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Chapter 5: Problem 56

Express each repeating decimal as a quotient of integers. If possible, reduce to lowest terms. \(0 . \overline{529}\)

Short Answer

Expert verified
The repeating decimal \(0 . \overline{529}\) expressed as a quotient of integers is \(\frac{529}{999}\)

Step by step solution

01

Let Variable

Let \(x = 0 . \overline{529}\). This decimal repeats every three digits, so multiply both sides by 1000 to shift the decimal point three places to the right: \(1000x = 529 . \overline{529}\)
02

Create an Equation

Next, subtract the original \(x\) from the multiplied equation: \(1000x - x = 529 . \overline{529} - 0 . \overline{529}\), which simplifies to: \(999x = 529\)
03

Solve for x

Solve the equation for x by dividing both sides by 999: \(x = \frac{529}{999}\)

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