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The concept of the greatest common divisor (GCD) is essential when simplifying fractions. The GCD of two numbers is the largest number that can perfectly divide both numbers without leaving a remainder.
For instance, in the case of simplifying the fraction \( \frac{65}{100} \), finding the GCD helps us reduce the fraction to its simplest form.
To determine the GCD of 65 and 100, you need to list the factors of each number:

• Factors of 65: 1, 5, 13, 65
• Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100

Shared factors are 1 and 5. The greatest is 5. Therefore, the GCD of 65 and 100 is 5. Once you have the GCD, it becomes possible to simplify the fraction, making it easier to work with.

Simplifying fractions is a process where a fraction is expressed in its simplest form. This means reducing the fraction as much as possible, so there's no common divisor between the numerator and the denominator other than 1.
For the fraction \( \frac{65}{100} \), we previously found that the greatest common divisor is 5. Using this, we divide both the numerator and the denominator by 5:

• Numerator: \( 65 \div 5 = 13 \)
• Denominator: \( 100 \div 5 = 20 \)

This results in the simplified fraction \( \frac{13}{20} \). Simplification is vital because it makes it easier to perform operations or compare fractions. It can also provide more clarity when interpreting the value represented by the fraction.

Writing percents as fractions is a helpful skill in math because it allows you to understand and work with percentages more precisely. When you see a percentage like 65%, it is a representation of how much out of a total of 100.
To convert 65% to a fraction, recall that "percent" essentially means "per 100." Thus, 65% becomes \( \frac{65}{100} \). This makes it easier to see the part-to-whole relationship explicitly.
Once you have it as a fraction, you can simplify it to make calculations smoother. The conversion from percent to fraction is often used in calculations involving discounts, interest rates, and probability, among other applications. Understanding each step of converting percentages to fractions and simplifying them is key to solving real-world problems efficiently.