Problem 22 - Elementary Algebra

Question

Reduce each fraction to lowest terms. $$\frac{-21 x y}{-14 a b}$$

Step-by-Step Solution

Answer

The fraction reduced to lowest terms is \( \frac{3xy}{2ab} \).

Step 1: Remove Negative Sign

The negative signs in both the numerator \( -21xy \) and the denominator \( -14ab \) cancel each other out. So the fraction can be simplified to \( \frac{21xy}{14ab} \).

Step 2: Find the Greatest Common Factor (GCF)

To reduce the fraction, we need to find the greatest common factor of 21 and 14. The GCF of 21 and 14 is 7.

Step 3: Divide by the GCF

Divide both the numerator and the denominator by the GCF (7). This gives us: \[ \frac{21xy}{14ab} = \frac{21 \div 7 \cdot xy}{14 \div 7 \cdot ab} = \frac{3xy}{2ab} \]

Step 4: Confirm Lowest Terms

Since 3 and 2 have no common factors other than 1, \( \frac{3xy}{2ab} \) is already in its lowest terms.

Key Concepts

Greatest Common Factor

The concept of the Greatest Common Factor (GCF) is central to simplifying fractions. The GCF of two numbers is defined as the largest number that divides both of them without leaving a remainder. It helps to bring down a fraction to its most simplified form. In our exercise, we looked at the numbers 21 and 14 in the numerator and denominator.
The method for finding the GCF is as follows:

• List the factors of each number.
• Identify the common factors.
• Choose the largest common factor.

For example: - Factors of 21: 1, 3, 7, 21. - Factors of 14: 1, 2, 7, 14. The common factor is 7, which is the greatest. Dividing both the numerator and the denominator by this number will simplify your fraction to its lowest terms.

Numerator and Denominator

The numerator and denominator play specific roles in a fraction. Simplifying a fraction means making these numbers as small as possible while retaining the original value of the fraction.

The **numerator** is the top part of the fraction. It shows how many parts of a whole are being considered. In our exercise, this was originally \(21xy\).

The **denominator** is the bottom part. It represents the total number of equal parts the whole is divided into. Here, that was initially \(14ab\).

When simplifying, the negative signs of both parts can cancel each other out. This makes the simplification process easier. Both parts can then be divided by their GCF, which in this exercise simplified the fraction from \(\frac{21xy}{14ab}\) to \(\frac{3xy}{2ab}\). Understanding this interplay helps prevent calculation mistakes during simplification.

Algebraic Expressions

Algebraic expressions in a fraction might seem confusing at first, but they function much like regular numbers when simplifying fractions. In our given fraction, we have algebraic expressions including variables like \(x\), \(y\), \(a\), and \(b\).

Key steps to simplifying fractions with algebraic terms include:

• Identify and cancel out common variables if possible, using the Greatest Common Factor.
• Focus on simplifying numerical parts first before involving the variables.

Algebraic expressions retain their variables after simplifying unless they are canceling terms with factors from both the numerator and the denominator. In this instance, the non-numeric parts \(xy\) and \(ab\) could not be simplified further as they are distinct expressions. Recognizing terms that can or cannot be reduced will prevent further errors in simplifying fractions.