Problem 23 - Elementary Algebra Within Reach

Question

Convert the percent to a fraction. $$\frac{1}{2} \%$$

Step-by-Step Solution

Answer

\(\frac{1}{2} \% = \frac{1}{200}\) as a fraction.

Step 1: Understand the percentage as a fraction

Every percentage can be written as a fraction by dividing it by 100. This is because 1 percent equals to 1/100.

Step 2: Convert the percentage fraction

We want to convert \( \frac{1}{2} \% \) to a fraction. Since 'percent' literally means 'for every hundred', we need to divide our fraction by 100. Therefore, the conversion will be \(\frac{1}{2} \div 100 \) or \(\frac{1}{2} \times \frac{1}{100} \).

Step 3: Simplify the Fraction

Multiplying the fractions, we get \(\frac{1}{2} \times \frac{1}{100} = \frac{1}{200} \). This cannot be simplified any further, so \(\frac{1}{200}\) is our final answer.

Key Concepts

Understanding Percentages

To grasp how percentages work, it's important to remember what a percentage represents. A percentage shows a part out of 100. The word itself comes from the Latin "per centum," which means "by the hundred." This is why every percentage can be turned into a fraction with a denominator of 100.
If you see 50%, it means 50 out of 100, which is just another way of saying 1/2 as a fraction. This basic principle helps us understand percentages more clearly and makes it easier to translate them into fractions.

Fraction Operations

Working with fractions involves a few simple arithmetic operations: addition, subtraction, multiplication, and division. When converting percentages to fractions, you often use multiplication and division.
For example, converting \(\frac{1}{2} \%\) into a fraction requires dividing by 100 or, equivalently, multiplying by \(\frac{1}{100}\). These operations are essential in fraction manipulation, and are the key to converting percentages effectively.

Simplifying Fractions

Once you convert a percentage to a fraction, simplifying it is the next step. Simplifying means adjusting the fraction to its smallest possible denominator while maintaining the same value.
Different techniques are used to simplify fractions:

• Find the greatest common divisor (GCD) of the numerator and denominator.
• Divide both by their GCD.

In this exercise, the fraction \(\frac{1}{200}\) is already in its simplest form since 1 and 200 share no common factors other than 1.

Percent to Fraction Conversion

Converting percentages like \(\frac{1}{2} \%\) to fractions involves understanding "percent" as "per hundred." When the percentage is a fraction itself, such as \(\frac{1}{2} \%\), we need to adjust by scaling down suitably.
Here's how it's done: take \(\frac{1}{2}\), multiply it by \(\frac{1}{100}\). This technique allows you to convert any percentage into a fraction simply by considering it 'per hundred.' The fraction obtained, \(\frac{1}{200}\), cannot be reduced further, marking the end of the conversion process. This simple yet effective approach is core to understanding percent to fraction conversion.