Problem 24 - Prealgebra

Question

Express each percent as a fraction or mixed number in simplest form and as a decimal. $$18 \%$$

Step-by-Step Solution

Answer

18% equals \(\frac{9}{50}\) as a fraction and 0.18 as a decimal.

Step 1: Converting Percent to Fraction

To convert a percentage to a fraction, you divide the percentage by 100. Thus, 18% becomes \( \frac{18}{100} \).

Step 2: Simplifying the Fraction

We simplify the fraction \( \frac{18}{100} \) by finding the greatest common divisor (GCD) of 18 and 100, which is 2. So, divide both the numerator and the denominator by 2: \( \frac{18 \div 2}{100 \div 2} = \frac{9}{50} \). Thus the simplest form as a fraction is \( \frac{9}{50} \).

Step 3: Converting Percent to Decimal

To convert a percent into a decimal, divide by 100. So, 18% as a decimal is \( \frac{18}{100} = 0.18 \).

Key Concepts

Simplifying Fractions

Simplifying fractions is an essential skill in mathematics that involves reducing fractions to their simplest form. A fraction is composed of a numerator (top number) and a denominator (bottom number). The goal of simplifying is to make both numbers as small as possible while keeping the same value as the original. This is done by dividing both the numerator and the denominator by a common factor they share.
Finding such a common factor becomes easier once you understand the Greatest Common Divisor (GCD). When a fraction is simplified, it becomes more straightforward to understand and use in further calculations or comparisons.

• For example, with the fraction \(\frac{18}{100}\), you find that both numbers can be divided by 2.
• This simplifies it down to \(\frac{9}{50}\), which cannot be simplified further since 9 and 50 do not have any common factors other than 1.

Thus, \(\frac{9}{50}\) is the simplest form of the fraction.

Converting Percents to Decimals

Converting percents to decimals is a straightforward process that involves dividing the percentage by 100. This method reflects the definition of a percent, which essentially means per hundred. With this in mind, converting a percent to a decimal simply means moving the decimal point two places to the left.
Taking 18% as an example:

• Start with the percent value: 18%.
• To convert it to a decimal, divide by 100 or simply move the decimal from 18.0 two places to the left.
• You get \(0.18\) as the decimal equivalent.

Understanding this conversion between percents and decimals is useful in various real-world applications, such as calculating interest rates, comparing data, and performing mathematical operations.

Greatest Common Divisor (GCD)

The Greatest Common Divisor (GCD) is a fundamental concept in number theory used widely to simplify fractions. GCD is the largest positive integer that can divide two or more integers without leaving a remainder. Knowing how to find the GCD can help you efficiently reduce fractions to their simplest form.
Let's look at how it works with the numbers 18 and 100 from our fraction example:

• List the factors of 18 (1, 2, 3, 6, 9, 18) and 100 (1, 2, 4, 5, 10, 20, 25, 50, 100).
• Identify the largest common factor, which is 2.
• Divide both the numerator, 18, and the denominator, 100, by 2 to simplify the fraction \(\frac{18}{100}\) to \(\frac{9}{50}\).

This process results in fractions that are easier to interpret and compare, making GCD an invaluable tool in mathematics.