Problem 40 - Prealgebra

Question

Change each percent to a fraction in lowest terms. $$25 \%$$

Step-by-Step Solution

Answer

\( 25\% \) as a fraction in lowest terms is \( \frac{1}{4} \).

Step 1: Convert Percent to Fraction

To convert a percentage to a fraction, start by writing the percent as a fraction with 100 as the denominator. So, \( 25\% \) becomes \( \frac{25}{100} \).

Step 2: Simplify the Fraction

Simplify the fraction \( \frac{25}{100} \) by finding the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 25 and 100 is 25.

Step 3: Divide by GCD

Divide both the numerator and the denominator by their GCD (which is 25): \( \frac{25 \div 25}{100 \div 25} = \frac{1}{4} \). Hence, \( 25\% \) as a fraction in lowest terms is \( \frac{1}{4} \).

Key Concepts

Simplifying Fractions

Simplifying fractions is the process of reducing a fraction to its simplest form. A fraction is considered simplified when the numerator (top number) and the denominator (bottom number) have no common factors other than 1. This process helps in making calculations easier and simplifying comparisons.
To simplify a fraction:

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

For example, if you have the fraction \( \frac{25}{100} \), you would first determine that the GCD is 25. Dividing both the numerator and the denominator by their GCD results in the simplified fraction \( \frac{1}{4} \). This means that 25 parts out of 100 are the same as 1 part out of 4.

Greatest Common Divisor

The greatest common divisor, often abbreviated as GCD, is the largest positive integer that evenly divides two or more integers. Finding the GCD is crucial for simplifying fractions.
To find the GCD:

• List the factors of both the numerator and the denominator.
• Identify the largest factor that these numbers share.

In the case of the fraction \( \frac{25}{100} \), the factors of 25 are 1, 5, and 25, and the factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, and 100. The greatest number that divides both is 25, which becomes the GCD. Knowing how to find the GCD makes handling fractions much smoother and is a valuable skill to master in arithmetic.

Percent to Fraction Conversion

Converting a percent to a fraction is an essential skill in mathematics; it's both simple and logical. Percentages denote parts per hundred, which makes them straightforward to convert into fractions.
To convert a percent, follow these simple steps:

• Write the percentage number over 100, as every percent is a fraction with a denominator of 100. For example, 25% becomes \( \frac{25}{100} \).
• Simplify the resulting fraction by finding and dividing by the GCD of the numerator and the denominator. In our example, the GCD is 25, simplifying \( \frac{25}{100} \) to \( \frac{1}{4} \).

This method helps to make percentages more manageable, especially when performing calculations or comparisons, by expressing them as fractions in their simplest form.