Express 0.8$$ \overline{33} $$ in the form of a/b where a and b are integers and b $$ \neq0 $$

To express $ 0.8\overline{33} $ as a fraction in the form $ \frac{a}{b} $, we can follow these steps:

1. **Define the repeating decimal**:
   Let
   \[
   x = 0.8333333\ldots
   \]
   where $ 3 $ repeats indefinitely.

2. **Separate the non-repeating and repeating parts**:
   We can write $ x $ as:
   \[
   x = 0.8 + 0.0333333\ldots
   \]
   The first part, $ 0.8 $, is a terminating decimal, while $ 0.0333333\ldots $ (denoted as $ 0.\overline{03} $) is the repeating part.

3. **Convert the repeating part to a fraction**:
   Let $ y = 0.0333333\ldots $.
   Then $ y = 0.\overline{03} $, which we can rewrite as follows:
   
   Multiply both sides by 10:
   \[
   10y = 0.3333333\ldots
   \]
   
   Since $ y = 0.0333333\ldots $, subtract the original equation from this one:
   \[
   10y - y = 0.3333333\ldots - 0.0333333\ldots
   \]
   \[
   9y = 0.3
   \]
   \[
   y = \frac{0.3}{9} = \frac{1}{30}
   \]

4. **Combine the two parts**:
   Now we can write $ x $ as:
   \[
   x = 0.8 + \frac{1}{30}
   \]

5. **Convert $ 0.8 $ to a fraction and add**:
   $ 0.8 = \frac{8}{10} = \frac{4}{5} $.
   
   Now add $ \frac{4}{5} $ and $ \frac{1}{30} $:
   \[
   x = \frac{4}{5} + \frac{1}{30}
   \]

6. **Find a common denominator**:
   The common denominator of $ 5 $ and $ 30 $ is $ 30 $:
   \[
   x = \frac{4 \times 6}{5 \times 6} + \frac{1}{30} = \frac{24}{30} + \frac{1}{30} = \frac{25}{30} = \frac{5}{6}
   \]

Thus, the fractional form of $ 0.8\overline{33} $ is:
\[
\frac{5}{6}
\]

Would you like further clarification on any step?

---

1. How do we separate the non-repeating and repeating parts in similar problems?
2. Why does multiplying by powers of 10 help in converting repeating decimals?
3. Can all repeating decimals be converted into fractions?
4. What is the significance of finding a common denominator?
5. How can we verify that $ 0.8\overline{33} = \frac{5}{6} $ is correct?

**Tip:** Always break down repeating decimals into non-repeating and repeating parts for easier conversion into fractions.

Learn how to convert the repeating decimal 0.8\overline{33} into a fraction. The step-by-step solution involves separating the non-repeating and repeating parts, using multiplication by powers of 10, and finding a common denominator. Suitable for students in Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation