Problem 115 - Introductory and Intermediate Algebra for College Students 4th

Question

Perform the indicated operation. Write the answer as an algebraic expression. $$\frac{3}{4} \cdot \frac{a}{5}$$

Step-by-Step Solution

Answer

\(\frac{3a}{20}\)

Step 1: Identify the numerators and denominators

The numerators are \(3\) and \(a\) and the denominators are \(4\) and \(5\).

Step 2: Multiply the numerators

Multiply \(3\) and \(a\) to get \(3a\), which will be the numerator of the resulting fraction.

Step 3: Multiply the denominators

Multiply \(4\) and \(5\) to get \(20\), which will be the denominator of the resulting fraction.

Step 4: Write the answer in the form of a fraction

Our answer should be a fraction with numerator and denominator obtained from previous steps. So, the answer is \(\frac{3a}{20}\).

Key Concepts

Fraction Multiplication

Fraction multiplication involves multiplying two fractions together. To do this, you multiply the numerators by each other and do the same for the denominators. For example, if you have two fractions \( \frac{3}{4} \) and \( \frac{a}{5} \), you multiply the numerators \(3\) and \(a\) to get \(3a\).
Then, you multiply the denominators \(4\) and \(5\) to get \(20\).

• Step 1: Multiply the numerators - \(3 \times a = 3a\).
• Step 2: Multiply the denominators - \(4 \times 5 = 20\).

Finally, put the resulting numerator and denominator together as a fraction \( \frac{3a}{20} \). This is how fraction multiplication simplifies the process by breaking it down into straightforward steps.

Numerator and Denominator

In any fraction, the fraction is made up of two main parts: the numerator and the denominator.
The numerator is the top part, which represents how many parts we have.
The denominator is the bottom part, indicating how many equal parts the whole is divided into.

• The numerator is written above the fraction line.
• The denominator is written below the fraction line.

For instance, in the fraction \( \frac{3}{4} \), \(3\) is the numerator and \(4\) is the denominator. Understanding these two essential components is critical when performing arithmetic operations with fractions, such as multiplication.

Simplifying Fractions

After performing arithmetic operations, sometimes you end up with fractions that can be simplified. Simplifying fractions means finding an equivalent fraction that has a smaller numerator and denominator.
This is done by finding the greatest common divisor (GCD) of the numerator and denominator, and then dividing both by this number.

• Check if the numerator and denominator have any common factors.
• Divide both by their greatest common factor if possible.

For example, suppose our fraction is \( \frac{6}{8} \). The GCD of \(6\) and \(8\) is \(2\). By dividing both by \(2\), we simplify it to \( \frac{3}{4} \). While \( \frac{3a}{20} \) can't be simplified further as \(3a\) and \(20\) don't have common factors, understanding this concept makes handling more complex fractions easier.