Problem 52 - Prealgebra

Question

Find each product. Write in simplest form. $$\frac{a b}{2} \cdot \frac{4}{b c}$$

Step-by-Step Solution

Answer

The simplest form is \( \frac{2a}{c} \).

Step 1: Understand the Problem

We need to multiply two fractions: \( \frac{ab}{2} \) and \( \frac{4}{bc} \). The expression should be simplified as much as possible.

Step 2: Multiply the Numerators

To multiply fractions, we start by multiplying the numerators (the top parts of the fractions). Multiply \( ab \) by \( 4 \), which gives us \( 4ab \).

Step 3: Multiply the Denominators

Multiply the denominators (the bottom parts of the fractions). Multiply \( 2 \) by \( bc \), which results in \( 2bc \).

Step 4: Form the New Fraction

The product of the two fractions is the new fraction, with the product of the numerators over the product of the denominators. So we have \( \frac{4ab}{2bc} \).

Step 5: Simplify the Fraction

To simplify, divide both the numerator and the denominator by their greatest common factor. Here, both \( 4ab \) and \( 2bc \) can be divided by 2 and the \( b \) terms cancel out: \( \frac{4ab}{2bc} = \frac{2a}{c} \).

Step 6: Write the Simplified Form

The simplest form of \( \frac{ab}{2} \cdot \frac{4}{bc} \) is \( \frac{2a}{c} \).

Key Concepts

Simplifying Fractions

Simplifying fractions is an essential step when dealing with fraction operations, ensuring that the results are in their simplest form. Simplifying involves reducing the fraction by canceling out common factors in the numerator and the denominator.

• First, identify the greatest common factor (GCF) of the numerator and denominator. This is the largest number that divides both terms without leaving a remainder.
• Divide both the numerator and the denominator by their GCF.

In the example problem, we had the fraction \( \frac{4ab}{2bc} \). By finding the common factor, which is 2 in this case (and recognizing that both the b terms are present in the numerator and denominator), it simplifies to \( \frac{2a}{c} \). Simplifying fractions not only aids in making the answers easier to interpret but also ensures calculations remain efficient when fractions are involved in larger algebraic expressions.

Algebraic Expressions

When working with expressions like \( \frac{ab}{2} \cdot \frac{4}{bc} \), it's important to understand how to manage the variables and constants within algebraic expressions. These expressions contain both numbers and variables, such as \( a, b, \) and \( c \).

• Variables represent unknown values and can stand for any number. They allow for flexibility in mathematical modeling.
• Constants are fixed numbers like 2 and 4 in the example, providing stability to expressions.

When multiplying algebraic fractions, treat variables as you would numerical factors. Combine like terms, and always look for opportunities to simplify the expression. Ultimately, clear management of variables and constants helps clarify the problem while guiding you towards the simplest form.

Fraction Operations

Fraction operations, particularly multiplication, require an understanding of how to handle both the numerators and denominators effectively.

• To multiply fractions, multiply the numerators together and the denominators together.
• Form a new fraction using these multiplication results, always simplifying where possible.
• When handling algebraic fractions, pay special attention to the variables as they can often provide additional simplification opportunities.

In our exercise, the multiplication of the fractions \( \frac{ab}{2} \) and \( \frac{4}{bc} \) was straightforward: numerators \( ab \times 4 \) gave \( 4ab \); denominators \( 2 \times bc \) gave \( 2bc \). Simplifying this resulted in \( \frac{2a}{c} \), by removing common factors. Effective fraction operations hinge on recognizing these simplification possibilities, which not only clarify the solution but also make working with more complex expressions easier.