Decimals

For a rational number to have a terminating decimal expansion, its denominator, when reduced to its simplest form, must only have 2 and 5 as its prime factors.
Now, let's check the prime factors of the denominators of each option:

: The prime factorization of 40 is . Since the denominator contains only 2 and 5, the decimal expansion will terminate.
: The prime factorization of 35 is . Since it contains a factor of 7, its decimal expansion will not terminate.
: The prime factorization of 21 is . Since it contains factors other than 2 and 5, its decimal expansion will not terminate.
: The prime factorization of 15 is . Since it contains a factor of 3, its decimal expansion will not terminate.

So, the correct option is (A):

1. The fraction can be simplified by finding the greatest common divisor (GCD) of 140 and 210. 2. The prime factorization of 140 is: The prime factorization of 210 is: 3. The GCD of 140 and 210 is 70. 4. Now, divide both the numerator and the denominator by 70: So, the simplified form of is .

This is a non-terminating and repeating decimal.