Problem 35 - Prealgebra

Question

Reduce each fraction to lowest terms. $$\frac{18 a^{2} b}{90 a b}$$

Step-by-Step Solution

Answer

The fraction simplifies to \( \frac{a}{5} \).

Step 1: Understanding the Problem

We are given the fraction \( \frac{18 a^{2} b}{90 a b} \) and need to simplify it to its lowest terms. This involves canceling out any common factors from the numerator and denominator.

Step 2: Factor Numerator and Denominator

First, identify the factors of the numerator and the denominator. For the numerator, \( 18a^2b = 2 \times 3^2 \times a^2 \times b \). For the denominator, \( 90ab = 2 \times 3^2 \times 5 \times a \times b \).

Step 3: Cancel Common Factors

Both the numerator and the denominator contain the factors \( 2 \), \( 3^2 \), \( a \), and \( b \). Cancel these common factors:\[\frac{18a^2b}{90ab} = \frac{2 \times 3^2 \times a^2 \times b}{2 \times 3^2 \times 5 \times a \times b} = \frac{a}{5}\]

Step 4: Write the Simplified Fraction

After canceling common factors, the fraction simplifies to \( \frac{a}{5} \).

Key Concepts

Common Factors

In mathematics, simplifying fractions involves identifying and removing common factors from the numerator and denominator.
Common factors are numbers or variables that can divide both the numerator and denominator evenly. They serve as the key to simplifying fractions effectively.
When you break down fractions, such as \( \frac{18a^2b}{90ab} \), you are looking for the common numbers or expressions that each part shares.

• In our example, both the numerator and the denominator share common factors such as \( 2 \), \( 3^2 \), \( a \), and \( b \).

Finding these allows us to reduce the fraction to its simplest form. Recognizing common factors is crucial, as it forms the basis for both simplifying and understanding the larger concepts in algebra.

Numerator and Denominator

Fractions consist of two main parts: the numerator and the denominator.
The numerator is the top part of the fraction and represents the number of parts we are considering. The denominator is the bottom part, which represents the total number of equal parts in a whole.

• In the fraction \( \frac{18a^2b}{90ab} \), \( 18a^2b \) is the numerator, and \( 90ab \) is the denominator.

Understanding these components is crucial when simplifying fractions, as we need to work with both to identify common factors and simplify the expression correctly.
Always remember: Changes to either part of the fraction alter its entire value. Thus, any simplification must consider both the numerator and the denominator.

Factorization

Factorization involves breaking down a number or expression into its simplest components or factors.
This process helps us identify parts of the numerator and denominator that can be cancelled out.
For fractions like \( \frac{18a^2b}{90ab} \), we factorize each part separately:

• The numerator \( 18a^2b \) factors as \( 2 \times 3^2 \times a^2 \times b \).
• The denominator \( 90ab \) factors as \( 2 \times 3^2 \times 5 \times a \times b \).

Factorization is essential for simplifying, as it lays the groundwork for finding and cancelling common factors.

Cancelling Common Terms

Once you've factorized the numerator and denominator, the next step is to cancel out the common terms.
This process is straightforward but crucial in reducing fractions to their lowest terms.
In the example \( \frac{18a^2b}{90ab} \), after factorization, common factors such as \( 2 \), \( 3^2 \), \( a \), and \( b \) appear in both parts of the fraction.

• By cancelling these out, you simplify the fraction to \( \frac{a}{5} \).

Cancelling common terms helps to express fractions more simply and elegantly, which is especially helpful in further mathematical calculations and analyses.