Problem 78 - Precalculus: Mathematics for Calculus

Question

Express the repeating decimal as a fraction. $$0.2 \overline{53}$$

Step-by-Step Solution

Answer

The fraction is \( \frac{251}{990} \).

Step 1: Identify Components

First, identify the non-repeating and repeating parts of the decimal. Here, 0.2 is the non-repeating part and 53 is the repeating part.

Step 2: Set Up Equation for Decimal

Let \( x = 0.2\overline{53} \). This implies \( x = 0.2535353\ldots \).

Step 3: Eliminate Decimal Part

Write two equations by shifting the decimal point to eliminate the repeating part. Multiply \( x\) by 1000 to move the decimal past the repeating part: \( 1000x = 253.535353\ldots \).

Step 4: Isolate Repeating Part

Also multiply \( x \) by 10 to move past the non-repeating part: \( 10x = 2.535353\ldots \).

Step 5: Subtract Equations

Subtract the second equation from the first to eliminate the repeating decimal: \( 1000x - 10x = 253.535353\ldots - 2.535353\ldots \) which simplifies to \( 990x = 251 \).

Step 6: Solve for x

Divide both sides of the equation by 990 to solve for \( x \): \( x = \frac{251}{990} \).

Step 7: Simplify the Fraction

Determine if \( \frac{251}{990} \) can be simplified. Here, both 251 and 990 are coprime, so \( \frac{251}{990} \) is already at its simplest form.

Key Concepts

Fraction Simplification

When converting a repeating decimal to a fraction, one important step is simplifying the resulting fraction. Simplification refers to reducing the fraction to its simplest form, where the numerator and the denominator no longer share a common factor other than 1. To simplify a fraction, you start by identifying any common factors between the numerator and the denominator. These common factors can be both prime numbers like 2, 3, or 5, and composite numbers. Once identified, you divide both the numerator and the denominator by their Greatest Common Divisor (GCD) to reduce the fraction.For example, if you end up with the fraction \( \frac{8}{12} \), you can simplify it because both numbers are divisible by 4. So you divide each by 4 to get \( \frac{2}{3} \), which is the simplified form. However, in our original problem, we found that the fraction was \( \frac{251}{990} \). Since 251 and 990 are coprime (they have no common factors other than 1), this fraction is already in its simplest form. This demonstrates the importance of checking if further simplification is needed, as it makes understanding and using the fraction much easier.

Non-Repeating and Repeating Decimals

Understanding the difference between non-repeating and repeating decimals is crucial when converting decimals to fractions. A non-repeating decimal is a decimal number that doesn't continue indefinitely or show any repeating pattern. On the other hand, a repeating decimal has digits after the decimal point that recur in a pattern.For instance, in the decimal 0.2\(\overline{53}\), 0.2 is the non-repeating part, and 53 is the repeating sequence. Recognizing this separation is essential because different techniques are used to convert these into fractions.When you encounter a repeating decimal, it's often represented with a bar over the repeating digits, like \(\overline{53}\). This notation helps differentiate repeating decimals from those that don't repeat. Identifying these sequences correctly helps in setting up equations to transform the decimal into a rational fraction efficiently.

Mathematics Problem-Solving

Approaching mathematics problems efficiently involves breaking them down into manageable steps, much like those used in our exercise. Problem-solving in math requires both a clear understanding of concepts and the ability to apply these concepts in a structured manner.Here’s how you can tackle math problems:

• Understand the Problem: Take your time to read the problem carefully. Identify the key components, such as what's being asked and the information provided.
• Plan Your Approach: Decide on the best method to solve the problem. Determine which rules, formulas, or theorems apply.
• Execute the Plan: Methodically apply your chosen approach. This includes writing down equations, performing calculations, and keeping track of your steps.
• Review Your Solution: Always verify your results. Check your calculations, revisit your steps, and ensure your final answer makes sense in the context of the problem.

In our solution, after setting up the equation for 0.2\(\overline{53}\), we systematically eliminated decimal parts by aligning equations, subtracting to remove repeating elements, and solving for \( x \). This systematic approach highlights how breaking down a problem into clear steps ensures you arrive at a correct and simplified solution.