$$ \text { QUESTION } 2 $$ Write the repeating decimal number $$ 2.7 \mathbf{1} \delta $$ as an improper fraction Write the repeating decimal number $$ 10, \overline{213} $$ as an improper fraction

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UpStudy AI · Accepted Answer

Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$x=10+0.\overline{213}$$
- step1: Simplify:
  $$x=\frac{3401}{333}$$
Solve the equation $$ x=2.7+0.0\overline{1} $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$x=2.7+0\overline{1}$$
- step1: Simplify:
  $$x=\frac{122}{45}$$
Let's convert the repeating decimal numbers into improper fractions step by step.

### 1. Converting $$ 2.7\overline{1} $$ to an improper fraction

Let $$ x = 2.7\overline{1} $$.

To express this as a fraction, we can separate the non-repeating and repeating parts:

1. **Separate the decimal**:
   $$
   x = 2.7 + 0.0\overline{1}
   $$

2. **Convert $$ 2.7 $$ to a fraction**:
   $$
   2.7 = \frac{27}{10}
   $$

3. **Convert $$ 0.0\overline{1} $$**:
   Let $$ y = 0.0\overline{1} $$.
   $$
   10y = 0.1\overline{1}
   $$
   Subtracting the original equation from this:
   $$
   10y - y = 0.1\overline{1} - 0.0\overline{1} \implies 9y = 0.1 \implies y = \frac{1}{90}
   $$

4. **Combine the fractions**:
   $$
   x = \frac{27}{10} + \frac{1}{90}
   $$
   To add these fractions, find a common denominator (which is 90):
   $$
   x = \frac{27 \times 9}{90} + \frac{1}{90} = \frac{243 + 1}{90} = \frac{244}{90}
   $$
   Simplifying gives:
   $$
   x = \frac{122}{45}
   $$

### 2. Converting $$ 10.\overline{213} $$ to an improper fraction

Let $$ x = 10.\overline{213} $$.

1. **Separate the decimal**:
   $$
   x = 10 + 0.\overline{213}
   $$

2. **Convert $$ 0.\overline{213} $$**:
   Let $$ y = 0.\overline{213} $$.
   $$
   1000y = 213.\overline{213}
   $$
   Subtracting the original equation from this:
   $$
   1000y - y = 213.\overline{213} - 0.\overline{213} \implies 999y = 213 \implies y = \frac{213}{999}
   $$
   Simplifying gives:
   $$
   y = \frac{71}{333}
   $$

3. **Combine the fractions**:
   $$
   x = 10 + \frac{71}{333} = \frac{10 \times 333 + 71}{333} = \frac{3330 + 71}{333} = \frac{3401}{333}
   $$

### Final Results

- The improper fraction for $$ 2.7\overline{1} $$ is $$ \frac{122}{45} $$.
- The improper fraction for $$ 10.\overline{213} $$ is $$ \frac{3401}{333} $$.
The repeating decimal $$ 2.7\overline{1} $$ is equal to the improper fraction $$ \frac{122}{45} $$, and the repeating decimal $$ 10.\overline{213} $$ is equal to the improper fraction $$ \frac{3401}{333} $$.

\[ \text { QUESTION } 2 \] Write the repeating decimal number \( 2.7 \mathbf{1} \delta \) as an improper fraction Write the repeating decimal number \( 10, \overline{213} \) as an improper fraction

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