Problem 48 - College Algebra

Question

Express each repeating decimal as a fraction in lowest terms. $$0 . \overline{83}=\frac{83}{100}+\frac{83}{10,000}+\frac{83}{1,000,000}+\cdots$$

Step-by-Step Solution

Answer

The repeating decimal 0.83 as a fraction in lowest terms is \(83/99\).

Step 1: Identifying repeating part and writing as series

We identify the repeating part of the decimal which is 83. We then write this as the sum of the series \(83/100 + 83/10000 + 83/1000000 + \ldots\). Each term in this series comes from dividing the previous term by 100, so this is a geometric series with a common ratio of \(1/100\).

Step 2: Summing the series

To find the sum of this infinite geometric series, we use the formula for the sum of a geometric series, which is \(S = a / (1-r)\), where 'a' is the first term of the series and 'r' is the common ratio. Substituting the values in the formula, we get \(S = 83/100 / (1 - 1/100) = 83/(100 - 1) = 83/99'.

Step 3: Simplifying to lowest terms

To express the fraction in lowest terms, we check to see if the numerator and the denominator have any common factors. If they do, we divide both by the common factor. In this case, 83 and 99 have no common factors, so the fraction is already in its simplest form, which is \(83/99\).