Problem 14 - Algebra 2 and Trigonometry

Question

In \(13-22,\) write each decimal as a common fraction. $$ 0 . \overline{6} $$

Step-by-Step Solution

Answer

The decimal \(0.\overline{6}\) is equal to the fraction \(\frac{2}{3}\).

Step 1: Define the Repeating Decimal

Let the repeating decimal be \( x = 0.\overline{6} \). This means \( x = 0.6666\ldots \), where 6 is the repeated digit.

Step 2: Multiply to Shift Decimal

Since the decimal repeats after one digit, multiply by 10 to align the repeating part with the original number: \( 10x = 6.6666\ldots \).

Step 3: Subtract the Equations

Now, subtract the original \( x \) from this new equation: \[ 10x = 6.6666\ldots \]\[ - x = 0.6666\ldots \]This gives us:\[ 9x = 6 \]

Step 4: Solve for x

To find \( x \), divide both sides of the equation by 9:\[ x = \frac{6}{9} \]

Step 5: Simplify the Fraction

Simplify the fraction \( \frac{6}{9} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 3:\[ x = \frac{6 \div 3}{9 \div 3} = \frac{2}{3} \]

Key Concepts

Repeating decimal

Repeating decimals are numbers that have a digit or a group of digits after the decimal point that continuously repeat without end.
For example, in the decimal number \( 0.\overline{6} \), the digit 6 repeats indefinitely. Identifying and understanding these patterns in numbers is crucial as it helps in converting the decimal into a more manageable and precise form, such as a fraction.

• Repetitions can start immediately after the decimal point or after a few digits.
• The repeated digit or group of digits in a repeating decimal is often denoted by an overline.
• Repeating decimals always have a corresponding fractional representation.

When dealing with repeating decimals, recognizing the pattern of repetition allows for effective algebraic manipulation to convert these numbers into fractions.

Fraction simplification

Fraction simplification is the process of reducing a fraction to its simplest form.
This involves dividing both the numerator (top number) and the denominator (bottom number) by their greatest common divisor (GCD). In the process of converting repeating decimals to fractions, simplification is often the final step to make the result as simple as possible.

• The simpler a fraction is, the easier it is to understand or work within further calculations.
• Simplification does not change the value of the fraction, only its appearance.
• Simplifying fractions is essential for achieving the most precise representation of a number.

For the example of \( \frac{6}{9} \), both 6 and 9 can be divided by 3, their greatest common divisor. Simplifying, we find \( \frac{6}{9} = \frac{2}{3} \).
This demonstrates the importance of simplification in obtaining the most refined representation of a number.

Algebraic manipulation

Algebraic manipulation involves using algebraic techniques to rearrange, solve, or transform mathematical expressions or equations.
It's a powerful tool in the conversion of repeating decimals into fractions.

• Setting the repeating decimal as a variable, often \( x \), helps us keep track of the number and its repeating pattern.
• Equating and then multiplying the variable by powers of ten positions the decimal sequence appropriately, aiding in elimination during subtraction.
• Through subtraction, constants and coefficients are simplified to extract the fraction form of the decimal.

For instance, when given \( x = 0.\overline{6} \), multiplying by 10 shifts the repeating sequence past the decimal,
creating two comparable equations. Subtracting the first equation from the second allows the repeating sequences to cancel out, leaving an algebraic equation that can be easily solved.With algebraic manipulation, complex-looking decimals are converted into simple fractions, making data easier to interpret and utilize.