Convert 0.18¯¯¯¯¯ to a rational number (fraction) in simplest form. an show dat work botty

Answer 1

To convert the repeating decimal \( 0.18\overline{18} \) into a fraction, we can use an algebraic approach.

Let \( x = 0.181818...\)

Step 1: Multiply by a power of 10

Since the repeating part has two digits ("18"), we multiply \( x \) by \( 100 \) to shift the decimal point two places to the right:

\[ 100x = 18.181818... \]

Step 2: Set up an equation

Now we have two equations:

1. \( x = 0.181818... \)
2. \( 100x = 18.181818... \)

Step 3: Subtract the first equation from the second

We can subtract the first equation from the second to eliminate the repeating part:

\[ 100x - x = 18.181818... - 0.181818... \] \[ 99x = 18 \]

Step 4: Solve for \( x \)

Now, divide both sides by 99:

\[ x = \frac{18}{99} \]

Step 5: Simplify the fraction

To simplify \( \frac{18}{99} \), we need to find the greatest common divisor (GCD) of 18 and 99.

The GCD of 18 and 99 is 9. Now we divide both the numerator and the denominator by their GCD:

\[ \frac{18 \div 9}{99 \div 9} = \frac{2}{11} \]

Conclusion

Thus, the repeating decimal \( 0.18\overline{18} \) can be expressed as the rational number:

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

Answer 2

is dis right botty?

Answer 3

Yes, the conversion I provided is correct. The repeating decimal \( 0.181818... \) is indeed equal to the fraction \( \frac{2}{11} \) in simplest form.

Here’s a recap of the process:

1. Let \( x = 0.181818... \).
2. Multiply by \( 100 \) to shift the decimal: \( 100x = 18.181818... \).
3. Subtract the original equation from this new equation: \[ 100x - x = 18.181818... - 0.181818... \] which simplifies to \( 99x = 18 \).
4. Solve for \( x \): \[ x = \frac{18}{99}. \]
5. Simplify \( \frac{18}{99} \) by dividing both the numerator and denominator by their GCD, which is \( 9 \): \[ \frac{18 \div 9}{99 \div 9} = \frac{2}{11}. \]

So \( 0.181818... = \frac{2}{11} \). If you have any further questions or need additional clarification, feel free to ask!