Question
                Find an equivalent fraction for the decimal number. In your final answer, include all of your work.
0.61 repeating
            
            
        0.61 repeating
Answers
                    Answer
            
    can you do it with less words and without all the symbols and slashes
    
                    Answer
            
    can you add a little more words
    
                    Answered by
            GPT-4o mini
            
    To convert the repeating decimal \( 0.61\overline{61} \) into a fraction, follow these steps:
1. **Define the repeating decimal:**
Let \( x = 0.61616161\ldots \)
2. **Multiply to eliminate the repeating part:**
Since the repeating block has 2 digits, multiply \( x \) by 100:
\[
100x = 61.616161\ldots
\]
3. **Set up the equation:**
Now we have two equations:
\[
x = 0.61616161\ldots \tag{1}
\]
\[
100x = 61.61616161\ldots \tag{2}
\]
4. **Subtract the first equation from the second:**
\[
100x - x = 61.61616161\ldots - 0.61616161\ldots
\]
This simplifies to:
\[
99x = 61
\]
5. **Solve for \( x \):**
\[
x = \frac{61}{99}
\]
6. **Final answer:**
Thus, the repeating decimal \( 0.61\overline{61} \) can be expressed as the fraction:
\[
\frac{61}{99}
\]
This fraction is already in simplest form, as 61 is a prime number and does not have any common factors with 99. Therefore, the final equivalent fraction for the decimal \( 0.61\overline{61} \) is:
\[
\frac{61}{99}
\]
    
1. **Define the repeating decimal:**
Let \( x = 0.61616161\ldots \)
2. **Multiply to eliminate the repeating part:**
Since the repeating block has 2 digits, multiply \( x \) by 100:
\[
100x = 61.616161\ldots
\]
3. **Set up the equation:**
Now we have two equations:
\[
x = 0.61616161\ldots \tag{1}
\]
\[
100x = 61.61616161\ldots \tag{2}
\]
4. **Subtract the first equation from the second:**
\[
100x - x = 61.61616161\ldots - 0.61616161\ldots
\]
This simplifies to:
\[
99x = 61
\]
5. **Solve for \( x \):**
\[
x = \frac{61}{99}
\]
6. **Final answer:**
Thus, the repeating decimal \( 0.61\overline{61} \) can be expressed as the fraction:
\[
\frac{61}{99}
\]
This fraction is already in simplest form, as 61 is a prime number and does not have any common factors with 99. Therefore, the final equivalent fraction for the decimal \( 0.61\overline{61} \) is:
\[
\frac{61}{99}
\]
                    Answered by
            GPT-4o mini
            
    Sure! Here’s a shorter version:
1. Let x equal the repeating decimal:
x = 0.616161...
2. Multiply by 100 to shift the decimal:
100x = 61.616161...
3. Subtract the first equation from the second:
100x - x = 61.616161... - 0.616161...
This gives:
99x = 61
4. Solve for x:
x = 61/99
So, the equivalent fraction for 0.61 repeating is 61/99.
    
1. Let x equal the repeating decimal:
x = 0.616161...
2. Multiply by 100 to shift the decimal:
100x = 61.616161...
3. Subtract the first equation from the second:
100x - x = 61.616161... - 0.616161...
This gives:
99x = 61
4. Solve for x:
x = 61/99
So, the equivalent fraction for 0.61 repeating is 61/99.
                    Answered by
            GPT-4o mini
            
    Of course! Here’s a more detailed explanation, while still keeping it concise:
1. Start by letting x equal the repeating decimal:
x = 0.616161...
2. To eliminate the repeating part, multiply both sides of the equation by 100 (since the repeating block "61" has 2 digits):
100x = 61.616161...
3. Now, subtract the first equation (x = 0.616161...) from the second equation (100x = 61.616161...):
100x - x = 61.616161... - 0.616161...
This simplifies to:
99x = 61
4. Next, solve for x by dividing both sides by 99:
x = 61/99
Thus, the repeating decimal 0.61 repeating can be expressed as the fraction 61/99. This fraction is in its simplest form since 61 is a prime number and doesn't share any common factors with 99.
    
1. Start by letting x equal the repeating decimal:
x = 0.616161...
2. To eliminate the repeating part, multiply both sides of the equation by 100 (since the repeating block "61" has 2 digits):
100x = 61.616161...
3. Now, subtract the first equation (x = 0.616161...) from the second equation (100x = 61.616161...):
100x - x = 61.616161... - 0.616161...
This simplifies to:
99x = 61
4. Next, solve for x by dividing both sides by 99:
x = 61/99
Thus, the repeating decimal 0.61 repeating can be expressed as the fraction 61/99. This fraction is in its simplest form since 61 is a prime number and doesn't share any common factors with 99.
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