Problem 94 - Elementary and Intermediate Algebra

Question

Multiply: \(\quad \frac{3}{4}\left(\frac{8}{5}\right)\)

Step-by-Step Solution

Answer

The product is \( \frac{6}{5} \).

Step 1: Identify the Fractions

First, we need to clearly identify the fractions we are going to multiply. In this problem, we are given two fractions: \( \frac{3}{4} \) and \( \frac{8}{5} \).

Step 2: Multiply the Numerators

Next, we multiply the numerators of the two fractions. The numerator of the first fraction is 3, and the numerator of the second fraction is 8. Multiply these values: \( 3 \times 8 = 24 \).

Step 3: Multiply the Denominators

Now we multiply the denominators of the fractions. The denominator of the first fraction is 4, and the denominator of the second fraction is 5. Multiply these values: \( 4 \times 5 = 20 \).

Step 4: Form the New Fraction

Using the results from Steps 2 and 3, form a new fraction. The numerator is 24 (from Step 2) and the denominator is 20 (from Step 3). So, the new fraction is \( \frac{24}{20} \).

Step 5: Simplify the Fraction

Finally, simplify the fraction \( \frac{24}{20} \). Both the numerator and the denominator can be divided by their greatest common divisor, which is 4. Divide both by 4: \( \frac{24}{4} = 6 \) and \( \frac{20}{4} = 5 \).The simplified fraction is \( \frac{6}{5} \).

Key Concepts

Numerator Multiplication

When it comes to multiplying fractions, the first step involves the multiplication of the numerators. A numerator is the top number in a fraction, representing how many parts of the whole are being considered. For example, in the fraction \( \frac{3}{4} \), '3' is the numerator, indicating three parts.

To multiply the numerators of two fractions, simply take the two top numbers and multiply them. Imagine you have \( \frac{3}{4} \) and \( \frac{8}{5} \); you multiply the numerators: \( 3 \) and \( 8 \). The equation would look like this: \( 3 \times 8 = 24 \).

• Identify the numerators of the given fractions.
• Multiply these numerators to get your result.
• Place this product at the top of your new fraction.

This part of fraction multiplication focuses solely on the top numbers, which is essential because the product of the numerators will be the numerator of the resultant fraction.

Denominator Multiplication

The process continues with the multiplication of the denominators, which are the bottom numbers in fractions indicating the total number of equal parts the whole is divided into. In our example, the denominators are '4' and '5' for the fractions \( \frac{3}{4} \) and \( \frac{8}{5} \), respectively.

When you multiply the denominators, the goal is to find the product that defines the new whole in terms of equal parts. Here’s how it’s done:

• Identify the denominators of the fractions you are dealing with.
• Multiply these denominators to calculate the new denominator.
• For our exercise, \( 4 \times 5 = 20 \).

In summary, multiplying denominators is crucial as it determines the size of the whole or the total number of parts in the resultant fraction. It's also a straightforward step, ensuring that the process remains simple and consistent.

Simplifying Fractions

Once you have formed a fraction by multiplying the numerators and denominators, it's time to simplify it. Simplifying a fraction makes it easier to interpret. This step involves reducing the fraction to its lowest terms by dividing both the numerator and denominator by their greatest common divisor (GCD).

For the fraction \( \frac{24}{20} \), the GCD of 24 and 20 is 4. Dividing both the numerator and the denominator by this number simplifies the fraction:

• Divide 24 by 4 to get 6.
• Divide 20 by 4 to get 5.
• The simplified fraction thus is \( \frac{6}{5} \).

Simplifying helps in expressing the fraction in a more refined form, making calculations and comparisons easier. It is an essential practice, especially in mathematical operations like equation solving or probability where elegant and manageable forms are preferred.