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Chapter 1: Problem 86

Use the definition of percent to convert to fractions. $$ 97 \% $$

Short Answer

Expert verified
97% is represented as \( \frac{97}{100} \) in fraction form.

Step by step solution

01

Understand the Percent Definition

Percent means per hundred. So, when we have a percentage, we look to express it as a part of 100.
02

Express 97% as a Fraction

Using the definition of percent, express 97% as the fraction \( \frac{97}{100} \).
03

Simplify the Fraction

Check if the fraction can be simplified further by finding the greatest common divisor (GCD) of 97 and 100. Since 97 is a prime number and does not divide 100, the fraction \( \frac{97}{100} \) is already in its simplest form.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Percent Definition
Percent, a term commonly encountered in both mathematics and daily life, originates from the Latin "per centum," which means "by the hundred." It is a way to express numbers as a part of a whole, specifically out of 100. When you see a percentage like 97%, it means 97 parts out of a total of 100 parts.

Understanding this concept helps in converting percentages to fractions, because it is all about recognizing that the percentage symbol (%) represents a denominator of 100. For instance, expressing 97% as a fraction would simply mean writing it as \( \frac{97}{100} \). This is because you are representing 97 parts out of the total 100 parts.
Prime Numbers and Their Role
Prime numbers are the building blocks in mathematics. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. This means it cannot be formed by multiplying two smaller natural numbers.

In the context of simplifying fractions, the role of prime numbers becomes crucial. They help in determining whether a fraction can be reduced. If the numerator or denominator is a prime number and does not divide evenly into the other, the fraction is already in its simplest form.
  • 97 is a prime number because it has no divisors other than 1 and 97.
  • It does not divide 100, which means \( \frac{97}{100} \) remains unsimplified.
When simplifying fractions, identifying prime numbers can quickly tell us if further simplification is possible.
Simplifying Fractions
Simplifying fractions, also known as reducing fractions, is the process of making a fraction as simple as possible. This involves finding the greatest common divisor (GCD) of the numerator and denominator and dividing both by that number.

Here's how to simplify a fraction:
  • Identify the numerator and denominator of the fraction.
  • Find their GCD.
  • Divide both the numerator and the denominator by this GCD.
If the GCD is 1, as in the case where one of the numbers is a prime that doesn't divide the other number, the fraction is already simplified. In our example, since 97 is a prime number and does not divide 100, the fraction remains unchanged as \( \frac{97}{100} \).

Understanding this process helps in making fractions more manageable and is a fundamental skill in various mathematical applications.

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