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
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:
\[
\frac{61}{99}
\]
Answer
can you do it with less words and without all the symbols and slashes
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.
Answer
can you add a little more words
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.