Problem 48 - Prealgebra

Question

Change each decimal to a fraction, and then reduce to lowest terms. $$0.0625$$

Step-by-Step Solution

Answer

0.0625 as a fraction in simplest form is \( \frac{1}{16} \).

Step 1: Write as a Fraction

To convert the decimal 0.0625 to a fraction, recognize it is in the ten-thousandths place. Therefore, 0.0625 can be written as \( \frac{625}{10000} \).

Step 2: Simplify the Fraction

Find the greatest common divisor (GCD) of 625 and 10000. The GCD is 625. Divide both the numerator and denominator by 625 to simplify: \( \frac{625 \div 625}{10000 \div 625} = \frac{1}{16} \).

Step 3: Verify the Simplified Fraction

Check that the simplified fraction \( \frac{1}{16} \) is indeed equivalent to the original decimal. Perform the division \( 1 \div 16 = 0.0625 \), confirming that the simplification is correct.

Key Concepts

Understanding the Greatest Common Divisor (GCD)

When simplifying fractions, the greatest common divisor (GCD) plays a crucial role. The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder. Finding the GCD helps to reduce fractions to their simplest form. Here's how you can determine the GCD:

• Identify Factors: Start by listing out all factors of both the numerator and the denominator.
• Find the Common Factors: Look for numbers that appear in both lists of factors.
• Choose the Greatest: The largest common factor is the GCD.

For example, to find the GCD of 625 and 10000, you can discover that their shared factors include 1, 5, 25, 125, and 625. Here, 625 is the largest, so it is the GCD. Once discovered, it can be used to simplify the fraction by dividing both the top and bottom by this number.

The Art of Simplifying Fractions

Simplifying fractions means reducing them to their simplest form, where the numerator and denominator have no common divisors other than 1. Here's a step-by-step way to simplify a fraction:

• Determine the GCD: Find the greatest common divisor of the numerator and denominator.
• Divide to Reduce: Divide both the numerator and the denominator by the GCD.
• Resulting Fraction: The fraction you now have is in its simplest form.

Take the fraction \(\frac{625}{10000}\) as an example. By dividing both parts by their GCD, 625, the fraction simplifies to \(\frac{1}{16}\). Remember, a simplified fraction is easier to work with and compare. Simplifying makes recognizing equivalent fractions straightforward too.

Understanding Equivalent Fractions

Equivalent fractions represent the same value or proportion, even if their numerators and denominators are different. When converting decimals to fractions and simplifying them, you're often looking at equivalent fractions in disguise.

How to Identify Equivalent Fractions

• Cross-Multiply: Compare two fractions by cross-multiplying. If the cross-products are equal, the fractions are equivalent.
• Common Denominators: Rewrite fractions with the same denominator. If the numerators are equal, the fractions are equivalent.

For instance, \(\frac{625}{10000}\) simplifies to \(\frac{1}{16}\). Both fractions are equivalent, meaning they both express the same value, 0.0625, but \(\frac{1}{16}\) is in its simplest form.