Rewrite as a simplified fraction. 0.67 where 7 repeats forever

The Correct Answer and Explanation is:

To convert the repeating decimal (0.67\overline{7}) into a simplified fraction, we can follow a systematic approach. The notation (0.67\overline{7}) means that the digits “67” are followed by an infinite series of the digit “7” (i.e., 0.677777…).

Step 1: Define the Decimal as a Variable

Let (x = 0.67\overline{7}).

Step 2: Eliminate the Repeating Part

To eliminate the repeating part, we can multiply (x) by a power of 10 that moves the decimal point to the right, covering one full cycle of the repeating digits. Since the “7” repeats after two digits (“67”), we multiply by (1000) (two for “67” and one for the repeat):

[
1000x = 677.7777\overline{7}
]

Step 3: Set Up a System of Equations

Next, we can create a second equation without the repeating part. Multiply (x) by (10) to cover just the “67”:

[
10x = 6.7777\overline{7}
]

Step 4: Subtract the Two Equations

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

[
1000x – 10x = 677.7777\overline{7} – 6.7777\overline{7}
]

This simplifies to:

[
990x = 670
]

Step 5: Solve for (x)

Next, divide both sides by (990):

[
x = \frac{670}{990}
]

Step 6: Simplify the Fraction

To simplify (\frac{670}{990}), we can find the greatest common divisor (GCD) of (670) and (990). The prime factorization shows:

(670 = 2 \times 5 \times 67)
(990 = 2 \times 3^2 \times 5 \times 11)

The common factors are (2) and (5). The GCD is (10). Now, we divide both the numerator and the denominator by (10):

[
\frac{670 \div 10}{990 \div 10} = \frac{67}{99}
]

Conclusion

Thus, the simplified fraction of the repeating decimal (0.67\overline{7}) is:

[
\boxed{\frac{67}{99}}
]

This fraction is now in its simplest form, representing the original repeating decimal accurately.