Problem 32 Simplify. $$ \frac{425}{525}... [FREE SOLUTION] (2024)

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To understand how to simplify a fraction, we start with the concept of the Greatest Common Divisor (GCD). The GCD of two numbers is the largest positive integer that divides both numbers without leaving a remainder.

For example, in the fraction \(\frac{425}{525}\), we need to find the GCD of 425 and 525. By calculating, we find that the GCD is 25. This means that 25 is the largest number that can divide both 425 and 525 perfectly. Knowing the GCD makes it easier to simplify fractions.

Steps to find the GCD:

• Find all factors of each number.
• Identify the common factors.
• Select the largest common factor. In this case, it's 25.

Understanding the GCD is crucial because it helps us reduce fractions to their simplest form.

Prime factorization breaks a number down into its prime number factors, which are numbers greater than 1 that cannot be divided evenly by any number other than 1 and themselves.

Let's perform prime factorization on 425 and 525:

For 425, the prime factors are:
\[ 425 = 5^2 \times 17 \]

For 525, the prime factors are:
\[ 525 = 5^2 \times 21 = 5^2 \times 3 \times 7 \]

Here, we notice the common prime factor is \(5^2 \), which helps us in determining the GCD. Understanding prime factorization not only aids in finding the GCD but also further develops your number sense and ability to simplify fractions.

A simplified fraction is one where the numerator and the denominator have no common factors other than 1. This means the fraction is reduced to its smallest possible form.

To simplify the given fraction \(\frac{425}{525}\):

• First, find the GCD of 425 and 525, which is 25.
• Next, divide both the numerator and the denominator by the GCD.
• Finally, we get \(\frac{425 \div 25}{525 \div 25} = \frac{17}{21}\).

The fraction \(\frac{17}{21}\) is now simplified, as 17 and 21 have no common factors other than 1.

Remember, a simplified fraction is easier to understand and work with, making calculations and comparisons much simpler.