The decimal $0.6\overline{7}$ represents a repeating decimal, where the digit "7" repeats indefinitely. Let's express this repeating decimal as a fraction.

Step 1: Let $x = 0.6\overline{7}$

This means that $x = 0.677777...$

Step 2: Multiply by 10 to shift the decimal point

Multiplying $x$ by 10 moves the decimal point one place to the right: $10x = 6.777777...$

Step 3: Subtract the original equation from the new one

Now subtract the original equation $x = 0.677777...$ from $10x = 6.777777...$: $10x - x = 6.777777... - 0.677777...$ This simplifies to: $9x = 6.1$

Step 4: Solve for $x$

Now solve for $x$ by dividing both sides by 9: $x = \frac{6.1}{9}$

Step 5: Simplify the fraction

Convert $6.1$ to a fraction: $6.1 = \frac{61}{10}$ Now substitute: $x = \frac{61}{10} \times \frac{1}{9} = \frac{61}{90}$

Thus, $0.6\overline{7} = \frac{61}{90}$.

Let me know if you need further clarification or have questions! Here are 5 related questions to practice:

1. How would you convert $0.4\overline{3}$ to a fraction?
2. What is the fraction form of $0.7\overline{5}$?
3. How can you simplify the fraction $\frac{122}{180}$?
4. How do you express $0.8\overline{2}$ as a fraction?
5. What is the decimal form of $\frac{61}{90}$?

Tip: Repeating decimals can often be turned into fractions using algebraic methods like subtraction after multiplying!