Find the period of decimals of `(22)/(7)` and the length of its period.

Answer

To find the period of the decimal representation of \( \frac{22}{7} \) and the length of its period, we will perform the long division of 22 by 7 step by step.

### Step 1: Start the Division
Divide 22 by 7.

- \( 7 \) goes into \( 22 \) three times (since \( 7 \times 3 = 21 \)).
- Write down \( 3 \) as the integer part of the quotient.
- Subtract \( 21 \) from \( 22 \) to get a remainder of \( 1 \).

**Hint:** When dividing, always multiply the divisor by the largest whole number that fits into the current dividend.

### Step 2: Introduce Decimal
Since we have a remainder, we will now add a decimal point to the quotient and bring down a zero.

- The current remainder is \( 1 \). Bring down a \( 0 \) to make it \( 10 \).

**Hint:** Adding a decimal point allows us to continue dividing into decimal places.

### Step 3: Continue Division
Now divide \( 10 \) by \( 7 \).

- \( 7 \) goes into \( 10 \) once (since \( 7 \times 1 = 7 \)).
- Write down \( 1 \) after the decimal point.
- Subtract \( 7 \) from \( 10 \) to get a remainder of \( 3 \).

**Hint:** Always keep track of the remainders, as they will help identify repeating patterns.

### Step 4: Repeat the Process
Now, we will bring down another \( 0 \) to the remainder \( 3 \).

- Now we have \( 30 \).
- \( 7 \) goes into \( 30 \) four times (since \( 7 \times 4 = 28 \)).
- Write down \( 4 \).
- Subtract \( 28 \) from \( 30 \) to get a remainder of \( 2 \).

**Hint:** Keep bringing down zeros to continue the division process.

### Step 5: Continue Dividing
Bring down another \( 0 \) to the remainder \( 2 \).

- Now we have \( 20 \).
- \( 7 \) goes into \( 20 \) two times (since \( 7 \times 2 = 14 \)).
- Write down \( 2 \).
- Subtract \( 14 \) from \( 20 \) to get a remainder of \( 6 \).

**Hint:** Each step gives you a new digit in the decimal representation.

### Step 6: More Division
Bring down another \( 0 \) to the remainder \( 6 \).

- Now we have \( 60 \).
- \( 7 \) goes into \( 60 \) eight times (since \( 7 \times 8 = 56 \)).
- Write down \( 8 \).
- Subtract \( 56 \) from \( 60 \) to get a remainder of \( 4 \).

**Hint:** The process continues until you notice a repeating remainder.

### Step 7: Identify Repeating Remainders
Bring down another \( 0 \) to the remainder \( 4 \).

- Now we have \( 40 \).
- \( 7 \) goes into \( 40 \) five times (since \( 7 \times 5 = 35 \)).
- Write down \( 5 \).
- Subtract \( 35 \) from \( 40 \) to get a remainder of \( 5 \).

**Hint:** If you see a remainder that has appeared before, it indicates the start of a repeating cycle.

### Step 8: Continue Until Repeat
Bring down another \( 0 \) to the remainder \( 5 \).

- Now we have \( 50 \).
- \( 7 \) goes into \( 50 \) seven times (since \( 7 \times 7 = 49 \)).
- Write down \( 7 \).
- Subtract \( 49 \) from \( 50 \) to get a remainder of \( 1 \).

**Hint:** Notice that we have returned to a remainder of \( 1 \), which we started with, indicating the decimal will repeat.

### Final Result
The decimal representation of \( \frac{22}{7} \) is \( 3.142857142857... \)

The repeating part (the period) is \( 142857 \), and its length is \( 6 \).

### Summary
- The period of the decimal of \( \frac{22}{7} \) is \( 142857 \).
- The length of the period is \( 6 \).