Problem 38 - Prealgebra

Question

Find each product. Write in simplest form. $$\frac{4 a}{5} \cdot \frac{3}{a}$$

Step-by-Step Solution

Answer

The product is \(\frac{12}{5}\).

Step 1: Multiply the Numerators

The numerators of the fractions are \(4a\) and \(3\). Multiply these together to get \(4a \times 3 = 12a\).

Step 2: Multiply the Denominators

The denominators of the fractions are \(5\) and \(a\). Multiply these together to get \(5 \times a = 5a\).

Step 3: Write the Fraction

Combine the results from Steps 1 and 2 to form the new fraction: \(\frac{12a}{5a}\).

Step 4: Simplify the Fraction

In \(\frac{12a}{5a}\), the \(a\) in the numerator and denominator cancel out. Simplify to \(\frac{12}{5}\).

Key Concepts

Simplifying Fractions

When working with fractions, simplifying is an essential step. Simplifying means reducing the fraction to its smallest or simplest form, which is where both the numerator and the denominator have no common factors other than 1.
For example, when you have a fraction like \(\frac{12a}{5a}\), you can simplify by canceling common terms. Here, the variable \(a\) in both the numerator and denominator can be canceled out, leaving \(\frac{12}{5}\).

• First, look for common factors in numbers or terms.
• Divide both the numerator and the denominator by their greatest common factor (GCF).
• If variables are present in both, like \(a\), cancel them out to fully simplify.

This process not only makes the fraction neater but also easier to understand or use in further calculations.

Numerators and Denominators

In any fraction, understanding numerators and denominators is crucial. The numerator is the top number, which indicates how many parts of the whole are being considered. The denominator, at the bottom, represents the total number of equal parts that make up the whole.
Consider the fraction \(\frac{4a}{5}\). Here, \(4a\) is the numerator and represents four parts of something over 5 parts total, indicated by the denominator \(5\).
When multiplying fractions, both numerators and denominators play different roles:

• Multiply the numerators: In our example, \(4a \times 3 = 12a\).
• Multiply the denominators: \(5 \times a = 5a\).

Afterward, you'll get a new fraction, like \(\frac{12a}{5a}\). Simplifying involves assessing both the numerator and denominator for common factors or terms.

Prealgebra Concepts

In prealgebra, concepts like fraction operations help build foundational math skills. Multiplying and simplifying fractions are key skills you will often use. Let's break down the basic prealgebra rules involved in our original exercise.
When multiplying fractions, the overall rule is to multiply straight across:

• Multiply the numerators together to get the numerator of the product.
• Multiply the denominators together to get the denominator of the product.

For our exercise, the final fraction after multiplying was \(\frac{12a}{5a}\), which simplifies to \(\frac{12}{5}\).
Prealgebra often involves dealing with variables within fractions, requiring you to know how to handle numerical parts and variable parts independently. By mastering these simple fraction operations in prealgebra, you'll be well-equipped for more advanced mathematics.