Problem 29 - Prealgebra

Question

Divide the numerator and the denominator of each of the following fractions by 3. $$\frac{39}{51}$$

Step-by-Step Solution

Answer

The fraction \( \frac{39}{51} \) simplified is \( \frac{13}{17} \).

Step 1: Identify the Numerator and Denominator

The given fraction is \( \frac{39}{51} \). The numerator is 39, and the denominator is 51. These are the values we need to divide by 3.

Step 2: Divide the Numerator by 3

Divide the numerator, 39, by 3. 39 divided by 3 equals 13. Therefore, the result for the numerator is 13.

Step 3: Divide the Denominator by 3

Now, divide the denominator, 51, by 3. 51 divided by 3 equals 17. Therefore, the result for the denominator is 17.

Step 4: Write the New Fraction

Combine the results from the previous steps to form the new fraction. The new fraction is \( \frac{13}{17} \), obtained by dividing both the numerator and denominator by 3.

Key Concepts

Understanding the Numerator

In a fraction, the top number is called the numerator. It represents how many parts you have. For example, in the fraction \( \frac{39}{51} \), 39 is the numerator. This number tells us that there are 39 parts of a total divided according to the denominator. When simplifying fractions, the numerator is crucial because it helps determine how much of the whole we possess. Typically, the goal is to make the numerator smaller, ideally without changing the overall value of the fraction.

• If the numerator can be evenly divided by a common factor, it simplifies the fraction.
• Always start simplification with the numerator for clear results.

First, identify if there are common divisors of both the numerator and the denominator, simplifying the process.

The Role of the Denominator

The denominator is the bottom number in a fraction. It shows us into how many equal parts the whole is divided. In \( \frac{39}{51} \), 51 is the denominator. Understanding the denominator is essential because it defines the proportion of the fraction. Simplifying the denominator is the next step after managing the numerator.

• The denominator helps us understand the size of the equal parts.
• We simplify the denominator by dividing it by a common factor, same as the numerator.
• For theoretical simplicity: a denominator of 1 indicates a whole number.

Always ensure that the denominator is never zero, as division by zero is undefined.

Mastering Division in Fractions

Division is a key operation in simplifying fractions. It involves breaking down or sharing equally among parts. To simplify \( \frac{39}{51} \), we apply division to both the numerator and the denominator. Here's why this works:

• Finding a common factor (like 3 in our example) and dividing both the numerator and denominator by it keeps the fraction equivalent.
• Division reduces the fraction to its simplest form by using shared factors.

Always verify that after division, no further common factors exist. This guarantees your fraction is in its simplest form. Remember that simplifying fractions makes them easier to understand and work with in math problems.