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Convert 0.18¯¯¯¯¯

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

7 answers

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

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

you sure

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!

ues less words and symbols

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:
\(\boxed{\frac{2}{11}}\)

less symbols

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