Problem 41 - Prealgebra

Question

Change each percent to a fraction in lowest terms. $$4 \%$$

Step-by-Step Solution

Answer

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

Step 1: Convert Percentage to Fraction

Start by understanding that percent means per hundred. Therefore, 4% is equal to \( \frac{4}{100} \).

Step 2: Simplify the Fraction

To simplify \( \frac{4}{100} \), find the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 4 and 100 is 4. Divide both the numerator and the denominator by 4: \( \frac{4}{100} = \frac{4 \div 4}{100 \div 4} = \frac{1}{25} \).

Step 3: Check the Simplification

Verify that \( \frac{1}{25} \) is in its simplest form by ensuring that the numerator and the denominator have no common divisors other than 1. Since there are none, \( \frac{1}{25} \) is the fraction in lowest terms.

Key Concepts

Fraction Simplification

When converting a percentage to a fraction, it's important to simplify the fraction to its lowest terms. This makes calculations easier and the results clearer.

Fraction simplification involves dividing both the numerator and the denominator by their greatest common divisor (GCD). By doing this, you are reducing the fraction until no further division is possible other than by 1.

• The numerator is the top number in a fraction.
• The denominator is the bottom number in a fraction.
• Example: In \( \frac{4}{100} \), 4 is the numerator and 100 is the denominator.

To simplify \( \frac{4}{100} \), identify the GCD of 4 and 100 and divide both by this number, resulting in \( \frac{1}{25} \). This ensures the fraction is in its lowest form.

Greatest Common Divisor

The Greatest Common Divisor, often abbreviated as GCD, is a crucial concept in fraction simplification. The GCD of two numbers is the largest positive integer that divides each of the numbers without leaving a remainder.

Finding the GCD helps simplify fractions efficiently.

• To find the GCD of two numbers, list their factors or use the Euclidean algorithm.
• Example: The factors of 4 are 1, 2, and 4; the factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, and 100.
• The highest factor they share is 4.

By dividing the numerator and denominator of \( \frac{4}{100} \) by their GCD (which is 4), you simplify it to \( \frac{1}{25} \). This process demonstrates how understanding and using the GCD can make working with fractions much simpler.

Percentages

Understanding percentages is key for converting them into fractions. A percentage represents a number out of 100. Therefore, converting percentages to fractions involves placing the percentage number over 100.

• For example, 4% becomes \( \frac{4}{100} \), where 4 is the percentage and 100 is the base.
• This fraction may then need simplification to express it in the lowest terms possible.

The reason we use 100 is due to historical trading and weights systems, but it's also a handy standard base that simplifies mathematical operations.

Once you have the fraction, you may simplify it as needed, which often involves dividing the numerator and denominator by their GCD. Converting percentages to fractions and simplifying them helps you understand and use numbers with different units effectively.