Question

Find an equivalent fraction for the decimal number. In your final answer, include all of your work.

0.61 repeating

Answer 1

can you do it with less words and without all the symbols and slashes

Answer 2

can you add a little more words

Answer 3

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:
\[
\frac{61}{99}
\]

Answer 4

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.

Answer 5

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.