Problem 35 - Algebra for College Students

Question

Change \(0 . \overline{36}\) to reduced \(a / b\) form, where \(a\) and \(b\) are integers and \(b \neq 0\). $$ \frac{4}{11} $$

Step-by-Step Solution

Answer

Repeating decimal \(0.\overline{36}\) equals \(\frac{4}{11}\).

Step 1: Understand the Problem

We want to convert the repeating decimal \(0.\overline{36}\) into the fraction \(\frac{a}{b}\) where \(a\) and \(b\) are integers, and the fraction is in simplest form.

Step 2: Set the Repeating Decimal as a Variable

Let \(x = 0.\overline{36}\). This represents the repeating decimal.

Step 3: Eliminate the Repeating Part

To eliminate the repeating part, multiply both sides of \(x = 0.\overline{36}\) by 100 (since the repeating sequence is two digits long):\[100x = 36.\overline{36}\]

Step 4: Form an Equation

Subtract the first equation from the second to eliminate the repeating part:\[ \begin{align*}100x &= 36.\overline{36} \x &= 0.\overline{36} \\hline99x &= 36\end{align*}\]

Step 5: Solve for x

Divide both sides by 99 to solve for \(x\):\[x = \frac{36}{99}\]

Step 6: Simplify the Fraction

Simplify \(\frac{36}{99}\) by finding the greatest common divisor of 36 and 99. The GCD is 9.Divide both the numerator and the denominator by 9:\[x = \frac{36 \div 9}{99 \div 9} = \frac{4}{11}\]

Step 7: Verify the Solution

Convert \(\frac{4}{11}\) back to a decimal to ensure it equals \(0.\overline{36}\). When you divide 4 by 11, you indeed get \(0.363636...\) confirming the solution is correct.

Key Concepts

Converting Decimals to Fractions

When you encounter a repeating decimal, the goal is often to convert it into a fraction. This process allows us to express the number in a more universally understood format.
To convert a repeating decimal into a fraction we:

• Assign a variable to the repeating decimal.
• Determine the repeating sequence of the digits.
• Multiply the decimal by a power of 10, equal to the number of repeating digits. This helps in aligning the decimals.
• Subtract the resulting equation from the original to eliminate the repeating part.

This step-by-step helps in transforming \(0.\overline{36}\) to \(\frac{36}{99}\). What remains is reducing this fraction.

Simplifying Fractions

Simplifying a fraction involves reducing it to its simplest form, which has the smallest possible numerator and denominator. This makes the fraction easier to work with or understand.
For instance, to simplify \(\frac{36}{99}\):

• Identify the greatest common divisor (GCD) of both 36 and 99.
• Divide both the numerator and denominator by this GCD.

The result is a fraction that cannot be reduced further, hence simpler.

Greatest Common Divisor

The GCD of two numbers is the largest integer that divides both numbers without leaving a remainder. This concept is critical in simplifying fractions.
For \(\frac{36}{99}\), the GCD is 9. Using the divisors of 36 and 99 helps us identify this:

• Divisors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
• Divisors of 99: 1, 3, 9, 11, 33, 99

The common divisors are 1, 3, and 9. The largest of these is 9, making it the GCD. It ensures the fraction is fully simplified by dividing both the numerator and the denominator.

Equivalent Fractions

Equivalent fractions represent the same value even though they may look different. They are generated by either multiplying or dividing both the numerator and denominator by the same number.
The fraction \(\frac{36}{99}\) is equivalent to \(\frac{4}{11}\) after simplification:

• The step involves dividing both 36 and 99 by their GCD of 9.
• This reduces \(\frac{36}{99}\) to \(\frac{4}{11}\).

These fractions, though different in appearance, have the same decimal representation of \(0.\overline{36}\).