Question

Which numbers are​ rational? The​ dots, . . . ​, indicate that the pattern continues.
I.
4.4444444 . . .
II.
4.123
III.
4.40440444 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.
III only
B.
I and II
C.
I only
D.
II only
E.
II and III
F.
None of these

Answer 1

To determine which of the given numbers are rational, we need to check if they can be expressed as a fraction \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \).

1. **I. \( 4.4444444 \ldots \)**: This number is a repeating decimal. It can be expressed as \( 4.\overline{4} \), which is a rational number.

2. **II. \( 4.123 \)**: This number is a terminating decimal, which can also be expressed as a fraction (e.g., \( 4123/1000 \)). Therefore, it is rational.

3. **III. \( 4.40440444 \ldots \)**: This number appears to be another repeating decimal (the part "404" repeats). This can also be expressed as a rational number.

Since all three numbers (I, II, and III) are rational, the answer is not listed as an option from the provided choices. However, given the choices, if the question is asking for which to select based on the rationality of the numbers:

- **I and II** are rational.
- **III** is also rational, but since the choices do not cater for all three correctly, the best option that includes the rational numbers appears to be:

**B. I and II** (although it misses III, but is the best match according to the available options).