Question

Express 7/11 as a repeating decimal. Mark the cycle of repeating digits with a bar.

Show your work.

Answer 1

To express \( \frac{7}{11} \) as a repeating decimal, we can perform long division. Here's a step-by-step breakdown:

### Step 1: Set up the long division
Divide 7 by 11:
- 11 goes into 7 zero times, so we write down 0. and add a decimal point.
- We then add a zero to 7, making it 70.

### Step 2: Perform the division
Now, divide 70 by 11:
- 11 goes into 70 six times, because \( 11 \times 6 = 66 \).
- Write down 6 after the decimal point.

### Step 3: Subtract
Now subtract 66 from 70:
\[
70 - 66 = 4
\]

### Step 4: Bring down the next 0
Next, bring down another 0, making it 40.

### Step 5: Divide again
Now, divide 40 by 11:
- 11 goes into 40 three times, because \( 11 \times 3 = 33 \).

### Step 6: Subtract again
Subtract 33 from 40:
\[
40 - 33 = 7
\]

### Step 7: Repeat the process
Now, if we bring down another 0, we again have 70, just like in step 2. This will lead us to the same numbers:
- Divide 70 by 11, which gives us 6,
- Subtract 66 from 70 to get 4,
- Bring down the next 0 to get 40,
- Divide 40 by 11, which gives us 3,
- Subtract 33 from 40 to get 7,
- And so on.

### Final Result
The repeating decimal pattern becomes clear:
- The steps above show that the result is \( 0.636363... \), where "63" is the repeating part.

Therefore, we can express \( \frac{7}{11} \) as:
\[
0.\overline{63}
\]

This notation indicates that "63" is the repeating sequence in the decimal representation of \( \frac{7}{11} \).