Question

Write each percent as a fraction in lowest terms. $$85 \%$$

Short answer

Answer: 85% is equivalent to the fraction \(\frac{17}{20}\) in its lowest terms.

Step-by-step solution

Convert the percentage to a fraction

Divide the given percentage (85%) by 100, and write it as a fraction. This gives: $$\frac{85}{100}$$

Simplify the fraction

Find the greatest common divisor (GCD) of the numerator (85) and the denominator (100), and then divide both numbers by the GCD to simplify the fraction. The GCD of 85 and 100 is 5. So, simplifying the fraction: $$\frac{85}{100} = \frac{85 \div 5}{100 \div 5} = \frac{17}{20}$$ The simplified fraction is \(\frac{17}{20}\).

Key concepts

Percent to Fraction Conversion

Understanding percent to fraction conversion is a key skill in mathematics and everyday life, as it allows you to express quantities in a different way, often making them easier to work with. A percentage represents a number out of 100. To convert a percentage to a fraction, you divide that number by 100.

For example, to convert 85% to a fraction, you divide 85 by 100, resulting in the fraction \(\frac{85}{100}\). This process transforms the percentage into a fraction with a denominator of a hundred, which is another way of saying 'per one hundred.' It's important to know that every percentage can be written as a fraction in this way, providing a simple method to make this conversion. When writing your answers, though, always make sure to simplify the fraction if possible, which leads us to our next key concept: simplifying fractions.

Simplifying Fractions

Simplifying fractions, also known as reducing fractions, is the process of making the fraction as simple as possible. This is done by finding the greatest common divisor (GCD), also known as the greatest common factor (GCF), of the numerator and the denominator, and then dividing both by this number. Simplifying a fraction does not change its value; it merely presents it in the simplest form.

Take the fraction \(\frac{85}{100}\); the GCD of 85 and 100 is 5. To simplify, divide both the numerator (85) and the denominator (100) by the GCD: \(\frac{85 \div 5}{100 \div 5} = \frac{17}{20}\). The fraction \(\frac{17}{20}\) is much easier to understand and work with since it has been reduced to its lowest terms. Whenever you convert a percent to a fraction, always remember to simplify to complete the process effectively.

Greatest Common Divisor

The Greatest Common Divisor (GCD) of two or more integers is the largest positive integer that divides each of the integers without leaving a remainder. Finding the GCD is an essential step in simplifying fractions, as it allows you to reduce them to their lowest terms. There are several methods to find the GCD, such as listing the factors of each number and choosing the highest one that appears in both lists, or using the Euclidean algorithm which is a more efficient process revolving around repeated division.

In our example with the numbers 85 and 100, the GCD is 5 because 5 is the largest number that divides both 85 and 100 without a remainder. Once the GCD is found, you divide both the numerator and the denominator by it to simplify the fraction: \(\frac{85}{100} = \frac{85 \div 5}{100 \div 5} = \frac{17}{20}\). The concept of the GCD is widely used in mathematics, not only in simplifying fractions but also in solving problems involving ratios, least common multiples, and even cryptography.