a. Find three rational numbers between $$ \frac{1}{5} $$ and $$ 0 . \overline{20} \ ). b. Find three rational numbers between $$ \frac{1}{3} $$ and $$ 0 . \overline{34} \ ).

Question

UpStudy AI · Accepted Answer

Let's solve each part step by step.

### Part a: Find three rational numbers between $$ \frac{1}{5} $$ and $$ 0.\overline{20} $$.

1. **Convert $$ 0.\overline{20} $$ to a fraction**:
   Let $$ x = 0.\overline{20} $$.
   Then, $$ 100x = 20.\overline{20} $$.
   Subtracting these two equations gives:
   $$
   100x - x = 20 \implies 99x = 20 \implies x = \frac{20}{99}
   $$

2. **Identify the two fractions**:
   - $$ \frac{1}{5} = \frac{20}{100} $$
   - $$ 0.\overline{20} = \frac{20}{99} $$

3. **Find a common denominator**:
   The least common multiple of 100 and 99 is 9900. We convert both fractions:
   - $$ \frac{1}{5} = \frac{20 \times 198}{100 \times 198} = \frac{3960}{9900} $$
   - $$ \frac{20}{99} = \frac{20 \times 100}{99 \times 100} = \frac{2000}{9900} $$

4. **Find three rational numbers between $$ \frac{3960}{9900} $$ and $$ \frac{2000}{9900} $$**:
   We can choose numbers like:
   - $$ \frac{2500}{9900} $$
   - $$ \frac{3000}{9900} $$
   - $$ \frac{3500}{9900} $$

Thus, three rational numbers between $$ \frac{1}{5} $$ and $$ 0.\overline{20} $$ are $$ \frac{2500}{9900}, \frac{3000}{9900}, \frac{3500}{9900} $$.

### Part b: Find three rational numbers between $$ \frac{1}{3} $$ and $$ 0.\overline{34} $$.

1. **Convert $$ 0.\overline{34} $$ to a fraction**:
   Let $$ y = 0.\overline{34} $$.
   Then, $$ 100y = 34.\overline{34} $$.
   Subtracting these two equations gives:
   $$
   100y - y = 34 \implies 99y = 34 \implies y = \frac{34}{99}
   $$

2. **Identify the two fractions**:
   - $$ \frac{1}{3} = \frac{33}{99} $$
   - $$ 0.\overline{34} = \frac{34}{99} $$

3. **Find three rational numbers between $$ \frac{33}{99} $$ and $$ \frac{34}{99} $$**:
   We can choose numbers like:
   - $$ \frac{33.1}{99} $$
   - $$ \frac{33.2}{99} $$
   - $$ \frac{33.3}{99} $$

Thus, three rational numbers between $$ \frac{1}{3} $$ and $$ 0.\overline{34} $$ are $$ \frac{33.1}{99}, \frac{33.2}{99}, \frac{33.3}{99} $$.

### Summary:
- **Part a**: Three rational numbers between $$ \frac{1}{5} $$ and $$ 0.\overline{20} $$ are $$ \frac{2500}{9900}, \frac{3000}{9900}, \frac{3500}{9900} $$.
- **Part b**: Three rational numbers between $$ \frac{1}{3} $$ and $$ 0.\overline{34} $$ are $$ \frac{33.1}{99}, \frac{33.2}{99}, \frac{33.3}{99} $$.
Three rational numbers between $$ \frac{1}{5} $$ and $$ 0.\overline{20} $$ are $$ \frac{2500}{9900}, \frac{3000}{9900}, \frac{3500}{9900} $$. Three rational numbers between $$ \frac{1}{3} $$ and $$ 0.\overline{34} $$ are $$ \frac{33.1}{99}, \frac{33.2}{99}, \frac{33.3}{99} $$.

a. Find three rational numbers between and and ( 0 . \overline{34} \ ).

Solución de inteligencia artificial de Upstudy

Responder

Solución

Part a: Find three rational numbers between and .

Part b: Find three rational numbers between and .

Summary:

Mind Expander

preguntas relacionadas

Latest Arithmetic Questions