Problem 62 - Algebra for College Students

Question

$$ 0 . \overline{273} \quad \frac{91}{333} $$

Step-by-Step Solution

Answer

The repeating decimal \(0.\overline{273}\) is equal to \(\frac{91}{333}\).

Step 1: Identify the Problem

We need to convert the given repeating decimal \( 0.\overline{273} \) into a fraction and verify if it simplifies to \( \frac{91}{333} \).

Step 2: Define the Repeating Decimal

Let \( x = 0.273273273\ldots \). This represents the repeating decimal \( 0.\overline{273} \).

Step 3: Eliminate the Repeating Part

Multiply \( x \) by 1000 to shift the decimal point three places to the right: \( 1000x = 273.273273\ldots \).

Step 4: Set Up the Equation

Now, subtract the original \( x = 0.273273\ldots \) from the equation in Step 3: \[ 1000x - x = 273.273273\ldots - 0.273273\ldots \]This simplifies to: \[ 999x = 273. \]

Step 5: Solve for x

Divide both sides of the equation by 999 to solve for \( x \): \[ x = \frac{273}{999}. \]

Step 6: Simplify the Fraction

Simplify \( \frac{273}{999} \) by dividing both the numerator and denominator by their greatest common divisor (GCD), which is 3:- Divide 273 by 3 to get 91.- Divide 999 by 3 to get 333.Thus, \( \frac{273}{999} = \frac{91}{333}. \)

Step 7: Verify the Solution

Since the simplified fraction \( \frac{273}{999} \) equals \( \frac{91}{333} \), the solution is correct.

Key Concepts

Simplifying Fractions

Simplifying fractions is an essential skill in mathematics to ensure numbers are represented in their simplest form. A fraction is simplified when the numerator and the denominator are as small as possible without changing the value of the fraction. This process often involves dividing them by their greatest common divisor (GCD).

For example, with the fraction \( \frac{273}{999} \), both numbers are divisible by 3. By dividing 273 by 3, you get 91, and by dividing 999 by 3, you get 333. This reduces the fraction to \( \frac{91}{333} \), which cannot be simplified further as 91 and 333 have no common factors other than 1.

By practicing simplifying fractions, you improve your understanding of their relationships and make calculations easier.

Greatest Common Divisor (GCD)

The Greatest Common Divisor, or GCD, is the largest number that divides both the numerator and the denominator of a fraction without leaving a remainder. It is fundamental in simplifying fractions since dividing both parts of a fraction by their GCD will result in the simplest form.

Finding the GCD involves several methods. One simple method is listing out the factors of each number and choosing the greatest factor they have in common. However, another more efficient method, especially for larger numbers, is the Euclidean algorithm, which involves dividing the larger number by the smaller number and using remainders to find the result.

Knowing the GCD helps not just in simplifying fractions but also in solving problems involving ratios, and it aids in understanding the concepts of multiples and factors.

Mathematical Proof

A mathematical proof is a logically sound argument that explains why a particular statement is true. In the context of converting repeating decimals to fractions, a mathematical proof often involves algebraic manipulation to show how the decimal can be represented as a fraction.

For example, to convert the repeating decimal \( 0.\overline{273} \) to a fraction, you can define it as \( x = 0.273273273\ldots \) and multiply by 1000 to shift the repeating digits, setting it into an equation such as \( 1000x = 273.273273\ldots \). By subtracting the original \( x \) from \( 1000x \), you eliminate the repeating part, simplifying to \( 999x = 273 \).

Solving this gives \( x = \frac{273}{999} \), which turn simplifies to \( \frac{91}{333} \) after finding the GCD. Each step demonstrates a clear logical process that forms the backbone of mathematical proof.