Problem 48 - Prealgebra

Question

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

Step-by-Step Solution

Answer

The simplified result is 1.

Step 1 - Analyze the Fraction

Look at the fraction \( \frac{3}{3} \). It is a common fraction where the numerator is equal to the denominator.

Step 2 - Understand the Concept of Simplification

To simplify a fraction, you divide the numerator and the denominator by their greatest common divisor (GCD), which in this case is 3.

Step 3 - Divide by the GCD

Divide the numerator and the denominator by 3: \[ \frac{3 \div 3}{3 \div 3} = \frac{1}{1} \]

Step 4 - Simplify the Result

The fraction \( \frac{1}{1} \) can be further simplified because dividing anything by 1 results in the numerator itself, so we get 1.

Key Concepts

greatest common divisor

The Greatest Common Divisor, often abbreviated as GCD, plays a crucial role in simplifying fractions. It refers to the largest number that divides both the numerator and the denominator without leaving a remainder. Finding the GCD is the first essential step when you're trying to simplify any fraction.

Here's how you can find the GCD:

• List the factors of both the numerator and the denominator.
• Identify the largest factor that appears in both lists.

Suppose you have two numbers, like 18 and 24. The factors of 18 are 1, 2, 3, 6, 9, and 18. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. In this case, the greatest common factor they share is 6, which is the GCD.When you apply this to simplify fractions, the process becomes straightforward. Divide both the numerator and the denominator by their GCD. For instance, if we had a fraction \( \frac{18}{24} \), dividing each by 6 gives \( \frac{3}{4} \), a simplified version of the original fraction.

numerator and denominator

Fractions consist of two parts: the numerator and the denominator. It's important to understand what each represents to simplify fractions effectively.

The **numerator** is the top part of the fraction, representing how many parts of a whole are considered. In the fraction \( \frac{3}{3} \), the numerator is 3. This means we are looking at 3 parts out of a whole.

The **denominator** is the bottom part and tells us the total number of equal parts the whole is divided into. Again, in \( \frac{3}{3} \), the denominator is also 3, indicating that the whole is divided into 3 equal parts.

Understanding this allows you to see that when the numerator and the denominator are equal, as in \( \frac{3}{3} \), the fraction actually represents a whole or 1. This is because you’re effectively dividing something into equal parts and then taking all those parts.

simplification process

The simplification process is essential for making fractions easier to work with. By simplifying fractions, you not only make them more visually pleasing, but calculations become simpler as well. Let's break this down further.

In the original exercise, the fraction \( \frac{3}{3} \) was simplified. Here's how simplification generally works:

• Find the GCD of the numerator and the denominator.
• Divide both the numerator and the denominator by this GCD.
• The result is a fraction in its simplest form.

For \( \frac{3}{3} \), since the numerator and the denominator are the same, the GCD is 3. Dividing both by 3, you get \( \frac{1}{1} \), which simplifies to 1.

Remember, a fraction doesn't always neatly resolve to a whole number, but simplifying ensures it cannot be reduced further. For instance, \( \frac{4}{8} \) simplifies to \( \frac{1}{2} \) after dividing both by their GCD, which is 4. Thus, mastering this simplification process will help make your mathematical life much easier.