2. Converting the following recurring decimals to common fractions in the simplest form: \( \begin{array}{lll}\text { (d) } 0,3 \dot{6} & \text { (e) } 0,01 \dot{6} & \text { (f) } 1,2 \dot{4}\end{array} \)

To convert the given recurring decimals to common fractions in their simplest form, we'll use algebraic methods. Let's address each part separately: ### (d) \( 0.\overline{3}6 \) **Interpretation:** The notation \( 0,\dot{3}6 \) typically means that the digit "6" is recurring after the digit "3". So, the decimal is \( 0.3666\ldots \). **Conversion Steps:** 1. Let \( x = 0.3666\ldots \). 2. Multiply both sides by 10 to shift the decimal point: \[ 10x = 3.6666\ldots \] 3. Subtract the original equation from this new equation: \[ 10x - x = 3.6666\ldots - 0.3666\ldots \] \[ 9x = 3.3 \] 4. Solve for \( x \): \[ x = \frac{3.3}{9} = \frac{33}{90} \] 5. Simplify the fraction: \[ \frac{33}{90} = \frac{11}{30} \] **Answer:** \[ \boxed{\dfrac{11}{30}} \] --- ### (e) \( 0.0\overline{1}6 \) **Interpretation:** The notation \( 0,01\dot{6} \) indicates that the digit "6" is recurring after "01". So, the decimal is \( 0.01666\ldots \). **Conversion Steps:** 1. Let \( x = 0.01666\ldots \). 2. Multiply both sides by 1000 to shift the decimal point: \[ 1000x = 16.6666\ldots \] 3. Multiply by 10 to create another equation: \[ 10x = 0.16666\ldots \] 4. Subtract the second equation from the first: \[ 1000x - 10x = 16.6666\ldots - 0.16666\ldots \] \[ 990x = 16.5 \] 5. Solve for \( x \): \[ x = \frac{16.5}{990} = \frac{165}{9900} = \frac{11}{660} = \frac{1}{60} \] **Answer:** \[ \boxed{\dfrac{1}{60}} \] --- ### (f) \( 1.2\overline{4} \) **Interpretation:** The notation \( 1,2\dot{4} \) signifies that the digit "4" is recurring after "2". So, the decimal is \( 1.24444\ldots \). **Conversion Steps:** 1. Let \( x = 1.24444\ldots \). 2. Subtract the integer part: \[ x - 1 = 0.24444\ldots \] 3. Let \( y = 0.24444\ldots \). 4. Multiply \( y \) by 10: \[ 10y = 2.4444\ldots \] 5. Subtract the original \( y \) from this: \[ 10y - y = 2.4444\ldots - 0.24444\ldots \] \[ 9y = 2.2 \] 6. Solve for \( y \): \[ y = \frac{2.2}{9} = \frac{22}{90} = \frac{11}{45} \] 7. Add back the integer part: \[ x = 1 + \frac{11}{45} = \frac{45}{45} + \frac{11}{45} = \frac{56}{45} \] **Answer:** \[ \boxed{\dfrac{56}{45}} \]