Problem 50 - Algebra and Trigonometry

Question

Express each repeating decimal as a fraction in lowest terms. $$ 0 . \overline{529} $$

Step-by-Step Solution

Answer

The repeating decimal \(0.\overline{529}\) can be expressed as a fraction in lowest terms as \(529/999\).

Step 1: Assign a variable to the repeating decimal

Let's take a variable \(x\) to represent the decimal \(0.529529...\), thus, \(x = 0.529529...\)

Step 2: Multiply the equation to shift the repeating part

Then, you need to shift the decimal point of \(x\) to right just before the repeating part starts. Since the repeating part '529' includes three digits, multiply both sides of the equation by \(1000. Thus, 1000x = 529.529529...\) now.

Step 3: Subtract the original equation from the new one

Subtract the original equation from the new equation (1000x - x = 529.529529... - 0.529529...), which simplifies to \(999x = 529\), thus, \(x = 529/999\)

Step 4: Simplify the fraction

The fraction can not be reduced further since the greatest common divider (GCD) of 529 and 999 is 1. Thus, the decimal is already expressed in lowest terms as \(529/999\).

Key Concepts

Expressing Decimals as Fractions

Understanding how to express repeating decimals as fractions is essential for solving a variety of mathematical problems. When dealing with a repeating decimal like 0.\(\overline{529}\), the goal is to find a fraction that represents the same value.

To approach this, you can assign a variable to the repeating decimal. For instance, let \(x = 0.\overline{529}\). By multiplying the equation by a power of 10 that aligns with the repeating cycle—here 1000, because there are three digits in the repeating segment—you effectively 'move' the decimal point so that it lines up in a way that subtraction will eliminate the repeating part. In this case, 1000x would equal 529.\(\overline{529}\).

Upon subtracting the original variable (x) from this new equation, the repeating decimals cancel out, leaving a simpler equation: \(999x = 529\). This can then be directly converted into a fraction.

Simplifying Fractions

Once you've expressed the repeating decimal as a fraction, the next step is to check if the fraction can be simplified. A fraction is considered to be in lowest terms when the numerator and the denominator have no common factor other than 1. In our example, the fraction obtained is \(\frac{529}{999}\).

To simplify a fraction, you would normally look for the greatest common divisor (GCD) of the numerator and the denominator and then divide both by that number. If the GCD is 1, it indicates that the fraction cannot be reduced any further, which is true for our example. Therefore, \(\frac{529}{999}\) is in its simplest form.

Greatest Common Divisor

The Greatest Common Divisor (GCD) of two numbers is the largest number that divides both of them without leaving a remainder. Finding the GCD is crucial when you're trying to simplify fractions. To find the GCD of two numbers, you can use different methods such as prime factorization, the Euclidean algorithm, or a GCD function in a calculator.

Why the GCD Matters

When you identify the GCD of the numerator and denominator in a fraction, you can divide both by this number to reduce the fraction to its simplest form. If the GCD is anything greater than 1, the fraction can be simplified; if it is 1, the fraction is already as simple as it gets. In our problem, the GCD of 529 and 999 is 1, confirming that the fraction \(\frac{529}{999}\) cannot be simplified further and ensuring the solution is in its simplest form.