Express `5/11` as a recurring decimal.

Answer

To express \( \frac{5}{11} \) as a recurring decimal, we will perform the division step by step.

### Step 1: Set up the division
We need to divide 5 by 11. Since 5 is less than 11, we cannot divide directly. We will start by adding a decimal point and a zero to 5, making it 5.0.

### Step 2: Perform the division
Now we can divide:
- 11 goes into 50 (the first two digits of 5.0) four times because \( 11 \times 4 = 44 \).
- Subtract 44 from 50, which leaves us with a remainder of 6.

### Step 3: Bring down the next zero
Next, we bring down another zero, making it 60.

### Step 4: Continue the division
- Now, we divide 60 by 11.
- 11 goes into 60 five times because \( 11 \times 5 = 55 \).
- Subtract 55 from 60, which leaves us with a remainder of 5.

### Step 5: Repeat the process
Next, we bring down another zero, making it 50 again. We notice that we are back to the same situation as in Step 2:
- 11 goes into 50 four times (as we calculated before).
- This gives us a remainder of 6 again.

Now, we can see that the pattern is repeating:
- \( 11 \) goes into \( 50 \) four times, then into \( 60 \) five times, and this cycle continues.

### Step 6: Write the recurring decimal
Since the digits "45" are repeating, we can express \( \frac{5}{11} \) as:
\[ 0.\overline{45} \]
This notation indicates that "45" is the repeating part of the decimal.

### Final Answer
Thus, \( \frac{5}{11} \) as a recurring decimal is \( 0.\overline{45} \).

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