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, $\frac{2.1\overline{6}}{1} = \frac{14}{3}$.
Rewrite as a simplified fraction.
\[\large{2.1\overline{6} = ?}\]
7 answers
wrong
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, $\frac{2.1\overline{6}}{1} = \frac{2}{1} = 2$.
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, $\frac{2.1\overline{6}}{1} = \frac{2}{1} = 2$.
wrong again
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, $\frac{2.1\overline{6}}{1} = \frac{2}{1} = 2$.
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, $\frac{2.1\overline{6}}{1} = \frac{2}{1} = 2$.
WONG WRONG WRONG
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:
$\frac{2.1\overline{6}}{1}=\boxed{\frac{21}{10}}$
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:
$\frac{2.1\overline{6}}{1}=\boxed{\frac{21}{10}}$