Problem 56 - Precalculus: Mathematical for Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem

We need to express the repeating decimal \(0.2\overline{53}\), which means \(0.2535353\ldots\), as a fraction.

Step 2: Set Up Equations

Let \(x = 0.2\overline{53}\). This implies \(x = 0.2535353\ldots\). Identify the repeating part of the decimal and move the decimal point appropriately to set up equations.

Step 3: Eliminate the Repeating Decimal

Multiply both sides by 1000 to move the repeating part completely to the right of the decimal point: \(1000x = 253.535353\ldots\).

Step 4: Create a Secondary Equation

Also multiply \(x\) by 10 to align with the repeating terms: \(10x = 2.535353\ldots\).

Step 5: Subtract Equations

Subtract the equation from Step 4 from the equation in Step 3:\(1000x - 10x = 253.535353\ldots - 2.535353\ldots\).

Step 6: Simplify the Result

The subtraction results in:\(990x = 251\).Divide by 990 to find \(x\):\(x = \frac{251}{990}\).

Step 7: Check and Simplify the Fraction

Check to see if \(\frac{251}{990}\) can be simplified further by finding the greatest common divisor (GCD) of 251 and 990, which turns out to be 1, so it cannot be simplified further.

Key Concepts

Fraction Conversion

Converting a repeating decimal to a fraction involves several key steps. The goal is to express the repeating decimal in a fraction form. Suppose you have a repeating decimal like \(0.2\overline{53}\). This notation indicates that the decimal \(253\) is repeating.

To convert this into a fraction, we start by assigning a variable to the repeating decimal. Let's say \(x = 0.2\overline{53}\).

• We multiply \(x\) by an appropriate power of 10 to shift the decimal point. Here, we use 1000 because the repeating part "53" has two digits.
• This multiplication results in the equation \(1000x = 253.535353\ldots\).
• To help eliminate the repeating part, create another equation by multiplying \(x\) by 10: \(10x = 2.535353\ldots\).

Subtracting these two equations will help in eliminating the repeating decimal, allowing us to solve for \(x\).

Decimal Representation

Understanding decimal representation is crucial when dealing with repeating decimals. A repeating decimal is one that has a sequence of digits that repeat indefinitely. These decimals can be expressed as fractions because they possess a regular pattern. For example, in \(0.2\overline{53}\), the digits "53" repeat forever.

When you perform operations like multiplication with a repeating decimal, you're essentially trying to isolate the repeating part. This helps in expressing the entire decimal as a fraction. Here, multiplying by 10 and 1000 shifts the decimal point, aligning numbers in such a way that subtraction will cancel out the repeating part.

Recognizing these repeating patterns is necessary to convert them into fractions effectively.

Mathematical Fractions

Fractions are a way to represent numbers that are not whole numbers. Converting a repeating decimal, like \(0.2\overline{53}\), into a fraction requires careful manipulation.

After setting up equations with a repeating decimal, subtract the simpler equation from the more complex one to eliminate the repeating decimal. In our case:

• The subtraction gives us \(990x = 251\).
• By solving, we find \(x = \frac{251}{990}\).

The result is a mathematical fraction, which represents the repeating decimal.

Sometimes it's possible to simplify fractions, but here, the greatest common divisor of 251 and 990 is 1.

This means \(\frac{251}{990}\) is already in its simplest form. Understanding fractions and their properties is essential in addressing these kinds of problems efficiently.