Question

Convert 0.18¯¯¯¯¯
0.
18
¯
to a rational number (fraction) in simplest form.(1 point)

Answer 1

you sure

Answer 2

ues less words and symbols

Answer 3

less symbols

Answer 4

To convert the repeating decimal \( 0.18\overline{18} \) to a fraction, we can use the following steps:

1. Let \( x = 0.181818\ldots \).

2. To eliminate the repeating part, we can multiply both sides of the equation by 100 (which shifts the decimal point two places to the right):
\[
100x = 18.181818\ldots
\]

3. Now, we can set up the equation:
\[
100x = 18 + x
\]

4. Subtract \( x \) from both sides:
\[
100x - x = 18
\]
\[
99x = 18
\]

5. Now, solve for \( x \):
\[
x = \frac{18}{99}
\]

6. To simplify \( \frac{18}{99} \), we can divide the numerator and the denominator by their greatest common divisor (GCD), which is 9:
\[
\frac{18 \div 9}{99 \div 9} = \frac{2}{11}
\]

Thus, the repeating decimal \( 0.18\overline{18} \) as a fraction in simplest form is

\[
\boxed{\frac{2}{11}}.
\]

Answer 5

Let's double-check the conversion of the repeating decimal \( 0.18\overline{18} \) to a fraction step by step to ensure accuracy:

1. **Set up the equation:**
Let \( x = 0.181818\ldots \).

2. **Multiply to eliminate the repeating part:**
Since the repeating part "18" has 2 digits, we can multiply by \( 100 \):
\[
100x = 18.181818\ldots
\]

3. **Set up the equation:**
Now subtract the original equation from this one:
\[
100x - x = 18.181818\ldots - 0.181818\ldots
\]
This simplifies to:
\[
99x = 18
\]

4. **Solve for \( x \):**
\[
x = \frac{18}{99}
\]

5. **Simplify the fraction:**
Now, we simplify \( \frac{18}{99} \). The GCD of 18 and 99 is 9:
\[
\frac{18 \div 9}{99 \div 9} = \frac{2}{11}
\]

After verifying the calculations, we conclude that the repeating decimal \( 0.18\overline{18} \) indeed simplifies to the fraction

\[
\boxed{\frac{2}{11}}.
\]

If your concern was about the notation or the calculations, please let me know!

Answer 6

Sure! Here’s a concise solution:

Let \( x = 0.181818\ldots \)
Multiply by 100:
\( 100x = 18.181818\ldots \)

Set up equation:
\( 100x - x = 18 \)
\( 99x = 18 \)
\( x = \frac{18}{99} \)

Simplifying:
\( \frac{18 \div 9}{99 \div 9} = \frac{2}{11} \)

So, \( 0.18\overline{18} = \frac{2}{11} \).
Final answer:
\(\boxed{\frac{2}{11}}\)

Answer 7

Sure! Here's the solution with minimal symbols:

Let \( x = 0.181818... \)
Multiply by 100:
\( 100x = 18.181818... \)

Then:
\( 99x = 18 \)
So, \( x = 18/99 \)

Simplify:
\( 18/99 = 2/11 \)

Final answer:
2/11