results in a positive product:
a multiplication rule where each negative sign "neutralizes" the other. This is why the result of a multiplication like a fraction action is positive, yielding a positive number that then stands as the numerator in our resulting fraction.

Fractional Multiplication Steps

Multiplying fractions involves a straightforward process consisting of several clear steps:

• Identify the Numerators and Denominators: Before multiplying, clearly recognize the numerators and denominators in each fraction.
• Multiply the Numerators: Multiply the numerators of the fractions. This is essentially your new numerator.
• Multiply the Denominators: Similarly, multiply the denominators to find your new denominator.
• Construct the Resulting Fraction: Form a new fraction using the products from the previous calculations: numerator over denominator.

In our exercise, the steps look like this:
Firstly, the numerators -3 and -4 are multiplied, resulting in a positive 12 due to the cancellation of the negative signs. Secondly, the denominators 5 and 7 produce 35 when multiplied. Finally, the resulting fraction is \(\frac{12}{35}\).This approach ensures accuracy and reinforces the systematic nature of fractional multiplication.