Problem 41 - Algebra 2

Question

Write each repeating decimal as a fraction \(0 . \overline{246}\)

Step-by-Step Solution

Answer

The fraction is \( \frac{82}{333} \).

Step 1: Express the repeating decimal using a variable

Let the repeating decimal be represented by a variable, say \( x \). So, \( x = 0.\overline{246} \).

Step 2: Eliminate the repeating part by multiplying

To eliminate the repeating decimal, multiply both sides of the equation by 1000 (since the repeating part is 3 digits long): \( 1000x = 246.\overline{246} \).

Step 3: Set up a subtraction equation

Subtract the original equation from the new equation: \( 1000x - x = 246.\overline{246} - 0.\overline{246} \). This simplifies to: \( 999x = 246 \).

Step 4: Solve for x

Now solve for \( x \) by dividing both sides of the equation by 999:\( x = \frac{246}{999} \).

Step 5: Simplify the fraction

To simplify \( \frac{246}{999} \), find the greatest common divisor (GCD) of 246 and 999. The GCD is 3. Divide the numerator and the denominator by their GCD:\( \frac{246 \div 3}{999 \div 3} = \frac{82}{333} \).

Key Concepts

Fractions

Fractions are an essential part of mathematics, representing a number that is not whole. A fraction has two parts: the numerator (the top number) and the denominator (the bottom number). The fraction indicates a division of these two values, effectively expressing part of a whole. For example, the fraction \( \frac{3}{4} \) represents three parts of a divided whole that has four parts in total.
In the context of converting repeating decimals into fractions, understanding fractions allows us to express numbers that might otherwise be complex in a straightforward way. Repeating decimals are numbers with a digit or a group of digits that repeats infinitely, like \( 0.\overline{246} \). These can always be expressed as fractions through a series of mathematical steps, despite their infinite nature.

Simplifying Fractions

Simplifying fractions is the process of making a fraction as simple as possible. This means reducing the numerator and denominator to the smallest numbers possible that represent the same value, without changing the actual value of the fraction.

To simplify a fraction, you divide both the numerator and the denominator by their greatest common divisor (GCD). For example, simplifying \( \frac{246}{999} \) involves dividing both 246 and 999 by their GCD, which is 3. This process results in the simplified fraction \( \frac{82}{333} \). This makes the fraction easier to work with and understand. Simplifying is an important skill because it helps compare fractions more easily and solve mathematical problems more efficiently.

Greatest Common Divisor

The Greatest Common Divisor, often abbreviated as GCD, is the largest positive integer that evenly divides each of the given numbers. It is crucial in simplifying fractions because it helps us find the largest number that both the numerator and the denominator can be divided by.

To find the GCD of two numbers, you can use different methods like listing out the factors of each number and selecting the largest one they share, or using the Euclidean algorithm which involves repeated division. For example, to find the GCD of 246 and 999 from our exercise, 3 is the biggest number they both can be divided by evenly. Once identified, using this GCD allows us to simplify the fraction effectively, making it easier to use in further calculations.

Variable Representation

In mathematics, using variables is a method to simplify and solve complex problems by representing unknown or changeable numbers with symbols, often letters like \( x \), \( y \), or \( z \). Variables allow us to create equations, which can then be solved using standard mathematical operations.

In the exercise of converting repeating decimals to fractions, a variable is used to represent the repeating decimal initially. By letting \( x = 0.\overline{246} \), we assign the repeating decimal a placeholder that is easier to manipulate through multiplication and subtraction. This transforms the complex repeating decimal process into a straightforward algebraic problem-solving exercise, ultimately allowing us to find the equivalent fraction.