Write the fraction for the recurring decimal… a 0.77777
The Correct Answer and Explanation is :
The correct answer is: ( \frac{7}{9} )
To convert the recurring decimal ( 0.77777\ldots ) (where the 7s repeat indefinitely) into a fraction, we can follow a systematic approach.
Step 1: Set Up an Equation
Let ( x = 0.77777\ldots ).
Step 2: Eliminate the Recurring Part
To isolate the repeating part, we can multiply both sides of the equation by 10. This will shift the decimal point one place to the right:
[
10x = 7.77777\ldots
]
Step 3: Subtract the Original Equation
Now, we have two equations:
- ( x = 0.77777\ldots ) (1)
- ( 10x = 7.77777\ldots ) (2)
Next, subtract equation (1) from equation (2):
[
10x – x = 7.77777\ldots – 0.77777\ldots
]
This simplifies to:
[
9x = 7
]
Step 4: Solve for ( x )
Now, we can solve for ( x ):
[
x = \frac{7}{9}
]
Conclusion
Thus, the fraction representation of the recurring decimal ( 0.77777\ldots ) is ( \frac{7}{9} ).
Verification
To verify that ( \frac{7}{9} ) is indeed equal to ( 0.77777\ldots ), we can perform the division ( 7 \div 9 ):
- 9 goes into 7 zero times. We write down a zero and a decimal point, and bring down a zero, making it 70.
- 9 goes into 70 seven times (since ( 9 \times 7 = 63 )).
- Subtracting gives us a remainder of 7.
- Bringing down another zero gives us 70 again, repeating the process.
This cycle will continue indefinitely, confirming that ( \frac{7}{9} = 0.77777\ldots ).
In summary, the recurring decimal ( 0.77777\ldots ) can be expressed as the fraction ( \frac{7}{9} ), and through algebraic manipulation and verification, we see that this conversion is accurate.