Problem 20 - Prealgebra

Question

Reduce each fraction to lowest terms. $$\frac{4}{10}$$

Step-by-Step Solution

Answer

The fraction \( \frac{4}{10} \) reduces to \( \frac{2}{5} \).

Step 1: Identify the Greatest Common Divisor (GCD)

To reduce a fraction to its lowest terms, the first step is to find the greatest common divisor (GCD) of the numerator and the denominator. For the fraction \( \frac{4}{10} \), the numbers are 4 and 10. List the factors of each number. The factors of 4 are 1, 2, and 4. The factors of 10 are 1, 2, 5, and 10. The common factors are 1 and 2, thus the GCD is 2.

Step 2: Divide by the GCD

Once the GCD is identified, divide both the numerator and denominator by this number to simplify the fraction. Divide the numerator, 4, and the denominator, 10, by the GCD, which is 2: \( \frac{4 \div 2}{10 \div 2} = \frac{2}{5} \).

Step 3: Verify the Final Fraction

The final step is to verify that the resulting fraction \( \frac{2}{5} \) is in its simplest form. Since 2 and 5 have no common factors other than 1, \( \frac{2}{5} \) cannot be simplified further. Thus, \( \frac{2}{5} \) is the reduced form of \( \frac{4}{10} \).

Key Concepts

Greatest Common Divisor (GCD)

When simplifying fractions, one of the most important steps is to find the greatest common divisor, or GCD. The GCD is the largest number that divides two numbers without a remainder. Identifying the GCD of the numerator and denominator helps in reducing fractions to their simplest form. For example, if we have the fraction \(\frac{4}{10}\), the process begins by listing the factors of each number.

• The factors of 4: 1, 2, 4
• The factors of 10: 1, 2, 5, 10

By comparing these lists, we find the largest common factor, which is 2. Therefore, 2 is the GCD of 4 and 10. With this information at hand, we can proceed to simplify the fraction easily.

Numerator and Denominator

Understanding the roles of the numerator and denominator is vital for fraction manipulation. In a fraction, the numerator is the top number and represents how many parts we have. The denominator is the bottom number, showing into how many equal parts the whole is divided.
In \(\frac{4}{10}\), 4 is the numerator and 10 is the denominator. To simplify this fraction:

• We apply the GCD to both components.
• Divide 4 (numerator) and 10 (denominator) by their GCD (2).

This results in \(\frac{2}{5}\), where both the numerator and denominator are reduced to their lowest terms. Knowing how these two elements of a fraction interact makes reducing and working with fractions much simpler.

Fraction Reduction

Fraction reduction, also known as simplifying fractions, is the process of bringing the fraction to its simplest form. This is accomplished by dividing the numerator and the denominator by their GCD.
In our example, the initial fraction \(\frac{4}{10}\) is reduced by dividing both the numerator and denominator by 2, the GCD we identified:

• Divide the numerator 4 by 2 to get 2.
• Divide the denominator 10 by 2 to get 5.

Hence, \(\frac{4}{10}\) is simplified to \(\frac{2}{5}\). This ensures the fraction represents the same value in its simplest form. A fraction is considered fully reduced when no number other than 1 can divide both the numerator and denominator evenly. Simplifying fractions aids in clearer and more efficient mathematical calculations.