If the recurring decimal \(0.841 \overline{81}\) be written as a fraction in its lowest terms, the difference between the denominator and the numerator is

Given:

Recurring decimal = \(0.841 \overline{81}\)

Formula used:

Algebraic method for converting recurring decimals to fractions.

Calculations:

Let X = \(0.841 \overline{81}\) --- (1)

⇒ 1000X = \(841. \overline{81}\) --- (2)

Now, multiply equation (2) by 100 (since there are 2 repeating digits) to shift one block of repeating digits past the decimal point:

⇒ 100 × 1000X = 100 × \(841. \overline{81}\)

⇒ 100000X = \(84181. \overline{81}\) --- (3)

Subtract equation (2) from equation (3) to eliminate the repeating part:

⇒ 100000X - 1000X = \(84181. \overline{81}\) - \(841. \overline{81}\)

⇒ 99000X = 84181 - 841

⇒ 99000X = 83340

⇒ X = \(\dfrac{83340}{99000}\)

⇒ X = \(\dfrac{8334}{9900}\)

⇒ X = \(\dfrac{4167}{4950}\)

⇒ X = \(\dfrac{1389}{1650}\)

⇒ X = \(\dfrac{463}{550}\)

The fraction in its lowest terms is \(\dfrac{463}{550}\).

Numerator = 463

Denominator = 550

Difference between the denominator and the numerator = Denominator - Numerator

⇒ Difference = 550 - 463

⇒ Difference = 87

∴ The difference between the denominator and the numerator is 87.