Question

0.85 with the 5 repeating converted into a fraction in simplest form

Answer 1

To convert the repeating decimal \( 0.85\overline{5} \) (which can also be written as \( 0.855555... \)) into a fraction, let's denote it as \( x \):

\[
x = 0.85\overline{5}
\]

Next, we multiply both sides by 10 to shift the decimal point one place to the right:

\[
10x = 8.5\overline{5}
\]

Now, we multiply both sides by 10 again to shift the decimal point one more place to the right:

\[
100x = 85.5\overline{5}
\]

Now, we have two equations:

1. \( 10x = 8.5\overline{5} \)
2. \( 100x = 85.5\overline{5} \)

Next, we can subtract the first equation from the second:

\[
100x - 10x = 85.5\overline{5} - 8.5\overline{5}
\]

This simplifies to:

\[
90x = 77
\]

Now, we solve for \( x \):

\[
x = \frac{77}{90}
\]

To ensure that the fraction is in simplest form, we check if there are any common factors between 77 and 90. The prime factorization reveals:

- \( 77 = 7 \times 11 \)
- \( 90 = 2 \times 3^2 \times 5 \)

Since the prime factors of 77 and 90 have no common elements, the fraction \( \frac{77}{90} \) is already in simplest form.

Thus, the repeating decimal \( 0.85\overline{5} \) converted to a fraction in simplest form is:

\[
\boxed{\frac{77}{90}}
\]