what is 2.267 with the 67 repeating as a fraction

To express \( 2.2\overline{67} \) (where the digits 67 repeat) as a fraction, let's go through the steps:

Let \( x = 2.2\overline{67} \).

### Step 1: Express the repeating decimal part separately
To isolate the repeating part, multiply both sides of the equation by 100 (since there are two digits repeating).

\[
100x = 226.7\overline{67}
\]

### Step 2: Subtract the original equation from this new equation
Now, subtract the original \( x = 2.2\overline{67} \) from the equation \( 100x = 226.7\overline{67} \):

\[
100x - x = 226.7\overline{67} - 2.2\overline{67}
\]

\[
99x = 224.5
\]

### Step 3: Solve for \( x \)
Now, solve for \( x \):

\[
x = \frac{224.5}{99}
\]

To simplify this, multiply both numerator and denominator by 10 to remove the decimal:

\[
x = \frac{2245}{990}
\]

### Step 4: Simplify the fraction
Find the greatest common divisor (GCD) of 2245 and 990. Using the Euclidean algorithm, the GCD is 5. Now, divide both the numerator and the denominator by 5:

\[
x = \frac{2245 \div 5}{990 \div 5} = \frac{449}{198}
\]

Thus, the fraction representation of \( 2.2\overline{67} \) is \( \frac{449}{198} \).

Would you like more details on how I simplified the fraction, or any other questions? Here are some follow-up questions to consider:

1. How do you find the GCD using the Euclidean algorithm?
2. What are some methods for simplifying fractions?
3. How can repeating decimals be represented in terms of geometric series?
4. What happens when more digits are repeating in a decimal?
5. Can all repeating decimals be converted to fractions?

**Tip**: When converting repeating decimals to fractions, always multiply by powers of 10 that match the number of repeating digits.

Learn how to convert 2.267 with repeating digits 67 into a fraction. Follow the step-by-step solution using algebraic methods, multiplication by powers of 10, and fraction simplification techniques. This example explains the use of the Euclidean Algorithm for GCD and is suitable for Grades 9-12.

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