Problem 72 - Algebra for College Students

Question

Find each quotient. $$\frac{-12 a b c^{2}}{12 b c}$$

Step-by-Step Solution

Answer

-ac

Step 1: Simplify the Fractions

Start by writing the fraction given in the problem: \( \frac{-12abc^{2}}{12bc} \). To simplify this fraction, we'll need to divide both the numerator and the denominator by their greatest common factor.

Step 2: Cancel Common Factors

Identify the common factors in the numerator and the denominator. Both the numerator \(-12abc^{2}\) and the denominator \(12bc\) have the terms \(12\), \(b\), and one \(c\) in common. We can cancel these terms to simplify the fraction.

Step 3: Simplify the Resulting Expression

After canceling the common terms, we are left with \( \frac{-a c}{1} \). Any fraction over 1 is just the numerator itself, so the expression simplifies to \(-ac\).

Step 4: Finalize the Solution

Conclude that the simplified form of the quotient is \(-ac\). This is the final answer.

Key Concepts

Understanding the Greatest Common Factor (GCF)

The Greatest Common Factor (GCF) is the largest factor that divides two numbers or algebraic terms without leaving a remainder. It allows us to simplify algebraic expressions by removing redundant elements. For instance, when looking at the fraction \( \frac{-12abc^2}{12bc} \), both the numerator and denominator share common factors. Here, the GCF is the number 12, along with the algebraic terms \( b \) and one \( c \). This means we can effectively divide the entire fraction by \( 12bc \), thereby simplifying it. Finding the GCF is crucial as it aids in reducing algebraic fractions to their simplest form, making computation and understanding much more manageable.To identify the GCF:

• List down the factors of each term separately.
• Find the common factors between these terms.
• Select the largest factor that is common among them.

Numerators and Denominators in Fractions

In any fraction, the numerator is the term or number found above the division line, while the denominator is below. The fraction \( \frac{-12abc^2}{12bc} \) has a numerator of \(-12abc^2\) and a denominator of \(12bc\). These two parts represent how many parts of a whole are being considered (numerator) and the total number of parts (denominator).Breaking it down:

• Numerator: In our exercise, \(-12abc^2\), which includes the variable terms \(a\), \(b\), \(c^2\), and the number \(-12\).
• Denominator: In our case, \(12bc\), comprising the variables \(b\), \(c\), and the factor 12.

Understanding these parts is crucial for simplifying fractions, as it helps in identifying what terms can be reduced through methods such as canceling common factors.

Canceling Common Factors

Canceling common factors is a process where you eliminate the same elements from both the numerator and the denominator. It's a key step in simplifying algebraic fractions because it reduces complexity. For example, in \( \frac{-12abc^2}{12bc} \), the terms \( 12 \), \( b \), and \( c \) are present in both the numerator and denominator. By canceling these, you simplify the expression significantly.Here's how you do it:

• Divide the numerator and the denominator by their common terms.
• Remove or cancel these terms entirely to simplify the fraction, leaving only non-canceled components.

Once the common factors \( 12bc \) are canceled out, we are left with \( \frac{-ac}{1} \), simplifying the expression to \(-ac\). This makes it easier to work with and understand.

Decoding Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. They're used to represent relationships and expressions in algebraic language. For instance, \(-12abc^2\) is an algebraic expression, representing the product of \(-12\), \(a\), \(b\), and \(c^2\).Key aspects of algebraic expressions:

• Use of variables to denote unknown or changeable values.
• Operations between these variables and numbers.
• They can form equations which are equalities involving expressions.

In our exercise, algebraic expressions were simplified by removing common factors, demonstrating how they can be manipulated to achieve simpler, more solvable forms.