Problem 51 - Fundamentals of Mathematics

Question

Reduce, if possible, each fraction. $$\frac{15}{33}$$

Step-by-Step Solution

Answer

\( \frac{15}{33} \) reduces to \( \frac{5}{11} \).

Step 1: Identify the Greatest Common Divisor (GCD)

First, we need to find the greatest common divisor (GCD) of the numerator and the denominator. The numerator is 15, and the denominator is 33. We list the factors for each number. - Factors of 15: 1, 3, 5, 15. - Factors of 33: 1, 3, 11, 33. The greatest common factor between 15 and 33 is 3.

Step 2: Divide the Numerator and Denominator by the GCD

Next, we divide both the numerator and the denominator by their greatest common divisor, which is 3 in this case. - Divide 15 by 3 to get 5. - Divide 33 by 3 to get 11.

Step 3: Write the Reduced Fraction

The original fraction, \( \frac{15}{33} \), simplifies to \( \frac{5}{11} \) when both the numerator and denominator are divided by their GCD (3). Thus, the fraction is reduced to \( \frac{5}{11} \).

Key Concepts

Greatest Common Divisor

The Greatest Common Divisor (GCD), sometimes known as the greatest common factor, is the largest positive integer that divides two or more numbers without leaving a remainder. For instance, when reducing a fraction like \( \frac{15}{33} \), finding the GCD is a crucial initial step. But why do we need it?The GCD helps us simplify fractions to their most basic form. By identifying the GCD, we know the highest number by which both the numerator and the denominator can be evenly divided.Let's explore how to find the GCD:

• List the factors of each number separately.
• Identify the common factors between the two lists.
• Choose the largest common factor to use as the GCD.

In our example, both 15 and 33 share the factor 3, which is the largest, thus the GCD is 3. This means 3 is the biggest number that can divide both 15 and 33 with no leftovers.Finding and using the GCD in fraction reduction ensures that the fraction is expressed in its simplest possible form.

Numerator

The numerator of a fraction is the top number that represents how many parts of the whole or set we have. Understanding the numerator's role is vital in fraction reduction as it is one of the two numbers affected during the simplification process.Take the fraction \( \frac{15}{33} \). Here:

• 15 is the numerator,
• It tells us the number of parts we have out of the total described by the denominator.

When simplifying fractions, the numerator is divided by the GCD to reduce its value. In our example:

• Divide 15 by the GCD, which is 3.
• This division gives us 5 as the new numerator.

Dividing the numerator correctly allows the fraction to retain its proportional relationship to the denominator even in its reduced form.

Denominator

The denominator is the bottom number of a fraction and signifies the total number of equal parts that make up a whole. It's essential in understanding how many parts make up the complete set from which the numerator draws.For instance, in the fraction \( \frac{15}{33} \):

• 33 is the denominator,
• It represents the total number of equal divisions or parts of the whole.

The denominator, just like the numerator, must be divided by the same GCD to simplify the fraction completely. In our example:

• We divide 33 by the GCD (3).
• This results in 11 being the new denominator as the fraction simplifies to \( \frac{5}{11} \).

This process ensures the original ratio between the numerator and denominator remains intact, maintaining the fraction's value in a simpler form.