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Chapter 22: Problem 703

Convert the repeating decimal \(477477 \ldots\) to a fraction.

Short Answer

Expert verified
The repeating decimal 477.477477... can be converted to the fraction \(\frac{159000}{333}\).

Step by step solution

01

Assign a variable to the repeating decimal

Let the repeating decimal be represented by the variable x: \[x = 477.477477\ldots\]
02

Multiply by a power of 10 to shift the decimal point

We will multiply both sides of the equation by 1000 to shift the decimal point three places to the right: \[1000x = 477477.477477\ldots\]
03

Subtract the original equation from the multiplied equation

By subtracting the original equation from the multiplied equation, we eliminate the repeating decimal: \[1000x - x = 477477.477477\ldots - 477.477477\ldots\] \[999x = 477000\]
04

Solve for x

Divide by 999 on both sides to solve for x: \[\frac{999x}{999} = \frac{477000}{999}\] \[x = \frac{477000}{999}\]
05

Simplify the fraction

Now, we simplify the fraction: Find the greatest common divisor (GCD) of the numerator and the denominator. GCD(477000, 999) = 3 Divide both numerator and denominator by their GCD to get the simplified fraction: \[\frac{477000}{3} = 159000\] \[\frac{999}{3} = 333\] So, the simplified fraction is: \[x = \frac{159000}{333}\] The repeating decimal 477.477477... can be converted to the fraction \(\frac{159000}{333}\).

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