Express 0.$$ \overline{833} $$ in the form of a/b where a and b are integers and b $$ \neq0 $$integers

To express $0.\overline{833}$ (a repeating decimal) in the form $\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$, follow these steps:

Let:
\[
x = 0.\overline{833}
\]

This means that $x = 0.833833833\ldots$.

1. **Identify the repeating part:** The repeating part is "833," which has 3 digits.

2. **Multiply both sides by $10^3 = 1000$** to shift the decimal point 3 places to the right:
   \[
   1000x = 833.833833833\ldots
   \]

   Now we have two equations:
   \[
   x = 0.833833833\ldots
   \]
   \[
   1000x = 833.833833833\ldots
   \]

3. **Subtract the first equation from the second** to eliminate the repeating decimal:
   \[
   1000x - x = 833.833833833\ldots - 0.833833833\ldots
   \]
   \[
   999x = 833
   \]

4. **Solve for $x$:**
   \[
   x = \frac{833}{999}
   \]

5. **Simplify the fraction:** Find the greatest common divisor (GCD) of 833 and 999. Both are divisible by 37:
   \[
   \frac{833 \div 37}{999 \div 37} = \frac{833}{999} = \frac{83}{111}
   \]

So, the fraction form of $0.\overline{833}$ is:
\[
\frac{83}{111}
\]

Would you like further clarification on any step?

Here are 5 related questions you might find helpful:

1. How can other repeating decimals be converted to fractions?
2. What is the method for finding the GCD to simplify fractions?
3. Can you apply this method to a decimal with a different repeating part?
4. How would you handle a repeating decimal with both non-repeating and repeating sections?
5. What if the repeating section had more than 3 digits?

**Tip:** When working with repeating decimals, always identify the length of the repeating part first—this helps determine the power of 10 to multiply with.

Express 0.$$ \overline{833} $$ in the form of a/b where a and b are integers and b $$ \neq0 $$ integers.

Learn how to convert the repeating decimal 0.\overline{833} into a fraction in its simplest form. The detailed solution shows the step-by-step method, including the identification of the repeating part, the multiplication step, and the simplification of the resulting fraction. Suitable for students in Grades 7-9.

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