Problem 2 - Elementary Algebra

Question

For exercises 1-12, rewrite the decimal number as a fraction. Simplify the fraction to lowest terms. $$ 0.9 $$

Step-by-Step Solution

Answer

0.9 can be written as \( \frac{9}{10} \). This fraction is already simplified.

Step 1 - Write as a fraction

Rewrite the decimal number 0.9 as a fraction by recognizing that 0.9 is equivalent to 9 tenths. Therefore, write it as \( \frac{9}{10} \).

Step 2 - Simplify the fraction

Check if the fraction \( \frac{9}{10} \) can be simplified further. The greatest common divisor (GCD) of 9 and 10 is 1, which means \( \frac{9}{10} \) is already in its simplest form.

Key Concepts

Simplifying Fractions

When we talk about simplifying fractions, we mean reducing them to their simplest form. A fraction is in its simplest form when the numerator (the top number) and the denominator (the bottom number) have no common factors other than 1.
To simplify a fraction, follow these steps:

• Find the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by this GCD.

Let's take an example: Consider the fraction \(\frac{12}{16}\). First, we find the GCD of 12 and 16, which is 4. Next, we divide both the numerator (12) and the denominator (16) by 4. This gives us the simplified fraction \(\frac{3}{4}\). Always check your final fraction to ensure it cannot be simplified further.

Greatest Common Divisor

The greatest common divisor (GCD), also known as the greatest common factor (GCF), is the largest number that can exactly divide both the numerator and the denominator of a fraction. Finding the GCD is crucial for simplifying fractions.
Here's how to find the GCD:

• List all the factors of the numerator.
• List all the factors of the denominator.
• Identify the largest factor that appears in both lists.

For example, to find the GCD of 24 and 36:

• Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
• Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

Common factors: 1, 2, 3, 4, 6, 12. The largest of these is 12. Therefore, the GCD of 24 and 36 is 12.
Using the GCD to simplify fractions: If you have a fraction \(\frac{24}{36}\), you can divide the numerator and the denominator by their GCD (12), resulting in \(\frac{2}{3}\). The fraction is now simplified.

Decimal Equivalence

Decimals can easily be converted to fractions and vice versa. To rewrite a decimal as a fraction, you express the decimal number in the form of \(\frac{a}{10^n}\), where \(n\) is the number of decimal places. Let’s use some examples for better understanding:
Consider the decimal 0.9
1. Rewrite the decimal as a fraction: 0.9 is 9 tenths, which is \(\frac{9}{10}\)

Check if it can be simplified. In this case, \(\frac{9}{10}\) does not need further simplification as the GCD of 9 and 10 is 1.

Another example is 0.25
1. Rewrite the decimal as a fraction: 0.25 is 25 hundredths, which is \(\frac{25}{100}\)

Find the GCD of 25 and 100, which is 25.

Divide both the numerator and the denominator by 25, this gives us \(\frac{1}{4}\).

So, 0.25 is equivalent to \(\frac{1}{4}\). Remember, converting between decimals and fractions is a valuable skill that helps in different areas of math and real-life situations.