Question
What is 211
2
11
written as a decimal?(1 point)
Responses
0.2¯¯¯
0.
2
¯
0 point repeating 2
0.8¯¯¯
0.
8
¯
0 point repeating 8
0.02¯¯¯¯¯
0.
02
¯
0 point repeating 0 2
0.18¯¯¯¯¯
2
11
written as a decimal?(1 point)
Responses
0.2¯¯¯
0.
2
¯
0 point repeating 2
0.8¯¯¯
0.
8
¯
0 point repeating 8
0.02¯¯¯¯¯
0.
02
¯
0 point repeating 0 2
0.18¯¯¯¯¯
Answers
Answer
show you work
Answer
dumb the steps down
Answer
less words and symbols
Answered by
GPT-4o mini
To convert the fraction \( \frac{211}{211} \) into a decimal, we note that:
\[
\frac{211}{211} = 1
\]
Now, let's analyze the other fractions.
1. \( \frac{2}{11} \):
To convert \( \frac{2}{11} \) into a decimal, we can perform long division. Dividing 2 by 11 gives approximately 0.181818... which can be represented as:
\[
0.18\overline{18}
\]
2. \( \frac{2}{2} \):
This equals 1.
3. \( \frac{1}{11} \):
Performing long division again, \( \frac{1}{11} \) gives approximately 0.090909... or:
\[
0.09\overline{09}
\]
4. Putting it all together, we focus on the fractions provided in the responses. The correct conversion from \( \frac{2}{11} \) would correspond to
\[
0.18\overline{18}
\]
However, based on your direct response options, it seems you may just want the \( 0.18\overline{2} \) which closely resembles \( 0.18\overline{18} \).
So if you have to choose from the options provided, it suggests a misunderstanding in fractions representation.
Thus, the answer among your options that closely aligns with
the fraction \( \frac{2}{11} \) is:
**0.18̅ (0.18 repeating)**.
\[
\frac{211}{211} = 1
\]
Now, let's analyze the other fractions.
1. \( \frac{2}{11} \):
To convert \( \frac{2}{11} \) into a decimal, we can perform long division. Dividing 2 by 11 gives approximately 0.181818... which can be represented as:
\[
0.18\overline{18}
\]
2. \( \frac{2}{2} \):
This equals 1.
3. \( \frac{1}{11} \):
Performing long division again, \( \frac{1}{11} \) gives approximately 0.090909... or:
\[
0.09\overline{09}
\]
4. Putting it all together, we focus on the fractions provided in the responses. The correct conversion from \( \frac{2}{11} \) would correspond to
\[
0.18\overline{18}
\]
However, based on your direct response options, it seems you may just want the \( 0.18\overline{2} \) which closely resembles \( 0.18\overline{18} \).
So if you have to choose from the options provided, it suggests a misunderstanding in fractions representation.
Thus, the answer among your options that closely aligns with
the fraction \( \frac{2}{11} \) is:
**0.18̅ (0.18 repeating)**.
Answered by
GPT-4o mini
To convert the fraction \(\frac{2}{11}\) into a decimal, we can perform long division. Here's how that process works:
### Long Division of \(2\) by \(11\)
1. **Set Up the Division:**
- We want to divide \(2.000000...\) by \(11\). We can add decimal places to \(2\) to assist with the division.
2. **First Division:**
- **11 into 20**: \(11\) goes into \(20\) \(1\) time.
- Write \(1\) above the line.
- Multiply \(1 \times 11 = 11\).
- Subtract \(11\) from \(20\) to get \(9\).
\[
\begin{array}{r|l}
& 0.1 \\
\hline
11 & 2.000000 \\
& -11 \\
\hline
& 9 \\
\end{array}
\]
3. **Bring Down the Next Digit:**
- Bring down a \(0\) to make it \(90\).
4. **Second Division:**
- **11 into 90**: \(11\) goes into \(90\) \(8\) times.
- Write \(8\) above the line.
- Multiply \(8 \times 11 = 88\).
- Subtract \(88\) from \(90\) to get \(2\).
\[
\begin{array}{r|l}
& 0.18 \\
\hline
11 & 2.000000 \\
& -11 \\
\hline
& 90 \\
& -88 \\
\hline
& 2 \\
\end{array}
\]
5. **Bring Down the Next Digit:**
- Bring down another \(0\) to make \(20\).
6. **Repeat:**
- Now, we see that we are back where we started!
- **11 into 20**: \(11\) goes into \(20\) \(1\) time again.
- This means the decimals will start repeating: thus, we have \(0.181818...\).
### Conclusion
The repeating decimal can be represented as:
\[
0.181818\ldots = 0.18\overline{18}
\]
### Final Answer Selection
Given the options you provided:
- **0.2̅ (0 point repeating 2)**
- **0.8̅ (0 point repeating 8)**
- **0.02̅̅̅̅ (0 point repeating 0 2)**
- **0.18̅̅̅̅ (0 point repeating 18)**
The correct decimal representation of \(\frac{2}{11}\) is:
**0.18̅ (0 point repeating 18)**.
This work confirms that the choice is indeed correct based on the initial conversion through long division.
### Long Division of \(2\) by \(11\)
1. **Set Up the Division:**
- We want to divide \(2.000000...\) by \(11\). We can add decimal places to \(2\) to assist with the division.
2. **First Division:**
- **11 into 20**: \(11\) goes into \(20\) \(1\) time.
- Write \(1\) above the line.
- Multiply \(1 \times 11 = 11\).
- Subtract \(11\) from \(20\) to get \(9\).
\[
\begin{array}{r|l}
& 0.1 \\
\hline
11 & 2.000000 \\
& -11 \\
\hline
& 9 \\
\end{array}
\]
3. **Bring Down the Next Digit:**
- Bring down a \(0\) to make it \(90\).
4. **Second Division:**
- **11 into 90**: \(11\) goes into \(90\) \(8\) times.
- Write \(8\) above the line.
- Multiply \(8 \times 11 = 88\).
- Subtract \(88\) from \(90\) to get \(2\).
\[
\begin{array}{r|l}
& 0.18 \\
\hline
11 & 2.000000 \\
& -11 \\
\hline
& 90 \\
& -88 \\
\hline
& 2 \\
\end{array}
\]
5. **Bring Down the Next Digit:**
- Bring down another \(0\) to make \(20\).
6. **Repeat:**
- Now, we see that we are back where we started!
- **11 into 20**: \(11\) goes into \(20\) \(1\) time again.
- This means the decimals will start repeating: thus, we have \(0.181818...\).
### Conclusion
The repeating decimal can be represented as:
\[
0.181818\ldots = 0.18\overline{18}
\]
### Final Answer Selection
Given the options you provided:
- **0.2̅ (0 point repeating 2)**
- **0.8̅ (0 point repeating 8)**
- **0.02̅̅̅̅ (0 point repeating 0 2)**
- **0.18̅̅̅̅ (0 point repeating 18)**
The correct decimal representation of \(\frac{2}{11}\) is:
**0.18̅ (0 point repeating 18)**.
This work confirms that the choice is indeed correct based on the initial conversion through long division.
Answered by
GPT-4o mini
Sure! Let's break it down into simpler, more straightforward steps for converting \(\frac{2}{11}\) into a decimal using long division.
### Step-by-Step Long Division of \(2\) by \(11\)
1. **Set It Up:**
- We want to divide \(2\) by \(11\).
- Since \(2\) is smaller than \(11\), we can't divide it yet, so we add a decimal point and some zeros: \(2.000000...\).
2. **First Division:**
- **How many times does \(11\) go into \(20\)**?
- It goes **1** time. Write **1** above the line.
\[
11 \, \, | \, \, 2.0
\]
- Write \(1\) above \(20\).
3. **Subtract:**
- Now, multiply \(1\) (the number you wrote above) by \(11\) to get \(11\).
- Subtract \(11\) from \(20\):
\[
20 - 11 = 9
\]
4. **Bring Down the Next \(0\):**
- Bring down the next \(0\) to make it \(90\).
5. **Second Division:**
- **How many times does \(11\) go into \(90\)**?
- It goes **8** times. Write **8** above the line next to \(1\).
\[
11 \, \, | \, \, 2.0
\]
- Now, you have \(0.18\).
6. **Subtract Again:**
- Multiply \(8\) by \(11\) to get \(88\).
- Subtract \(88\) from \(90\):
\[
90 - 88 = 2
\]
7. **Repeat: Bring Down the Next \(0\):**
- Bring down another \(0\) to make it \(20\) again.
8. **Continue the Division:**
- **How many times does \(11\) go into \(20\)**?
- It goes **1** time (like in step 2).
- You add another \(1\) above the line.
9. **Subtract Again:**
- Multiply \(1\) by \(11\) to get \(11\), and subtract it from \(20\) to get \(9\) again, then bring down the next \(0\) to make it \(90\).
10. **Notice the Pattern:**
- If you continue this process, you will keep getting the same results:
- \(1\) goes into \(20\) (making \(9\)), then bring down the \(0\) to \(90\),
- \(8\) goes into \(90\) (making \(2\)), and you repeat.
### Final Result
After going through these steps, you can see that the digits **18** keep repeating.
So, \(\frac{2}{11}\) is approximately:
\[
0.181818\ldots \text{ (which can be written as } 0.18\overline{18}\text{)}
\]
### Correct Answer
From your options, the correct answer for \(\frac{2}{11}\) is:
**0.18̅ (0 point repeating 18)**.
This means the decimal representation of \(\frac{2}{11}\) is \(0.181818...\) with \(18\) repeating forever!
### Step-by-Step Long Division of \(2\) by \(11\)
1. **Set It Up:**
- We want to divide \(2\) by \(11\).
- Since \(2\) is smaller than \(11\), we can't divide it yet, so we add a decimal point and some zeros: \(2.000000...\).
2. **First Division:**
- **How many times does \(11\) go into \(20\)**?
- It goes **1** time. Write **1** above the line.
\[
11 \, \, | \, \, 2.0
\]
- Write \(1\) above \(20\).
3. **Subtract:**
- Now, multiply \(1\) (the number you wrote above) by \(11\) to get \(11\).
- Subtract \(11\) from \(20\):
\[
20 - 11 = 9
\]
4. **Bring Down the Next \(0\):**
- Bring down the next \(0\) to make it \(90\).
5. **Second Division:**
- **How many times does \(11\) go into \(90\)**?
- It goes **8** times. Write **8** above the line next to \(1\).
\[
11 \, \, | \, \, 2.0
\]
- Now, you have \(0.18\).
6. **Subtract Again:**
- Multiply \(8\) by \(11\) to get \(88\).
- Subtract \(88\) from \(90\):
\[
90 - 88 = 2
\]
7. **Repeat: Bring Down the Next \(0\):**
- Bring down another \(0\) to make it \(20\) again.
8. **Continue the Division:**
- **How many times does \(11\) go into \(20\)**?
- It goes **1** time (like in step 2).
- You add another \(1\) above the line.
9. **Subtract Again:**
- Multiply \(1\) by \(11\) to get \(11\), and subtract it from \(20\) to get \(9\) again, then bring down the next \(0\) to make it \(90\).
10. **Notice the Pattern:**
- If you continue this process, you will keep getting the same results:
- \(1\) goes into \(20\) (making \(9\)), then bring down the \(0\) to \(90\),
- \(8\) goes into \(90\) (making \(2\)), and you repeat.
### Final Result
After going through these steps, you can see that the digits **18** keep repeating.
So, \(\frac{2}{11}\) is approximately:
\[
0.181818\ldots \text{ (which can be written as } 0.18\overline{18}\text{)}
\]
### Correct Answer
From your options, the correct answer for \(\frac{2}{11}\) is:
**0.18̅ (0 point repeating 18)**.
This means the decimal representation of \(\frac{2}{11}\) is \(0.181818...\) with \(18\) repeating forever!
Answered by
GPT-4o mini
Got it! Here’s a simpler, shorter version:
### Long Division of \(2\) by \(11\)
1. Write it as \(2.000...\).
2. **First Division:**
- \(11\) into \(20\) goes \(1\) time.
- Write \(1\) above the line.
3. **Subtract:**
- \(20 - 11 = 9\).
4. **Bring down \(0\):**
- Now you have \(90\).
5. **Second Division:**
- \(11\) into \(90\) goes \(8\) times.
- Write \(8\) above the line.
6. **Subtract:**
- \(90 - 88 = 2\).
7. **Bring down another \(0\):**
- Now you have \(20\) again.
8. **Repeat:**
- \(11\) into \(20\) goes \(1\) time.
- \(11\) into \(90\) goes \(8\) times again.
### Result
You will keep getting \(0.181818...\).
### Final Answer
So, \(\frac{2}{11} = 0.18\overline{18}\).
**Correct choice:** **0.18̅** (0 point repeating 18).
### Long Division of \(2\) by \(11\)
1. Write it as \(2.000...\).
2. **First Division:**
- \(11\) into \(20\) goes \(1\) time.
- Write \(1\) above the line.
3. **Subtract:**
- \(20 - 11 = 9\).
4. **Bring down \(0\):**
- Now you have \(90\).
5. **Second Division:**
- \(11\) into \(90\) goes \(8\) times.
- Write \(8\) above the line.
6. **Subtract:**
- \(90 - 88 = 2\).
7. **Bring down another \(0\):**
- Now you have \(20\) again.
8. **Repeat:**
- \(11\) into \(20\) goes \(1\) time.
- \(11\) into \(90\) goes \(8\) times again.
### Result
You will keep getting \(0.181818...\).
### Final Answer
So, \(\frac{2}{11} = 0.18\overline{18}\).
**Correct choice:** **0.18̅** (0 point repeating 18).