Problem 13 - Algebra 2 and Trigonometry

Question

In \(13-22,\) write each decimal as a common fraction. $$ 0.125 $$

Step-by-Step Solution

Answer

0.125 as a fraction is \( \frac{1}{8} \).

Step 1: Identify the Decimal Place Value

The number 0.125 is a decimal where the last digit, 5, is in the thousandths place. This means 0.125 can be expressed as \( \frac{125}{1000} \).

Step 2: Simplify the Fraction

Now, simplify \( \frac{125}{1000} \). Find the greatest common divisor (GCD) of 125 and 1000. The GCD is 125. Divide both the numerator and the denominator by this GCD: \( \frac{125 \div 125}{1000 \div 125} = \frac{1}{8} \).

Step 3: Verify the Simplified Fraction

Check that \( \frac{1}{8} \) is in its simplest form. The numerator and denominator are coprime (no common factors except 1), confirming \( \frac{1}{8} \) is simplified.

Key Concepts

Simplification of Fractions

Simplifying fractions might seem like a challenging process at first, but it's actually straightforward once you grasp the steps involved. The principal goal here is to express the fraction with the smallest possible whole numbers.
Start by identifying the greatest common divisor (GCD) of the numerator and the denominator. The GCD is the largest number that divides both numbers without leaving a remainder.

• For example, in the fraction \(\frac{125}{1000}\), both 125 and 1000 can be divided by 125. Therefore, 125 is the GCD for these numbers.
• Once the GCD is identified, divide both the numerator and the denominator by this number.
• In our example, dividing both 125 and 1000 by 125 results in the fraction \(\frac{1}{8}\).

By doing this, you ensure that the fraction is in its simplest form, where the numerator and denominator share no further common denominators other than 1.

Place Value in Decimals

Understanding place value in decimals is crucial when converting decimals to fractions. Each digit in a decimal number has a particular place value that determines its worth relative to the other digits.
Consider the decimal 0.125:

• The digit '1' is in the tenths place, representing \(\frac{1}{10}\).
• The digit '2' is in the hundredths place, representing \(\frac{2}{100}\).
• The digit '5' is in the thousandths place, representing \(\frac{5}{1000}\).

Summing these values directly provides us the fraction \(\frac{125}{1000}\).
Recognizing these place values makes it easier to convert decimals into fractions and effectively streamlines the initial step in the conversion process.

Greatest Common Divisor

The greatest common divisor (GCD) is central to simplifying fractions, making it a vital concept in mathematics. It is the largest number that can evenly divide two numbers.
To find the GCD, you can use several methods:

• Listing out all the divisors of both numbers and identifying the largest shared divisor. For instance, the divisors of 125 include 1, 5, 25, and 125. The divisors of 1000 include 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, and 1000. Therefore, the GCD is 125.
• Using the Euclidean algorithm, a quicker method that uses successive divisions.

Understanding and finding the GCD allows you to reduce fractions efficiently. In our exercise, dividing both the numerator and the denominator by the GCD, which is 125, transformed \(\frac{125}{1000}\) into \(\frac{1}{8}\), a far simpler and cleaner fraction.