2.6 repeated simplified as a fraction Please help asap!!!

2.6 repeated simplified as a fraction Please help asap!!!

The Correct Answer and Explanation is :

To convert the repeating decimal ( 2.6\overline{6} ) (which means ( 2.66666… )) into a fraction, we can use a method involving algebra.

Step 1: Set Up the Equation

Let ( x = 2.6\overline{6} ). This gives us the equation:

[
x = 2.66666\ldots
]

Step 2: Eliminate the Repeating Part

To isolate the repeating part, multiply both sides of the equation by 10. This shifts the decimal point one place to the right:

[
10x = 26.66666\ldots
]

Step 3: Set Up Another Equation

Now, we have two equations:

1. ( x = 2.66666\ldots )
2. ( 10x = 26.66666\ldots )

Step 4: Subtract the Two Equations

Next, subtract the first equation from the second to eliminate the repeating part:

[
10x – x = 26.66666\ldots – 2.66666\ldots
]

This simplifies to:

[
9x = 24
]

Step 5: Solve for ( x )

Now, divide both sides by 9:

[
x = \frac{24}{9}
]

Step 6: Simplify the Fraction

To simplify ( \frac{24}{9} ), find the greatest common divisor (GCD) of 24 and 9. The GCD is 3. Now divide both the numerator and denominator by 3:

[
x = \frac{24 \div 3}{9 \div 3} = \frac{8}{3}
]

Conclusion

Thus, the repeating decimal ( 2.6\overline{6} ) can be expressed as the fraction ( \frac{8}{3} ). This method of converting repeating decimals to fractions can be applied to other repeating decimals as well by using a similar algebraic approach.