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$Convert the following repeating decimal to a fraction in simplest form. .6 bar(8) Answer:$

Convert the following repeating decimal to a fraction in simplest form. $\newline$ $.6 \overline{8}$ $\newline$ Answer:

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Full solution

Q. Convert the following repeating decimal to a fraction in simplest form.

\newline

.6 \overline{8}

\newline

Answer:

Denote Decimal as $x$ : Let's denote the repeating decimal $0.68\overline{68}$ as $x$ .
$x = 0.68\overline{68}$
To convert a repeating decimal to a fraction, we need to create an equation that when solved, will eliminate the repeating part.
Multiply by $10$ : First, we multiply $x$ by $10$ to shift the decimal point one place to the right, since the repeating part is one digit long. $\newline$ $10x = 6.\overline{8}$ $\newline$ Now we have two equations: $\newline$ $1$ ) $x = 0.\overline{68}$ $\newline$ $2$ ) $10x = 6.\overline{8}$
Subtract Equations: Next, we subtract equation $1$ from equation $2$ to get rid of the repeating part. $\newline$ $10x - x = 6.8\overline{8} - 0.6\overline{8}$ $\newline$ This subtraction will give us a non-repeating decimal result.
Perform Subtraction: Performing the subtraction: $\newline$ $9x = 6.8\overline{ } - 0.68\overline{ }$ $\newline$ $9x = 6.8 - 0.68$ $\newline$ $9x = 6.12$
Solve for x: Now, we solve for x by dividing both sides of the equation by $9$ . $x = \frac{6.12}{9}$
Divide by GCD: Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD). In this case, the GCD of $612$ and $900$ is $3$ . $\newline$ $x = \frac{612 \div 3}{900 \div 3}$ $\newline$ $x = \frac{204}{300}$
Simplify Fraction: We can simplify the fraction further by dividing both the numerator and the denominator by their GCD again, which is now $4$ . $\newline$ $x = (204 \div 4) / (300 \div 4)$ $\newline$ $x = 51 / 75$
Final Simplification: Finally, we simplify the fraction one last time by dividing both the numerator and the denominator by their GCD, which is now $3$ . $\newline$ $x = (51 \div 3) / (75 \div 3)$ $\newline$ $x = 17 / 25$ $\newline$ This is the fraction in its simplest form.