Express each of the following as a recurring decimal :
`17/90`

Answer

To express \( \frac{17}{90} \) as a recurring decimal, we will perform the division step by step.

### Step 1: Set up the division
We need to divide 17 by 90. Since 17 is less than 90, we start with 0 and add a decimal point.

\[ 0. \]

### Step 2: Add zeros to 17
To continue the division, we can add a decimal point and zeros to 17. This gives us 170.

### Step 3: Divide 170 by 90
Now we divide 170 by 90.

\[ 90 \text{ goes into } 170 \text{ one time.} \]

So we write 1 in the quotient:

\[ 0.1 \]

### Step 4: Calculate the remainder
Now, we calculate the remainder:

\[ 170 - 90 = 80 \]

### Step 5: Bring down another 0
Next, we bring down another 0, making it 800.

### Step 6: Divide 800 by 90
Now we divide 800 by 90.

\[ 90 \text{ goes into } 800 \text{ eight times.} \]

So we write 8 in the quotient:

\[ 0.18 \]

### Step 7: Calculate the new remainder
Now we calculate the new remainder:

\[ 800 - (90 \times 8) = 800 - 720 = 80 \]

### Step 8: Recognize the repeating pattern
We notice that we have a remainder of 80 again. This means that the next step will repeat the previous steps.

Thus, we can conclude that the decimal representation of \( \frac{17}{90} \) is:

\[ 0.1\overline{8} \]

### Final Answer
The recurring decimal representation of \( \frac{17}{90} \) is:

\[ 0.1\overline{8} \]

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