Problem 2 - Prealgebra

Question

Write each fraction in simplest form. If the fraction is already in simplest form, write simplified. $$\frac{9}{15}$$

Step-by-Step Solution

Answer

The fraction \( \frac{9}{15} \) simplifies to \( \frac{3}{5} \).

Step 1: Identify the Greatest Common Divisor (GCD)

To simplify the fraction, we first need to find the greatest common divisor of the numerator and the denominator. The greatest common divisor is the largest number that can divide both numbers. Here, the numbers are 9 and 15.

Step 2: List the Divisors

List the divisors for each number: - Divisors of 9: 1, 3, 9 - Divisors of 15: 1, 3, 5, 15 The common divisors are 1 and 3.

Step 3: Determine the Greatest Common Divisor

From the list of common divisors (1 and 3), the greatest common divisor is 3.

Step 4: Divide Both the Numerator and the Denominator

Divide both the numerator and the denominator of the fraction by their greatest common divisor to simplify the fraction:\[\frac{9 \div 3}{15 \div 3} = \frac{3}{5}\]

Step 5: Finalize the Simplification

After performing the division, the fraction simplifies to \( \frac{3}{5} \). There are no common divisors for the numbers 3 and 5 other than 1, which means \( \frac{3}{5} \) is in its simplest form.

Key Concepts

Greatest Common Divisor (GCD)

The greatest common divisor, or GCD, is a key concept in simplifying fractions. It's the largest number that divides two numbers without leaving a remainder. When you simplify a fraction, you are trying to find the GCD of the numerator and the denominator.
For example, in the fraction \( \frac{9}{15} \), you need to identify the GCD of 9 and 15 to simplify it. To do this, list out all the divisors of each number. Divisors of a number are integers that can divide that number without leaving a remainder. For 9, they are 1, 3, and 9. For 15, they are 1, 3, 5, and 15.
Since 1 is always a divisor for any number, look for the largest common number that divides both 9 and 15, which in this case is 3. Use this GCD to divide both the numerator and the denominator, in order to simplify the fraction.

Numerator and Denominator

In any fraction, you'll find two parts: the numerator and the denominator. The numerator is the top number in the fraction, and it represents how many parts of a whole are being considered. The denominator, on the other hand, is the bottom number, signifying the total number of equal parts the whole is divided into.
In the fraction \( \frac{9}{15} \), 9 is the numerator and 15 is the denominator. Understanding these terms is crucial because when you simplify a fraction, you are actually trying to make the relationship between the numerator and denominator more straightforward.
By the end of the simplification process, you want a fraction where the numerator and the denominator are as small as possible, while still retaining the same value as the original fraction. This is achieved by dividing both the numerator and the denominator by their greatest common divisor.

Mathematical Divisors

Divisors are numbers you can divide another number by, without any remainder. Understanding divisors plays a vital role in simplifying fractions because they help find the greatest common divisor. Once you have the divisors listed for both numbers in a fraction, identifying the common ones becomes easy.
For the numbers 9 and 15 in the exercise, the divisors are:

• Divisors of 9: 1, 3, 9
• Divisors of 15: 1, 3, 5, 15

The common divisors shared by these numbers are 1 and 3. By picking the largest of these common divisors, which is 3, you can simplify the fraction. Dividing both the numerator and denominator by their GCD, 3, turns \( \frac{9}{15} \) into \( \frac{3}{5} \), which is its simplest form.