Express the rational number `1//13` in decimal form and hence, find the decimal expansion of `4(4)/(13)` .

Answer

To solve the problem of expressing the rational number \( \frac{1}{13} \) in decimal form and then finding the decimal expansion of \( 4 \frac{4}{13} \), we can follow these steps:

### Step 1: Convert \( \frac{1}{13} \) to Decimal Form
To convert \( \frac{1}{13} \) into decimal form, we perform the division of 1 by 13.

1. Divide 1 by 13:
- 1.000000 (add decimal and zeros for division)
- 13 goes into 1, 0 times.
- 13 goes into 10, 0 times.
- 13 goes into 100, 7 times (since \( 13 \times 7 = 91 \)).
- Subtract 91 from 100, which gives 9.
- Bring down the next 0 to make it 90.
- 13 goes into 90, 6 times (since \( 13 \times 6 = 78 \)).
- Subtract 78 from 90, which gives 12.
- Bring down the next 0 to make it 120.
- 13 goes into 120, 9 times (since \( 13 \times 9 = 117 \)).
- Subtract 117 from 120, which gives 3.
- Bring down the next 0 to make it 30.
- 13 goes into 30, 2 times (since \( 13 \times 2 = 26 \)).
- Subtract 26 from 30, which gives 4.
- Bring down the next 0 to make it 40.
- 13 goes into 40, 3 times (since \( 13 \times 3 = 39 \)).
- Subtract 39 from 40, which gives 1.
- Bring down the next 0 to make it 10.
- Repeat the process.

The decimal expansion of \( \frac{1}{13} \) is \( 0.076923 \) (and it repeats).

### Step 2: Find the Decimal Expansion of \( 4 \frac{4}{13} \)
Next, we need to find the decimal expansion of \( 4 \frac{4}{13} \).

1. Convert \( 4 \frac{4}{13} \) into an improper fraction:
\[ 4 \frac{4}{13} = 4 + \frac{4}{13} = \frac{4 \times 13 + 4}{13} = \frac{52 + 4}{13} = \frac{56}{13} \]

2. Now, we will convert \( \frac{56}{13} \) into decimal form by performing the division of 56 by 13:
- 56 divided by 13:
- 13 goes into 56, 4 times (since \( 13 \times 4 = 52 \)).
- Subtract 52 from 56, which gives 4.
- Bring down the next 0 to make it 40.
- 13 goes into 40, 3 times (since \( 13 \times 3 = 39 \)).
- Subtract 39 from 40, which gives 1.
- Bring down the next 0 to make it 10.
- 13 goes into 10, 0 times.
- Bring down the next 0 to make it 100.
- 13 goes into 100, 7 times (since \( 13 \times 7 = 91 \)).
- Subtract 91 from 100, which gives 9.
- Bring down the next 0 to make it 90.
- 13 goes into 90, 6 times (since \( 13 \times 6 = 78 \)).
- Subtract 78 from 90, which gives 12.
- Bring down the next 0 to make it 120.
- 13 goes into 120, 9 times (since \( 13 \times 9 = 117 \)).
- Subtract 117 from 120, which gives 3.
- Bring down the next 0 to make it 30.
- 13 goes into 30, 2 times (since \( 13 \times 2 = 26 \)).
- Subtract 26 from 30, which gives 4.
- Bring down the next 0 to make it 40.
- 13 goes into 40, 3 times (since \( 13 \times 3 = 39 \)).
- Subtract 39 from 40, which gives 1.
- Bring down the next 0 to make it 10.
- Repeat the process.

The decimal expansion of \( 4 \frac{4}{13} \) is \( 4.307692 \) (and it repeats).

### Final Answer
- The decimal form of \( \frac{1}{13} \) is \( 0.076923 \).
- The decimal expansion of \( 4 \frac{4}{13} \) is \( 4.307692 \).