Question

Question content area top
Part 1
Which numbers are​ rational? The​ dots, . . . ​, indicate that the pattern continues.
I.
7.7777777 . . .
II.
7.70770777 font size decreased by 4 . font size decreased by 4 . font size decreased by 4 .
III.
7.123
Question content area bottom
Part 1
Which numbers are​ rational?
A.
II only
B.
II and III
C.
III only
D.
I only
E.
I and III
F.
None of these

Answer 1

Answer: E (I and III)

Reason:
- A decimal is rational exactly when it terminates or eventually repeats a fixed block of digits.
- I. 7.7777777... = 7.\overline{7} is a repeating decimal (equals 70/9), so rational.
- II. 7.70770777... (the groups of 7s keep increasing) does not settle into a fixed repeating block, so it is nonterminating and nonrepeating — not rational.
- III. 7.123 is a terminating decimal (7123/1000), so rational.

Thus I and III are rational.