Which statement is true when a rational number in fractional form is converted to a decimal?

Question

1. The decimal repeats.
II. The decimal terminates.

Answer 1

When a rational number in fractional form is converted to a decimal, the following statements can be true:

1. The decimal repeats.
2. The decimal terminates.

The truth of these statements depends on the specific rational number:

• A rational number will result in a terminating decimal if its denominator (when in lowest terms) has only the prime factors 2 or 5. For example, \( \frac{1}{4} = 0.25 \).

• A rational number will result in a repeating decimal if its denominator (when in lowest terms) has prime factors other than 2 or 5. For example, \( \frac{1}{3} = 0.333...\), which repeats.

So, both statements can be true under the right conditions.

Answer 2

is 7/8 terminating

Answer 3

Yes, \( \frac{7}{8} \) is a terminating decimal.

When you convert \( \frac{7}{8} \) to a decimal, you can perform the division:

\[ 7 \div 8 = 0.875 \]

Since the decimal representation of \( \frac{7}{8} \) is \( 0.875 \), which does not repeat and ends after three decimal places, it is considered a terminating decimal.

Additionally, the denominator \( 8 \) can be factored into \( 2^3 \), which only has the prime factor \( 2 \), confirming that \( \frac{7}{8} \) produces a terminating decimal.