Problem 42 - Prealgebra

Question

Change each percent to a fraction in lowest terms. $$2 \%$$

Step-by-Step Solution

Answer

The fraction equivalent of 2% in lowest terms is \(\frac{1}{50}\).

Step 1: Understand the Percent

The first step is to understand that the percent symbol (%) means out of 100. Therefore, \(2\%\) is interpreted as \(\frac{2}{100}\).

Step 2: Simplify the Fraction

The next step is to simplify the fraction \(\frac{2}{100}\). We do this by finding the greatest common divisor (GCD) of 2 and 100, which is 2. We divide both the numerator and the denominator by their GCD: \(\frac{2 \div 2}{100 \div 2} = \frac{1}{50}\).

Step 3: Verify in Simplest Form

Finally, verify that \(\frac{1}{50}\) is in its simplest form. Since the greatest common divisor of 1 and 50 is 1, the fraction \(\frac{1}{50}\) is already in lowest terms.

Key Concepts

Understanding Fractions

Fractions are way to express parts of a whole or a division of quantities. Every fraction is composed of two parts: the numerator and the denominator. The numerator is the top number, which indicates the number of equal parts being considered. The denominator is the bottom number, representing the total number of equal parts the whole is divided into.
For instance, when you see the fraction \(\frac{2}{100}\), it means that you have 2 parts out of a total of 100 parts. This concept helps in expressing values that are less than one, allowing for easy comparison of quantities.
Fractions are foundational in mathematics as they enable calculations involving division, which is essential for solving many real-world problems like dividing resources, splitting bills, or converting percentages.

Simplifying Fractions

Simplifying fractions is the process of reducing a fraction to its simplest form so that the numerator and denominator are as small as possible and have no common factors other than 1. This is important because it makes fractions easier to understand and compare.
For example, simplifying \(\frac{2}{100}\) involves finding a number that divides both 2 and 100 evenly. Once we find the greatest common divisor (GCD), which in this case is 2, we divide both the numerator and the denominator by this number. This gives us \(\frac{1}{50}\).
A fraction is considered to be in its simplest form when the only common factor between the numerator and denominator is 1. Simplifying fractions helps in performing arithmetic operations more efficiently and facilitates better understanding.

Greatest Common Divisor (GCD)

The greatest common divisor (GCD) is a key concept when working with fractions, particularly in the process of simplification. The GCD of two numbers is the largest number that divides both of them without leaving a remainder.
To find the GCD of 2 and 100, you consider all the divisors of these two numbers and choose the greatest one. In our example, 1 and 2 are the divisors of 2, while the divisors of 100 include 1, 2, 4, 5, 10, and so on, up to 100. The common divisor in both lists is 1 and 2, and among them, 2 is the greatest.
Using the GCD to simplify fractions ensures they are reduced to their simplest form, which is crucial for clarity in solving mathematical problems involving fractions.