Problem 50 - Prealgebra

Question

Simplify by dividing the numerator by the denominator. $$\frac{12}{3}$$

Step-by-Step Solution

Answer

The simplified form of \( \frac{12}{3} \) is 4.

Step 1: Understand Division

To simplify the fraction \( \frac{12}{3} \), we need to divide the numerator (12) by the denominator (3). Division is the process of determining how many times the denominator can fit into the numerator.

Step 2: Compute the Division

Perform the division operation: 12 divided by 3 equals 4. This is because 3 fits into 12 exactly 4 times without any remainder.

Step 3: Conclude the Simplification

Since the division results in a whole number, the simplified form of \( \frac{12}{3} \) is 4. There is no need for further simplification.

Key Concepts

Division

When simplifying a fraction, division plays a key role. It helps us find out how many times the denominator fits into the numerator. In simple terms, when you perform division, you're trying to group the numerator into equal parts as described by the denominator. For example, with the fraction \( \frac{12}{3} \), division answers the question: "How many groups of 3 are there in 12?"

Here’s how it works step by step:

• First, identify the numerator, which is the top number of the fraction.
• Next, look for the denominator, the bottom number of the fraction.
• Perform the division by dividing the numerator by the denominator.

In our specific example: Divide 12 by 3, which gives us 4. Thus, division helps us reduce fractions to simpler forms when possible.

Numerator and Denominator

In a fraction, understanding the roles of the numerator and denominator is crucial in performing any operations like simplification. The numerator is the top number; it represents how many parts of the whole are being considered. The denominator, on the other hand, is the bottom number; it tells us into how many equal parts the whole is divided.

For example, take the fraction \( \frac{12}{3} \):

• The numerator is 12, meaning we are considering 12 parts of something.
• The denominator is 3, telling us that the whole is divided into 3 equal parts.

Considering these elements, we use them in division to simplify the fraction. So with \( \frac{12}{3} \), we interpret it as 12 parts divided into groups of 3, making it clearer how the simplification works.

Whole Numbers

Once you execute division on a fraction and obtain a result, the outcome can be a whole number, fraction, or mixed number. A whole number appears when the numerator is exactly divisible by the denominator with no remainder. The fraction \( \frac{12}{3} \) offers a perfect example of this.

Through division, 12 divided by 3 yields 4, which is a whole number because there's no leftover. Here’s how you recognize a whole number from fractions:

• Check if dividing the numerator by the denominator results in a number without fractions or decimals.
• If there is no remainder, your result is a whole number.
• Whole numbers include numbers like 0, 1, 2, 3, and so on.

Thus, converting fractions to whole numbers simplifies equations and clarifies how many whole sets or units you're dealing with. Simple fractions like \( \frac{12}{3} \) illustrate just how effective this simplification can be.