Problem 25 - Prealgebra

Question

Write each decimal as a fraction in lowest terms. $$0.08$$

Step-by-Step Solution

Answer

0.08 as a fraction in lowest terms is \( \frac{2}{25} \).

Step 1: Convert Decimal to Fraction

First, write the decimal number as a fraction with 1 as the denominator. For the decimal 0.08, this can be expressed as \( \frac{0.08}{1} \).

Step 2: Remove the Decimal Point

To remove the decimal point, multiply both the numerator and the denominator by 100 (since there are two decimal places in 0.08). This gives:\[ \frac{0.08 \times 100}{1 \times 100} = \frac{8}{100} \]

Step 3: Simplify the Fraction

To simplify the fraction \( \frac{8}{100} \), find the greatest common divisor (GCD) of 8 and 100. The GCD is 4. Divide both the numerator and the denominator by 4: \[ \frac{8 \div 4}{100 \div 4} = \frac{2}{25} \]

Step 4: Final Step: Verify the Simplification

Re-check if \( \frac{2}{25} \) is in its lowest terms. The numerator 2 and the denominator 25 have no common divisors other than 1. Therefore, \( \frac{2}{25} \) is the simplest form.

Key Concepts

Lowest Terms

Converting a decimal into a fraction in its lowest terms means simplifying it as much as possible, so the fraction cannot be reduced further. When a fraction is in lowest terms, the only common divisor of the numerator and the denominator should be 1.

To achieve this, follow these steps:

• First, convert the decimal to a fraction. For instance, 0.08 can be written as \( \frac{8}{100} \).
• Check both numbers for any common factors.
• If the only common factor is 1, then the fraction is already in its lowest terms. If they have a greater common factor, further simplification is required.

This ensures that the fraction is in its simplest form, making it easier to work with in calculations and most succinctly representing the numerical value.

Simplifying Fractions

Simplifying a fraction involves reducing it to its most basic form, such that the numerator and denominator are the smallest possible whole numbers. This process often reveals the inherent simplicity of mathematical relationships.

Here's how you simplify fractions:

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

For example, with the fraction \( \frac{8}{100} \), the GCD is 4. Dividing both by 4 simplifies it to \( \frac{2}{25} \).

By simplifying fractions, we make them more convenient for further mathematical operations and comparison with other fractions.

Greatest Common Divisor (GCD)

The greatest common divisor (GCD) is the largest whole number that divides two numbers without leaving a remainder. It's a key tool in simplifying fractions and ensuring they are in their lowest terms.

To find the GCD:

• List the factors of both the numerator and the denominator.
• Identify the largest number that appears in both lists.

For instance, when simplifying \( \frac{8}{100} \), the factors of 8 include 1, 2, 4, 8 and the factors of 100 include 1, 2, 4, 5, 10, 20, 25, 50, 100. The biggest common factor is 4—the GCD. Thus, dividing both by 4 gives \( \frac{2}{25} \).

Finding the GCD not only aids in simplification but also ensures the fraction is represented as cleanly as possible.