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!

Learn how to convert the repeating decimal 0.6̅7 into a fraction using algebra. Follow a step-by-step process, including multiplying and subtracting equations, to express 0.6̅7 as the fraction 61/90. Suitable for middle school students in Grades 6-8.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation