Problem 60 - Prealgebra

Question

Explain why percents are rational numbers.

Step-by-Step Solution

Answer

Percents can be expressed as fractions of integers, making them rational numbers.

Step 1: Understanding Percents

Percents represent parts out of a hundred. For example, 45% is equivalent to 45 out of 100.

Step 2: Converting Percent to Fraction

To show that a percent is a rational number, convert it to a fraction. For example, 45% can be written as \( \frac{45}{100} \).

Step 3: Simplifying the Fraction

Further simplify the fraction \( \frac{45}{100} \) by dividing the numerator and the denominator by their greatest common divisor, which is 5, resulting in \( \frac{9}{20} \).

Step 4: Defining a Rational Number

A rational number is any number that can be expressed as the fraction of two integers (\( a \) and \( b \)) where \( b eq 0 \).

Step 5: Validating the Criteria for a Rational Number

Since \( \frac{9}{20} \) (derived from 45%) is a fraction composed of two integers, it meets the definition of a rational number.

Key Concepts

Understanding Percentages

Percentages are a way of expressing a number as a fraction of 100. This makes it easy to compare different quantities as they all relate to the same base number, which is 100. When you see a percentage, for example, 45%, it simply means 45 out of 100.

• The word "percent" comes from the Latin phrase "per centum," which means "by the hundred."
• Percentages are used widely in statistics, finance, and other fields to express proportions and relationships.

Understanding this basic principle helps in converting percentages to other forms, such as fractions or decimals, which can further aid in various mathematical and real-world applications.

Fractions and Their Role

A fraction represents a part of a whole and consists of two numbers: a numerator and a denominator. In the case of converting percentages, the fraction simply represents the same part of a hundred.
Let's consider the percentage 45%. When converted into a fraction, it is written as \( \frac{45}{100} \). This representation allows us to see the percentage as a ratio or relationship between two numbers.
Fractions are powerful tools in mathematics because they can easily be added, subtracted, multiplied, and divided. Moreover, they provide a clear representation of numbers in terms of parts of a whole.

Simplifying Fractions Made Easy

Simplifying fractions is the process of making them as simple as possible by finding the greatest common divisor (GCD) of the numerator and the denominator.

• Take the fraction \( \frac{45}{100} \), derived from 45%.
• Find the largest number that can divide both 45 and 100 without a remainder; this is 5.
• Divide both the numerator and denominator by 5 to simplify:\[ \frac{45 \div 5}{100 \div 5} = \frac{9}{20} \]

By simplifying, we present the fraction in its most concise form but retain its original value, making calculations and comparisons easier. This step is crucial in validating that percentages are indeed rational numbers.

Mathematical Conversion: Linking Concepts

Mathematical conversion involves changing one form of a number, such as a percentage, into another form, like a fraction or decimal. This ability to convert between different representations is vital in mathematics for problem-solving and understanding relationships.

• Start with your percentage, such as 45%.
• Convert it to a fraction by placing it over 100: \( \frac{45}{100} \).
• Simplify the fraction if possible, as shown: \( \frac{9}{20} \).
• Recognize that this results in a rational number.

By understanding and practicing these conversions, you strengthen your ability to look at numbers from different perspectives, each useful in specific situations. This not only demystifies percentages but also enhances numerical problem-solving skills across various disciplines.