Example Question - decimal fraction conversion

Converting a Repeating Decimal to a Fraction

The image displays a handwritten mathematical question which asks to express the number 0.282323... in the form \(\frac{a}{b}\), where \(a\) and \(b\) are integers. The given number is a repeating decimal: 0.282323... with the pattern "23" repeating indefinitely. To convert this repeating decimal to a fraction, follow these steps: 1. Let \(x = 0.282323...\) 2. Notice that the repeating block "23" starts after the hundredths place. 3. To shift the repeating block to start immediately after the decimal, multiply \(x\) by 1000: \(1000x = 282.323...\) 4. Now subtract \(x\) from \(1000x\) to get a number where the decimal part cancels out. 5. \(1000x - x = 282.323... - 0.282323...\) 6. \(999x = 282\) 7. Divide by 999 to solve for \(x\): \(x = \frac{282}{999}\) 8. Simplify the fraction: \(\frac{282}{999} = \frac{94}{333}\) (dividing numerator and denominator by 3) The answer is \(x = \frac{94}{333}\).