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Chapter 9: Problem 53

Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers). $$0.00 \overline{952}=0.00952952 \ldots$$

Short Answer

Expert verified
Question: Represent the repeating decimal $$0.00952952952\ldots$$ as a fraction. Answer: The repeating decimal $$0.00952952952\ldots$$ can be represented as the fraction $$\frac{952}{999}$$.

Step by step solution

01

Identify the repeating decimal pattern

The repeating pattern in the given decimal is 952, and it starts from the third decimal place.
02

Rewrite the repeating decimal as a geometric series

Now we will rewrite the repeating decimal as the sum of a geometric series with first term $$a_1$$ and common ratio $$r$$. In this case, $$a_1 = 0.00952$$ and $$r = 0.001$$. So, the geometric series representation is: $$0.00952952\ldots = 0.00952 + 0.00952\cdot0.001 + 0.00952\cdot(0.001)^2 + \cdots$$
03

Calculate the sum of the geometric series

To find the sum of the infinite geometric series, we use the formula: $$S = \frac{a_1}{1-r}$$ $$S = \frac{0.00952}{1 - 0.001} = \frac{0.00952}{0.999}$$
04

Convert the decimal to a fraction

Multiply the numerator and denominator of the fraction by a suitable power of 10 to eliminate the decimal points and simplify it to get the final fraction representation: $$\frac{0.00952}{0.999} \times \frac{1000}{1000} = \frac{952}{999}$$ So, the given repeating decimal can be represented as the fraction $$\frac{952}{999}.$$

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Most popular questions from this chapter

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