Problem 1 - Algebra A Combined Function

Question

Write each decimal as a fraction. $$ 0.6 $$

Step-by-Step Solution

Answer

0.6 as a fraction is \( \frac{3}{5} \).

Step 1: Understanding the Decimal

Recognize that the decimal 0.6 is in the tenths place. This means it can be represented as 6 tenths.

Step 2: Set Up the Initial Fraction

Write 0.6 as a fraction over 10. This represents the decimal as a fraction: \( \frac{6}{10} \).

Step 3: Simplify the Fraction

Check if the fraction \( \frac{6}{10} \) can be simplified by finding the greatest common divisor (GCD) of 6 and 10. The GCD is 2. Divide both the numerator and the denominator by 2: \( \frac{6 \div 2}{10 \div 2} = \frac{3}{5} \).

Step 4: Conclusion

The decimal 0.6 as a simplified fraction is \( \frac{3}{5} \).

Key Concepts

Converting Decimals to Fractions

Converting decimals into fractions is simpler than you might think! Every decimal number tells us something about its place value, and that's the key to converting it. For example, consider 0.6. The digit 6 is in the "tenths" position, meaning it represents 6 tenths. This can be written as the fraction \( \frac{6}{10} \). This step is crucial because it translates the decimal into a fraction form.

• Identify the place value of the decimal.
• Write the digit over the respective place value. For 0.6, it's \( \frac{6}{10} \).

Switching from a decimal to a fraction is essentially just a matter of understanding and using place values! This method remains consistent regardless of whether you are dealing with 0.6 or any other decimal number.

Simplifying Fractions

Once a decimal has been converted to a fraction, the next crucial step is simplifying it. Simplifying makes the fraction as easy to read and work with as possible. For example, the fraction \( \frac{6}{10} \) can be simplified. But how do we simplify? First, we need to find a number that both the numerator (6) and denominator (10) can be divided by evenly. That's where the Greatest Common Divisor (GCD) comes into play.

• Check if there's a smaller fraction that represents the same value.
• Divide both the numerator and denominator by the same number to simplify.

The simplified form of \( \frac{6}{10} \) is \( \frac{3}{5} \) because 2 is the largest number that divides both 6 and 10 evenly. Remember, a fraction is fully simplified when its numerator and denominator share no common factors other than 1.

Greatest Common Divisor and Simplifying

The Greatest Common Divisor (GCD) is integral when simplifying fractions. It is the largest number that divides two numbers without leaving a remainder. Determining the GCD helps reduce a fraction to its simplest form. Here's how you can find it:Start by listing the factors of both numbers. The factors of 6 are 1, 2, 3, and 6, while for 10, they are 1, 2, 5, and 10. The largest common factor, aside from 1, is 2.

• Find all factors of both numbers.
• Select the largest factor that both numbers share as the GCD.

Once you have the GCD, divide both the numerator and the denominator by this number to simplify the fraction. It ensures that the fraction is in its simplest form and easy to handle mathematically. Applying this to \( \frac{6}{10} \), the GCD, as noted, is 2, resulting in the simplified fraction \( \frac{3}{5} \).