Question

Write the decimal as a fraction.
0.8520.852
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Part 1
The

Answer 1

To convert the repeating decimal \( 0.8520\overline{852} \) into a fraction, we will define the decimal as a variable and use algebraic methods.

Let \( x = 0.8520\overline{852} \).

To eliminate the repeating decimal, we can multiply \( x \) by a power of 10 that shifts the decimal point to the right of the repeating part. Since "852" has three digits, we multiply by \( 1000 \):

\[
1000x = 852.8520\overline{852}
\]

Next, we also express \( x \) again:

\[
1000x = 852.8520\overline{852}
\]
\[
x = 0.8520\overline{852}
\]

Now, we subtract the second equation from the first:

\[
1000x - x = 852.8520\overline{852} - 0.8520\overline{852}
\]
\[
999x = 852
\]

Now, we can solve for \( x \):

\[
x = \frac{852}{999}
\]

Next, we simplify the fraction \( \frac{852}{999} \). We can find the greatest common divisor (GCD) of the numerator and the denominator.

The GCD of 852 and 999 is 3. We divide both the numerator and denominator by their GCD:

\[
x = \frac{852 \div 3}{999 \div 3} = \frac{284}{333}
\]

Thus, the decimal \( 0.8520\overline{852} \) can be expressed as the fraction:

\[
\frac{284}{333}
\]