Problem 11 - Elementary Algebra

Question

For exercises 1-12, rewrite the decimal number as a fraction. Simplify the fraction to lowest terms. $$ 0.002 $$

Step-by-Step Solution

Answer

0.002 as a fraction is \(\frac{1}{500}\).

Step 1: Identify the decimal

First, identify the given decimal number. Here, the decimal number is 0.002.

Step 2: Write the decimal as a fraction

To convert a decimal to a fraction, place the decimal number over its place value. The number 0.002 is in the thousandths place, so you can write it as \(\frac{2}{1000}\).

Step 3: Simplify the fraction

Simplify the fraction \(\frac{2}{1000}\) by dividing both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 2 and 1000 is 2. When you divide both by 2, you get \(\frac{1}{500}\).

Key Concepts

Simplifying Fractions

Simplifying a fraction means reducing it to its simplest form. This happens when the numerator (the top number) and the denominator (the bottom number) can no longer be divided by the same number, except for 1. For example:

• We start with the fraction \(\frac{2}{1000}\).
• Identify the greatest common divisor (GCD) of 2 and 1000, which is 2.
• We divide both the numerator and the denominator by their GCD: \(\frac{2 \, / \, 2}{1000 \, / \, 2} = \frac{1}{500}\).

So, \(\frac{2}{1000}\) simplifies to \(\frac{1}{500}\), meaning it cannot be simplified further. Simple steps like these make understanding fractions much easier.

Place Value

Place value is crucial when converting decimals to fractions. Each digit in a decimal number represents a certain place value:

• Decimal 0.1 is 'one-tenth', which can be written as \(\frac{1}{10}\).
• Decimal 0.01 is 'one-hundredth', which can be written as \(\frac{1}{100}\).
• And decimal 0.001 is 'one-thousandth', represented as \(\frac{1}{1000}\).

In our example, the decimal 0.002 is in the thousandths place. This means we write it as \(\frac{2}{1000}\). Understanding place value helps to easily convert any decimal into a fraction.

Greatest Common Divisor (GCD)

The Greatest Common Divisor (GCD) is important for simplifying fractions. The GCD of two numbers is the largest number that divides both of them without leaving a remainder. Here’s a quick guide to finding the GCD:

• List the factors of each number: For 2, the factors are 1, 2. For 1000, the factors are 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, 1000.
• Identify the common factors: For 2 and 1000, both share the factors 1 and 2.
• Select the largest common factor: The GCD is 2.

Use the GCD to simplify fractions: Divide both the numerator and the denominator by the GCD. For \(\frac{2}{1000}\), the GCD is 2, so it becomes \(\frac{1}{500}\). Once simplified, the fraction represents the same value, just in its simplest form.