Question

Find the fraction equivalent of the number in simplest form.
1.overline235 =

Answer 1

To convert the repeating decimal \( 1.\overline{235} \) into a fraction, we can use the following steps:

1. Let \( x = 1.\overline{235} \).

2. To eliminate the repeating decimal, multiply \( x \) by \( 1000 \) (because 235 has 3 digits):
\[
1000x = 1235.\overline{235}
\]

3. Now, subtract the original \( x \) from this new equation:
\[
1000x - x = 1235.\overline{235} - 1.\overline{235}
\]
\[
999x = 1234
\]

4. Solve for \( x \):
\[
x = \frac{1234}{999}
\]

5. Now, we need to simplify the fraction \( \frac{1234}{999} \). We can find the greatest common divisor (GCD) of the numerator and the denominator.

6. The prime factorization of \( 1234 \) is \( 2 \times 617 \) (617 is a prime number), and the prime factorization of \( 999 \) is \( 3^3 \times 37 \). Since 1234 and 999 share no common factors, their GCD is 1.

7. Therefore, the fraction \( \frac{1234}{999} \) is already in simplest form.

Thus, the fraction equivalent of \( 1.\overline{235} \) in simplest form is:
\[
\frac{1234}{999}
\]