Problem 32 - Algebra A Combined Function

Question

Simplify each fraction. See Examples 8 through \(10 .\) $$ \frac{455}{525} $$

Step-by-Step Solution

Answer

The fraction \( \frac{455}{525} \) simplifies to \( \frac{13}{15} \).

Step 1: Identify the greatest common divisor

To simplify the fraction \( \frac{455}{525} \), we first need to find the greatest common divisor (GCD) of the numerator and the denominator. We list the factors of each number. The factors of 455 are 1, 5, 7, 13, 35, 65, 91, and 455. The factors of 525 are 1, 3, 5, 7, 15, 21, 25, 35, 75, 105, 175, 525. The largest common factor is 35.

Step 2: Divide both numerator and denominator by their GCD

Now that we know the GCD is 35, we divide both the numerator and the denominator of the fraction by 35. We perform the division as follows: \( \frac{455}{35} = 13 \) and \( \frac{525}{35} = 15 \).

Step 3: Rewrite the simplified fraction

After dividing both terms by the GCD, the fraction simplifies to \( \frac{13}{15} \). This is the simplest form of the original fraction.

Key Concepts

Greatest Common Divisor (GCD)

The greatest common divisor (GCD) is a key tool in simplifying fractions. It is the largest number that can evenly divide two numbers without leaving a remainder. Finding the GCD helps us reduce a fraction to its simplest form, maintaining its value while using smaller, more manageable numbers.

When simplifying a fraction like \( \frac{455}{525} \), we start by finding the GCD of its numerator and denominator. This involves:

• Finding all the factors of 455, which are: 1, 5, 7, 13, 35, 65, 91, and 455.
• Finding all the factors of 525, which are: 1, 3, 5, 7, 15, 21, 25, 35, 75, 105, 175, and 525.
• Identifying the largest factor common to both lists, which is 35 in this case.

By dividing the numerator and the denominator by the GCD, we simplify the fraction without changing its value. This process ensures efficient calculations and easier comprehension of fraction comparisons.

Numerator and Denominator

In any fraction, the two numbers involved are called the numerator and the denominator. These play distinct roles in representing parts of a whole.

- **Numerator**: This is the top number in a fraction. It represents how many parts of the whole we are considering. For \( \frac{455}{525} \), the numerator is 455.- **Denominator**: This is the bottom number. It signifies into how many equal parts the whole is divided. Here, the denominator is 525.

In simplifying a fraction, it's crucial to treat the numerator and the denominator appropriately, maintaining the fraction's equivalence by altering both equally. By dividing both 455 and 525 by their greatest common divisor, we convert \( \frac{455}{525} \) into \( \frac{13}{15} \), a more straightforward version of the same quantity.

Simplified Fraction

A simplified fraction is one where the numerator and denominator are as small as possible while preserving the fraction's value. Simplification involves dividing both the numerator and denominator by their greatest common divisor (GCD). For instance, \( \frac{455}{525} \) simplifies to \( \frac{13}{15} \) when both are divided by 35, their GCD.

The benefits of dealing with simplified fractions include:

• Making comparison between fractions easier.
• Simplifying arithmetic operations involving fractions.
• Allowing for clearer visualization of the parts of a whole.

Simplified fractions are often preferable in mathematics because they are easier to understand and work with, offering clear insights and reducing potential calculation errors.