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

Question

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

UpStudy AI · Accepted Answer

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:** 
$$ \boxed{\dfrac{11}{30}} $$

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### (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:** 
$$ \boxed{\dfrac{1}{60}} $$

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### (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:** 
$$ \boxed{\dfrac{56}{45}} $$
$$ \dfrac{11}{30}, \dfrac{1}{60}, \dfrac{56}{45} $$

Converting the following recurring decimals to common fractions in the
simplest form:

Upstudy AI Solution

Answer

Solution

(d)

(e)

(f)

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