Question

Rewrite as a simplified fraction.

\[\large{2.1\overline{6} = ?}\]

Answer 1

[AI-generated answer — Bot GPT 3.5]

To write the decimal as a fraction, we can call the repeating part $x$:

$2.1\overline{6} = 2 + \frac{x}{10}$

Since there is one digit repeating, multiplying by 10 shifts the repeating part one place to the left:

$10x = 26.\overline{6}$

Then, we subtract the original equation from the modified equation to eliminate the repeating part:

$10x - x = 26.\overline{6} - 2.\overline{6} \\ 9x = 24 \\ x = \frac{24}{9} \\ x = \frac{8}{3}$

Finally, we substitute this value back into our original equation:

$2 + \frac{8}{3} = \frac{2 \cdot 3}{1 \cdot 3} + \frac{8}{3} = \frac{6 + 8}{3} = \frac{14}{3}$

Therefore, $\frac{2.1\overline{6}}{1} = \frac{14}{3}$.

Answer 2

wrong

Answer 3

[AI-generated answer — Bot GPT 3.5]

I apologize for the mistake. Let's try again.

To write the decimal as a fraction, we can call the repeating part $x$:

$2.1\overline{6} = 2 + \frac{x}{10}$

To eliminate the repeating part, we can subtract the original equation from the equation with a shifted decimal:

$10(2.1\overline{6}) = 21.\overline{6}$

$10(2 + \frac{x}{10}) = 21.\overline{6}$

$20 + x = 21.\overline{6}$

Now, we subtract the original equation from this equation:

$21.\overline{6} - 2.1\overline{6} = 21.\overline{6} - 2 - \frac{x}{10}$

$19.\overline{6} = 19.\overline{6} - \frac{x}{10}$

To eliminate the repeating part on the right side, we can subtract the left side from both sides of the equation:

$0 = - \frac{x}{10}$

This means $x = 0$.

Now, we substitute this value back into our original equation:

$2 + \frac{0}{10} = \frac{2}{1} = 2$

Therefore, $\frac{2.1\overline{6}}{1} = \frac{2}{1} = 2$.

Answer 4

wrong again

Answer 5

[AI-generated answer — Bot GPT 3.5]

I apologize for the mistake. Let's try again.

To write the decimal as a fraction, we can call the repeating part $x$:

$2.1\overline{6} = 2 + \frac{x}{10}$

To eliminate the repeating part, we can subtract the original equation from the equation with a shifted decimal:

$10(2.1\overline{6}) = 21.\overline{6}$

$10(2 + \frac{x}{10}) = 21.\overline{6}$

$20 + x = 21.\overline{6}$

Now, we subtract the original equation from this equation:

$21.\overline{6} - 2.1\overline{6} = 21.\overline{6} - 2 - \frac{x}{10}$

$19.\overline{6} = 19.\overline{6} - \frac{x}{10}$

To eliminate the repeating part on the right side, we can subtract the left side from both sides of the equation:

$0 = - \frac{x}{10}$

This means $x = 0$.

Now, we substitute this value back into our original equation:

$2 + \frac{0}{10} = \frac{2}{1} = 2$

Therefore, $\frac{2.1\overline{6}}{1} = \frac{2}{1} = 2$.

Answer 6

WONG WRONG WRONG

Answer 7

[AI-generated answer — Bot GPT 3.5]

Apologies for the mistakes. Let's try again.

To write the decimal as a fraction, we can call the repeating part $x$:
$\frac{2.1\overline{6}}{1}=2+\frac{x}{10}$

Since there is only one repeating digit (the 6), we multiply both sides of the equation by 10 to shift the decimal point:
$10\cdot\frac{2.1\overline{6}}{1}=10\cdot\left(2+\frac{x}{10}\right)$

$21.\overline{6}=20+\frac{x}{1}$

Now, subtract the original equation from this equation to eliminate the repeating part on the left side:
$21.\overline{6}-2.1\overline{6}=20+\frac{x}{1}-\left(2+\frac{x}{10}\right)$

$19.\overline{6}=18+\frac{9x}{10}$

To eliminate the repeating part on the right side, subtract $18$ from both sides and multiply both sides by $10$:
$10\cdot(19.\overline{6}-18)=10\cdot\left(18+\frac{9x}{10}-18\right)$

$9.\overline{6}=9x$

Next, divide both sides by $9$ to solve for $x$:
$\frac{9.\overline{6}}{9}=\frac{9x}{9}$

$1.\overline{0}=x$

Finally, substitute this value back into the original equation:
$\frac{2.1\overline{6}}{1}=2+\frac{1.\overline{0}}{10}$

$\frac{2.1\overline{6}}{1}=\frac{20}{10}+\frac{1.\overline{0}}{10}$

$\frac{2.1\overline{6}}{1}=\frac{21.\overline{0}}{10}$

Simplifying the fraction, we get:
$\frac{2.1\overline{6}}{1}=\boxed{\frac{21}{10}}$