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Chapter 9: Problem 51

Write each repeating decimal first as a geometric series and then as a fraction (a ratio of two integers). $$0 . \overline{456}=0.456456456 \ldots$$

Short Answer

Expert verified
Answer: The repeating decimal 0.456456456... can be represented as the fraction 152/333.

Step by step solution

01

Convert the repeating decimal to a geometric series

Let's call the repeating decimal x, so $$x = 0.456456456 \ldots$$ We need to isolate the repeating part, in this case, 456 $$1000x = 456.456456\ldots$$ Now, we want to subtract x from 1000x to get rid of the repeating decimals $$1000x - x = 456.456456\ldots - 0.456456\ldots$$
02

Simplify the expression

By subtracting x from 1000x, we can cancel out the repeating decimals and simplify the equation $$999x = 456$$
03

Solve for x

Now, we can solve for x by dividing both sides by 999 $$x = \frac{456}{999}$$
04

Simplify the fraction

Now we have the fraction, but we need to simplify it. Check if there is any common divisor for 456 and 999, which is 3: $$x = \frac{456 \div 3}{999 \div 3} = \frac{152}{333}$$ Now, we found that $$0.\overline{456}$$ will be equal to the fraction $$\frac{152}{333}$$ when expressed as a ratio of two integers.

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