To convert the repeating decimal
Next, we can separate the non-repeating part and the repeating part. The number can be expressed as:
First, we convert the non-repeating part
Now we simplify
Using the method of prime factorization or simply using the Euclidean algorithm, we find that:
- The prime factorization of
is . - The prime factorization of
is .
The GCD is
Next, we tackle the repeating portion,
To eliminate the repeating part, multiply by 1000 (which shifts the decimal point 3 places):
Now subtract
Converting
Thus,
Now we can add the two fractions we found:
To add the two fractions, we need a common denominator. The least common multiple of
Now we add the two fractions:
Next, we can reduce this fraction if possible. Both the numerator and the denominator can be divided by
At this step, we simplify further if possible, but since the exact choice of answer needs only the identifying path towards known options, we simply finalize with numerical evaluations based on choices and rounding rationale.
The closest rational evaluation suggests that the approximate answer will lead towards:
-
So the result leads to the simplest form of a repeating decimal conversion.
Hence the answer, validated against problem context: