Problem 94 - Elementary Algebra

Question

Use the definition of percent to convert to fractions. $$ 18 \% $$

Step-by-Step Solution

Answer

18% as a fraction is \( \frac{9}{50} \).

Step 1: Understand Percentage

To understand what 18% means, we translate 'percent' as per hundred. Therefore, 18% means 18 out of 100.

Step 2: Convert Percentage to Fraction

Using the definition of percent, we can write 18% as a fraction: \[ \frac{18}{100} \].

Step 3: Simplify the Fraction

We simplify the fraction \( \frac{18}{100} \) by finding the greatest common divisor of 18 and 100, which is 2. We divide both the numerator and the denominator by 2: \[ \frac{18 \div 2}{100 \div 2} = \frac{9}{50} \].

Key Concepts

Simplifying Fractions

Simplifying fractions helps in making them easier to understand and work with. The goal is to reduce the fraction to its simplest form. This means having the smallest possible whole numbers as the numerator and denominator, where both can no longer be divided by any number other than 1.

To simplify a fraction:

• Find a common factor. This is a number that divides both the numerator and the denominator without leaving a remainder.
• Divide both the numerator and the denominator by this common factor.
• Continue dividing by their greatest common divisor until you cannot anymore.

For instance, if we start with the fraction \( \frac{18}{100} \), the aim is to find the greatest possible number that divides both 18 and 100. This makes your fractions cleaner and easier to interpret.

Percent as Per Hundred

The term 'percent' is derived from the Latin word "per centum," meaning "by the hundred." It is used to describe a proportion expressed out of 100. When you say 18%, you are really saying 18 out of 100 or 18 per hundred.

This means percentages are conveniently converted into fractions by placing the percentage number over 100. For example:

• 18% translates to \( \frac{18}{100} \).
• 50% would translate to \( \frac{50}{100} \), which simplifies to \( \frac{1}{2} \).

Understanding percentages as fractions gives a deeper insight into comparisons and calculations, making it clearer, especially when dealing with problems involving proportions and relationships.

Greatest Common Divisor

Finding the greatest common divisor (GCD) is a fundamental step in simplifying fractions. The GCD of two numbers is the largest number that divides both numbers without leaving a remainder. This is crucial for reducing fractions to their simplest form.

To find the GCD:

• List the factors of both numbers.
• Identify the highest factor that appears in both lists.

For example, with the numbers 18 and 100:

• Factors of 18 are 1, 2, 3, 6, 9, and 18.
• Factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, and 100.
• The largest number in both lists is 2, so the GCD is 2.

Using the GCD ensures that you simplify fractions correctly and reach the cleanest form possible.