To find the decimal representation of \( \frac{1}{11} \) and deduce the decimal representations of \( \frac{2}{11} \), \( \frac{3}{11} \), and \( \frac{4}{11} \), we can follow these steps:
### Step 1: Find the decimal representation of \( \frac{1}{11} \)
1. **Set up the division**: We need to divide 1 by 11.
2. **Perform the division**:
- Since 1 is less than 11, we write \( 0. \) and add a decimal point.
- We can think of 1 as 1.000000... (adding zeros).
- Now, we divide 10 by 11. Since 11 goes into 10 zero times, we write down 0.
- Bring down another 0, making it 100.
- Now, \( 11 \) goes into \( 100 \) \( 9 \) times (since \( 11 \times 9 = 99 \)).
- Subtract \( 99 \) from \( 100 \) to get \( 1 \).
- Bring down another 0 to make it \( 10 \) again.
- Repeat the process: \( 11 \) goes into \( 10 \) \( 0 \) times, then into \( 100 \) \( 9 \) times, and so on.
- This results in a repeating decimal: \( 0.090909...\).
Thus, we can write:
\[
\frac{1}{11} = 0.\overline{09}
\]
### Step 2: Find the decimal representation of \( \frac{2}{11} \)
1. **Set up the division**: We need to divide 2 by 11.
2. **Perform the division**:
- Since 2 is less than 11, we write \( 0. \) and add a decimal point.
- Think of 2 as \( 2.000000... \).
- Divide \( 20 \) by \( 11 \). \( 11 \) goes into \( 20 \) \( 1 \) time (since \( 11 \times 1 = 11 \)).
- Subtract \( 11 \) from \( 20 \) to get \( 9 \).
- Bring down another \( 0 \) to make it \( 90 \).
- \( 11 \) goes into \( 90 \) \( 8 \) times (since \( 11 \times 8 = 88 \)).
- Subtract \( 88 \) from \( 90 \) to get \( 2 \).
- Bring down another \( 0 \) to make it \( 20 \) again.
- Repeat the process: \( 11 \) goes into \( 20 \) \( 1 \) time, then into \( 90 \) \( 8 \) times, and so on.
- This results in a repeating decimal: \( 0.181818...\).
Thus, we can write:
\[
\frac{2}{11} = 0.\overline{18}
\]
### Step 3: Find the decimal representation of \( \frac{3}{11} \)
1. **Set up the division**: We need to divide 3 by 11.
2. **Perform the division**:
- Since 3 is less than 11, we write \( 0. \) and add a decimal point.
- Think of 3 as \( 3.000000... \).
- Divide \( 30 \) by \( 11 \). \( 11 \) goes into \( 30 \) \( 2 \) times (since \( 11 \times 2 = 22 \)).
- Subtract \( 22 \) from \( 30 \) to get \( 8 \).
- Bring down another \( 0 \) to make it \( 80 \).
- \( 11 \) goes into \( 80 \) \( 7 \) times (since \( 11 \times 7 = 77 \)).
- Subtract \( 77 \) from \( 80 \) to get \( 3 \).
- Bring down another \( 0 \) to make it \( 30 \) again.
- Repeat the process: \( 11 \) goes into \( 30 \) \( 2 \) times, then into \( 80 \) \( 7 \) times, and so on.
- This results in a repeating decimal: \( 0.272727...\).
Thus, we can write:
\[
\frac{3}{11} = 0.\overline{27}
\]
### Step 4: Find the decimal representation of \( \frac{4}{11} \)
1. **Set up the division**: We need to divide 4 by 11.
2. **Perform the division**:
- Since 4 is less than 11, we write \( 0. \) and add a decimal point.
- Think of 4 as \( 4.000000... \).
- Divide \( 40 \) by \( 11 \). \( 11 \) goes into \( 40 \) \( 3 \) times (since \( 11 \times 3 = 33 \)).
- Subtract \( 33 \) from \( 40 \) to get \( 7 \).
- Bring down another \( 0 \) to make it \( 70 \).
- \( 11 \) goes into \( 70 \) \( 6 \) times (since \( 11 \times 6 = 66 \)).
- Subtract \( 66 \) from \( 70 \) to get \( 4 \).
- Bring down another \( 0 \) to make it \( 40 \) again.
- Repeat the process: \( 11 \) goes into \( 40 \) \( 3 \) times, then into \( 70 \) \( 6 \) times, and so on.
- This results in a repeating decimal: \( 0.363636...\).
Thus, we can write:
\[
\frac{4}{11} = 0.\overline{36}
\]
### Summary of Results:
- \( \frac{1}{11} = 0.\overline{09} \)
- \( \frac{2}{11} = 0.\overline{18} \)
- \( \frac{3}{11} = 0.\overline{27} \)
- \( \frac{4}{11} = 0.\overline{36} \)
