Problem 24 - Basic College Mathematics

Question

Simplify. $$ \frac{15}{25} $$

Step-by-Step Solution

Answer

\( \frac{3}{5} \)

Step 1: Identify the Greatest Common Divisor (GCD)

Determine the greatest common divisor (GCD) of the numerator (15) and the denominator (25). To find the GCD, list the factors of each number: - Factors of 15: 1, 3, 5, 15 - Factors of 25: 1, 5, 25 The greatest common factor is 5.

Step 2: Divide the Numerator and Denominator by the GCD

Divide the numerator and the denominator of the fraction by their greatest common divisor (5): \ \[ \frac{15 \div 5}{25 \div 5} \]

Step 3: Simplify the Fraction

Perform the division from Step 2: \[ \frac{15 \div 5}{25 \div 5} = \frac{3}{5} \] Therefore, the simplified fraction is \( \frac{3}{5} \).

Key Concepts

Greatest Common Divisor

The Greatest Common Divisor (GCD) is crucial for simplifying fractions. It is the largest number that divides both the numerator and the denominator without leaving a remainder. To find the GCD of two numbers, list out all the factors of each number.

For example:

• Factors of 15: 1, 3, 5, 15
• Factors of 25: 1, 5, 25

The common factors between 15 and 25 are 1 and 5, and the greatest one is 5. Therefore, the GCD of 15 and 25 is 5. This means that both 15 and 25 can be divided evenly by 5.
Using the GCD, we can proceed to simplify the fraction more easily.

Numerator and Denominator

Understanding the numerator and denominator is key to working with fractions. The numerator is the top number of a fraction, while the denominator is the bottom number.

For example, in the fraction \( \frac{15}{25} \) , 15 is the numerator and 25 is the denominator.

• The numerator represents the number of equal parts you have.
• The denominator represents the total number of equal parts into which the whole is divided.

To simplify the fraction, both the numerator and the denominator should be divided by their GCD. This will help reduce the fraction to its simplest form.

Mathematical Division

Mathematical division is the process of determining how many times one number is contained within another. When simplifying fractions, division is used to reduce the numerator and the denominator by their GCD.

For instance, in the given problem:

• Divide the numerator 15 by the GCD 5: \( 15 \div 5 = 3 \)
• Divide the denominator 25 by the GCD 5: \( 25 \div 5 = 5 \)

So, \( \frac{15\div 5}{25\div 5} \) becomes \( \frac{3}{5} \). This simplifies the fraction to \( \frac{3}{5} \).
Division ensures each part of the fraction is minimized appropriately, making it easier to understand and work with.