Problem 67 - Algebra 1: Concepts and Skills

Question

Write the percent as a fraction or as a mixed number in simplest form. (Skills Review p. 768 ) $$ 25 \% $$

Step-by-Step Solution

Answer

The percentage 25% can be expressed as a fraction in simplest form as \(\frac{1}{4}\)

Step 1: Write the Percent as a Fraction

Express the percentage as a fraction with the percent value as the numerator and 100 as the denominator. So 25% becomes \(\frac{25}{100}\)

Step 2: Simplify the Fraction

Simplify the fraction by identifying the greatest common divisor (GCD) of the numerator and the denominator. In this case, the GCD of 25 and 100 is 25. Divide the numerator and the denominator by the GCD, so \(\frac{25}{100}\) simplifies to \(\frac{1}{4}\)

Key Concepts

Fractions

Fractions are an essential part of mathematics, and they allow us to express quantities that are less than a whole. A simple way to look at a fraction is to consider it as a part of a whole divided into equal pieces. Typically, a fraction consists of:

• A numerator, which is the top number, representing the parts considered.
• A denominator, the bottom number, indicating how many parts the whole is divided into.

For example, in the fraction \(\frac{25}{100}\), 25 is the numerator, showing how many parts we have out of 100, which is the denominator.
When converting a percentage to a fraction, it’s easiest to think of the percentage as "out of 100" since percent literally means "per one hundred." This makes the initial conversion straightforward: put the percent as the numerator and 100 as the denominator.

Greatest Common Divisor

The greatest common divisor (GCD) is crucial for simplifying fractions. It is the largest positive integer that divides both the numerator and the denominator without leaving a remainder. Finding the GCD helps in reducing fractions to their simplest form.

Let's say you want to simplify \(\frac{25}{100}\). You need to find the GCD of 25 and 100. Here's how you can do that:

• List the factors of each number. For 25, the factors are 1, 5, 25. For 100, they are 1, 2, 4, 5, 10, 20, 25, 50, 100.
• Identify the largest common factor from these lists, which in this case is 25.

The GCD of 25 and 100 is 25, allowing us to divide both the numerator and the denominator by this number to simplify the fraction.

Simplifying Fractions

Simplifying fractions is the process of reducing them to their simplest form where the numerator and the denominator share no common factors other than 1.
Here’s how you simplify \(\frac{25}{100}\):

• Identify the GCD of the numerator and the denominator, which we found to be 25.
• Divide both the numerator and the denominator by the GCD: \(\frac{25 \div 25}{100 \div 25} = \frac{1}{4}\).

Once simplified, \(\frac{1}{4}\) is in its simplest form because 1 and 4 have no common divisors besides 1.

Simplifying is an important skill as it makes fractions easier to understand and compare. It's always a good practice to express fractions in their simplest form to facilitate clearer communication and smoother calculations.