Problem 46 - College Algebra

Question

Express each repeating decimal as a fraction in lowest terms. $$0 . \overline{1}=\frac{1}{10}+\frac{1}{100}+\frac{1}{1000}+\frac{1}{10,000}+\cdots$$

Step-by-Step Solution

Answer

The repeating decimal \(0.1111...\) can be expressed as the fraction \(\frac{1}{9}\) in lowest terms.

Step 1: Formula Deduction

Let's assign the repeating decimal to a variable, \(x\). Then, \(x = 0.1111...\). If we multiply \(x\) by 10, we get \(10x = 1.1111...\). Here, \(10x\) and \(x\) have the same decimal points. Therefore, if we subtract \(x\) from \(10x\), we'll have \(9x = 1\).

Step 2: Solving the equation

Now, we need to solve the equation \(9x = 1\) to find \(x\). The solution is \(x = \frac{1}{9}\), when both sides are divided by 9.

Step 3: Verification

As a final step, let's verify the result. One way to verify is to divide 1 by 9. The answer should be the same as the original repeating decimal, \(0.1111...\), which it is.