Question

5
select the correct answer from each drop-down menu.
complete the statement about the decimal expansion of the fraction \\(\frac{7}{12}\\).
the decimal expansion of the fraction \\(\frac{7}{12}\\) is terminating because when using long division, the remainder never becomes repeating zeros.
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Explanation:

Step1: Analyze the fraction $\frac{7}{12}$

A terminating decimal has a denominator (in simplest form) with only factors of 2 and 5. The denominator 12 factors as $2^2\times3$, so it has a factor of 3. Thus, $\frac{7}{12}$ will have a non - terminating (repeating) decimal expansion.

Step2: Analyze the remainder in long division

When we do long division of 7 by 12, we get $7\div12 = 0.58333\cdots$. In long division, we keep getting a remainder that repeats (in this case, the remainder when we divide 70 by 12 is 10, then 100 by 12 is 4, then 40 by 12 is 4, and so on - the remainder repeats, not becomes repeating zeros). So the first drop - down should be "non - terminating (repeating)" and the second drop - down should be "never" is incorrect, but based on the problem's context of correcting the statement:

The correct first drop - down option: The decimal expansion of $\frac{7}{12}$ is non - terminating (repeating) because the denominator 12 has a prime factor (3) other than 2 and 5.

The second part: When using long division, the remainder never becomes repeating zeros (since the decimal is repeating, the remainders repeat, not become zeros). But to correct the first drop - down:

Answer:

First drop - down: non - terminating (repeating)
Second drop - down: never (this part is correct as the remainder doesn't become repeating zeros, but the first part was wrong. So the correct statement is: The decimal expansion of the fraction $\frac{7}{12}$ is non - terminating (repeating) because when using long division, the remainder never becomes repeating zeros.