What value should come in the place of question mark (?) in the following question:

\(0.5\overline{47}−0.3\overline{45}+0.2\overline{44}=?\)

Given:

The expression is \(0.5\overline{47}−0.3\overline{45}+0.2\overline{44}=?\)

Calculations:

For \(0.5\overline{47}\): Let x = 0.5474747...

Multiply by 10: 10x = 5.474747... (Equation 1)

Multiply by 1000: 1000x = 547.474747... (Equation 2)

Subtract Equation 1 from Equation 2

990x = 542

\(x = \frac{542}{990} = \frac{271}{495}\)

For \(0.3\overline{45}\): Let y = 0.3454545...

Multiply by 10: 10y = 3.454545... (Equation 3)

Multiply by 1000: 1000y = 345.454545... (Equation 4)

Subtract Equation 3 from Equation 4: 990y = 342

\(y = \frac{342}{990}\)

For \(0.2\overline{44}\): Let z = 0.2444444...

Multiply by 10: 10z = 2.444444... (Equation 5)

Multiply by 100: 100z = 24.444444... (Equation 6)

Subtract Equation 5 from Equation 6: 100z - 10z = 24.4444... - 2.4444...

90z = 22

\(z = \frac{22}{90}\)

The expression is \(\frac{542}{990} - \frac{342}{990} + \frac{22}{90}\)

LCM(990, 990, 90) = 990

\(\frac{542}{990} - \frac{342}{990} + \frac{242}{990}\)

⇒ \(\frac{542 - 342 + 242}{990}\)

⇒ \(\frac{200 + 242}{990}\)

⇒ \(\frac{442}{990}\)

∴ The value of the expression is \(\frac{442}{990}\).