Problem 12 - Elementary Algebra

Question

For exercises 1-12, rewrite the decimal number as a fraction. Simplify the fraction to lowest terms. $$ 0.004 $$

Step-by-Step Solution

Answer

\( \frac{1}{250} \)

Step 1: Understand the Decimal

The number given is 0.004. Identify its place value. Since the last digit is in the thousandths place, we can represent it as a fraction.

Step 2: Write as a Fraction

Write the decimal 0.004 as a fraction: \( \frac{4}{1000} \).

Step 3: Simplify the Fraction

Simplify the fraction by finding the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 4 and 1000 is 4. \( \frac{4 \div 4}{1000 \div 4} = \frac{1}{250} \).

Step 4: Verify the Simplified Fraction

Ensure the fraction \( \frac{1}{250} \) cannot be simplified further. Since 1 and 250 have no common factors other than 1, the fraction is in its lowest terms.

Key Concepts

Simplifying Fractions

To simplify a fraction, you need to find the largest number that divides both the numerator and the denominator evenly. This number is called the Greatest Common Divisor (GCD). For example, to simplify \( \frac{4}{1000} \), we need to find the GCD of 4 and 1000.
In our case, the GCD is 4. So, you divide both the top and bottom numbers by 4: \( \frac{4 \div 4}{1000 \div 4} = \frac{1}{250} \)
Here are the steps:

• Find the GCD.
• Divide the numerator by the GCD.
• Divide the denominator by the GCD.
• Write the simplified fraction.

The key task in simplification is identifying the GCD accurately.

Place Value

Understanding place value is crucial when converting decimals to fractions. In a number like 0.004, each digit has a specific place value. The rightmost '4' is in the thousandths place, meaning it is 4 positions after the decimal point.
Here's a breakdown:

• The digit 0 before the decimal is in the units place.
• The first digit after the decimal is in the tenths place.
• The second digit is in the hundredths place.
• The third digit, which is '4' in our example, is in the thousandths place.

As we move right from the decimal point, each place value is one-tenth of the previous one. So, 0.004 can be written as \( \frac{4}{1000} \). Recognizing place values helps in correctly setting up the fraction.

Greatest Common Divisor (GCD)

Finding the GCD is essential for simplifying fractions. The GCD of two numbers is the largest number that divides both without leaving a remainder.
Here’s how you find the GCD:

• List the factors of each number.
• Identify the common factors.
• Choose the largest common factor.

For example, for numbers 4 and 1000:

• Factors of 4: 1, 2, 4
• Factors of 1000: 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, 1000

The greatest common factor here is 4.
Hence, for the fraction \( \frac{4}{1000} \), we divide both the numerator and the denominator by 4 to get \( \frac{1}{250} \). This is the simplest form of the fraction.