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Chapter 9: Problem 54

Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers). $$5.12 \overline{83}=5.12838383 \ldots$$

Short Answer

Expert verified
To convert the repeating decimal 5.12(83) into a fraction, we follow these steps: 1. Identify the non-repeating and repeating parts of the decimal: 5.12 and 83. 2. Rewrite the decimal as the sum of the non-repeating part and the repeating part: 5.12 + 0.0083(83). 3. Write the repeating part as a geometric series: 0.0083 + 0.0083(1/100) + 0.0083(1/100)^2 + ... 4. Use the formula for the sum of an infinite geometric series (S = a/(1-r)) to find the sum of the series, which is 0.83/99. 5. Add the non-repeating part (5.12) to the sum of the geometric series to get the fraction: (512/100) + (83/99). 6. Find a common denominator (9900), and simplify the fraction, which results in the final answer of 14747/2475. So, the repeating decimal 5.12(83) can be expressed as the fraction 14747/2475.

Step by step solution

01

Identify the non-repeating and the repeating parts of the decimal

In the given decimal, the non-repeating part is '5.12' and the repeating part is '83'. We can rewrite the decimal as: $$5.12 \overline{83}=5.12+0.0083 \overline{83}$$
02

Write the repeating part as a geometric series

The repeating part '83' can be written as a geometric series with a common ratio of $$\frac{1}{100}$$, because there are two decimal places in '83'. So, we have: $$0.0083 \overline{83}=0.0083 + 0.0083\left(\frac{1}{100}\right) + 0.0083\left(\frac{1}{100}\right)^2 + \cdots$$
03

Use the formula for sum of an infinite geometric series

To find the sum of this geometric series, we can use the formula for the sum of an infinite geometric series: $$S = \frac{a}{1-r}$$ where 'S' is the sum, 'a' is the first term, and 'r' is the common ratio. For our series, the first term 'a' is 0.0083 and the common ratio 'r' is $$\frac{1}{100}$$. So, we have: $$S = \frac{0.0083}{1-\frac{1}{100}} = \frac{0.0083}{\frac{99}{100}}$$
04

Solve for the sum of the geometric series

Now, we can solve for the sum of the given geometric series by multiplying both numerator and denominator by 100: $$S = \frac{0.0083 \times 100}{\frac{99}{100} \times 100} = \frac{0.83}{99}$$
05

Add the non-repeating part to the sum of the geometric series

Finally, we will add the non-repeating part (5.12) to the sum of the geometric series (0.83/99) to get the answer as a fraction: $$5.12 \overline{83} = 5.12 + \frac{0.83}{99} = \frac{512}{100} + \frac{83}{99}$$
06

Find a common denominator and simplify the fraction

To sum the fractions, we need a common denominator, which is the least common multiple (LCM) of 100 and 99. The LCM of 100 and 99 is 9900. So, we can rewrite the fractions with a denominator of 9900 and sum them: $$\frac{512 \times 99}{9900} + \frac{83 \times 100}{9900} = \frac{50688 + 8300}{9900} = \frac{58988}{9900}$$ Now, we need to simplify the fraction by finding the greatest common divisor (GCD) of the numerator and denominator. The GCD of 58988 and 9900 is 4. We can divide both numerator and denominator by 4 to get the simplified fraction: $$\frac{58988}{9900} = \frac{14747}{2475}$$ So, the final answer is: $$5.12 \overline{83}= \frac{14747}{2475}$$

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