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Chapter 9: Problem 47

Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers). $$0 . \overline{037}=0.037037 \ldots$$

Short Answer

Expert verified
Question: Convert the repeating decimal \(0.037037\ldots\) into a fraction. Answer: The repeating decimal \(0.037037\ldots\) can be expressed as the fraction \(\frac{37}{999}\).

Step by step solution

01

Express the repeating decimal as a geometric series

Let \(x = 0.037037\ldots\). We can express the repeating part as a geometric series by multiplying \(x\) by increasing powers of \(10\) and then subtracting it to eliminate the repeating part. For this task, the terms of the repeating part \(037\) would be expressed as: $$0.037 + 0.00037 + 0.0000037 + \ldots$$ This is a geometric series with the first term \(a = 0.037\) and common ratio \(r = \frac{1}{1000}\).
02

Calculate the sum of the infinite geometric series

Now, we need to calculate the sum of the infinite geometric series. The formula for the sum of an infinite geometric series is given by: $$S = \frac{a}{1 - r}$$ In our series, we have \(a = 0.037\) and \(r = \frac{1}{1000}\). Hence, we will put these values in the formula to find the sum.
03

Substitute the values and solve

Now we substitute our values into the formula: $$S = \frac{0.037}{1 - \frac{1}{1000}}$$ Now, we simply compute the sum of the series: $$S = \frac{0.037}{\frac{999}{1000}} = \frac{0.037\times1000}{999} = \frac{37}{999}$$
04

Write the repeating decimal as a fraction

Thus, the repeating decimal \(0.037037\ldots\) can be expressed as the fraction \(\frac{37}{999}\). So, $$0 . \overline{037}=0.037037 \ldots = \frac{37}{999}$$

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