Problem 62 - Prealgebra

Question

Convert the given decimal to a fraction. Reduce your answer to lowest terms. 0.38

Step-by-Step Solution

Answer

The fraction is \( \frac{19}{50} \).

Step 1: Write the Decimal as a Fraction

Start by expressing the decimal 0.38 as a fraction. The number 0.38 can be written as \( \frac{38}{100} \) because it has two decimal places, which means it is out of 100.

Step 2: Find the Greatest Common Divisor (GCD)

Find the greatest common divisor of the numerator (38) and the denominator (100). We list the factors of each. - Factors of 38: 1, 2, 19, 38 - Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100 The greatest common divisor is 2.

Step 3: Simplify the Fraction

Divide both the numerator and the denominator by their greatest common divisor, 2, to simplify the fraction: \[\frac{38}{100} = \frac{38 \div 2}{100 \div 2} = \frac{19}{50} \] The fraction \( \frac{19}{50} \) is in its simplest form.

Key Concepts

Greatest Common Divisor

When you want to simplify a fraction, finding the greatest common divisor (GCD) is essential. The GCD is the largest number that can equally divide both the numerator and the denominator without leaving a remainder.
It serves as a key tool in reducing fractions to their simplest form. Let's take a closer look at how to find the GCD:

• List the factors: First, list all the factors of the numerator and the denominator. For example, with 38, the factors are 1, 2, 19, and 38. For 100, the factors are 1, 2, 4, 5, 10, 20, 25, 50, and 100.
• Identify common factors: Then, identify the common factors among these lists. Here, the common factors of 38 and 100 are 1 and 2.
• Select the greatest common factor: The greatest among these factors is the GCD. In our example, it is 2.

Understanding the GCD helps you simplify calculations and handle fractions more efficiently. It’s a valuable concept in not just math problems but in various real-world applications as well.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form so that the numerator and denominator are as small as possible while still having the same value. This usually makes the fraction easier to work with and understand.Steps for Simplification:

• Find the GCD: As outlined in the previous section, determine the GCD of the fraction’s numerator and denominator. In this case, we’ve identified it as 2 for the fraction \( \frac{38}{100} \).
• Divide both terms: You then divide the numerator and denominator by this GCD. So, divide 38 and 100 each by 2 to get \( \frac{19}{50} \).
• Check the result: Ensure that the simplified fraction has whole numbers and cannot be reduced further.

This simplification process is crucial in various mathematical problems, helping to keep expressions neat and avoid unnecessary complexities.

Fractions in Lowest Terms

Fractions expressed in their lowest terms are those that cannot be simplified further. This means the numerator and denominator have no greater common factor other than 1.Why Use Lowest Terms?

• Clarity: They offer a clear and concise representation of the fraction’s value, making it easier to compare with other fractions.
• Efficiency: Working with fractions in their simplest form often simplifies mathematical computations and reduces errors.
• Universality: Simplified fractions are standardized, allowing for easier communication and understanding, especially in educational settings and assessments.

When you convert a decimal to a fraction, reducing it to its lowest terms ensures that you are working with the most accurate and simplified version of that number. In the case of \( 0.38 \), we simplify \( \frac{38}{100} \) to \( \frac{19}{50} \), ensuring the fraction reflects its simplest absolute form.