Problem 27 - Prealgebra

Question

Divide the numerator and the denominator of each of the following fractions by 3. $$\frac{12}{9}$$

Step-by-Step Solution

Answer

The fraction \( \frac{12}{9} \) simplifies to \( \frac{4}{3} \) by dividing both the numerator and denominator by 3.

Step 1: Identify the Fraction

First, identify the given fraction which is \( \frac{12}{9} \). This fraction can be simplified by dividing both the numerator and the denominator by a common factor, which in this case is 3.

Step 2: Divide the Numerator

The numerator is 12. Divide 12 by 3 to simplify it. Calculation: \( 12 \div 3 = 4 \). So, the new numerator is 4.

Step 3: Divide the Denominator

The denominator is 9. Divide 9 by 3 to simplify it. Calculation: \( 9 \div 3 = 3 \). So, the new denominator is 3.

Step 4: Write the Simplified Fraction

Combine the simplified numerator and denominator to write the new fraction. Thus, \( \frac{12}{9} \) simplifies to \( \frac{4}{3} \).

Key Concepts

Understanding the Numerator

In the context of a fraction, the numerator is the top number. It represents the part of the whole that you have. For example, in the fraction \( \frac{12}{9} \), the numerator is 12. This number tells us how many parts of the total we are considering. For instance:

• If you have \( \frac{12}{9} \) of a cake, you have 12 parts of something divided into 9 equal sections total (more than a whole).
• The numerator describes quantity or count regarding the division.

When simplifying a fraction, as in the example, you start by checking what number can divide the numerator evenly. In this case, both 12 and 9 can be divided perfectly by 3, a common factor. The new numerator after dividing by 3 becomes 4. This simplification means we still have 4 parts of the cake, but measured differently. Understanding the role of the numerator and how it is simplified helps in grasping the basics of fractions effectively.
Simplifying it makes the fraction easier to understand or compare with others.

Grasping the Denominator

The denominator of a fraction sits at the bottom and plays a crucial role. It shows into how many equal parts the whole is divided. Take the fraction \( \frac{12}{9} \), where 9 is the denominator.

• The denominator helps in understanding the size of each part.
• In \( \frac{12}{9} \), it implies each part is one-ninth of the whole.

When we simplify a fraction, we look to divide the denominator by the same number as the numerator. For instance, dividing 9 by 3 results in 3. This operation adjusts the size of each part by making them larger, thereby keeping the fraction value unchanged despite numerically altering numerator and denominator. Knowing how a denominator works and how to simplify it deepens your understanding of fractions as relative numbers that show proportionate quantities.

The Role of a Common Factor

A common factor is a number that divides two numbers without leaving a remainder. In fractions, identifying a common factor is vital for simplification.In the fraction \( \frac{12}{9} \), the common factor is 3. Both the numerator (12) and the denominator (9) are divisible by this common factor. Here’s how why this matters:

• Dividing both numbers by the same factor keeps the fraction equivalent.
• The goal is to make the numbers smaller and easier to compare or use.

Finding a common factor involves simply looking for the largest number that can divide both the numerator and denominator evenly. By reducing the fraction \( \frac{12}{9} \) with the common factor 3, we get \( \frac{4}{3} \) which is more manageable and interpretable. Recognizing and using common factors are essential skills for simplifying fractions effectively and helps in understanding its size more intuitively.