Question

Which numbers are​ rational? The​ dots, . . . ​, indicate that the pattern continues.
I.
7.7777777 . . .
II.
7.234
III.
7.70770777 font size decreased by 4 . font size decreased by 4 . font size decreased by 4 .
Question content area bottom
Part 1
Which numbers are​ rational?
A.
II only
B.
I only
C.
II and III
D.
III only
E.
I and II
F.
None of these

Answer 1

Rational numbers have decimal representations that either terminate or eventually repeat a fixed block.

I. 7.7777777... = 7.\overline{7} is a repeating decimal → rational.
II. 7.234 is a terminating decimal → rational.
III. 7.70770777... shows no fixed repeating block (the runs of 7's grow), so it is nonterminating and nonrepeating → not rational.

Therefore I and II are rational. Answer: E.