Final answer:
The only pair where both decimals represent rational numbers is A. 0753 and 0.232323…, where the first is a terminating decimal and the second is a repeating decimal.
Step-by-step explanation:
The subject of this question is in the field of Mathematics and focuses on identifying which pairs of decimals represent rational numbers. A rational number is a number that can be expressed as a fraction with integers in the numerator and the denominator and the denominator not equal to zero. Looking at the options given in the question, we can assess each pair:
- A. 0753 and 0.232323… - 0753 (assuming the intended decimal is 0.753) is rational, and 0.232323… is also rational because it's a repeating decimal (which can be written as the fraction 23/99).
- B. 8.3182706837521… and 0.752 - The first number has an ellipsis without a clear pattern, suggesting it's not repeating, thus not rational, but 0.752 is a terminating decimal, which means it's rational.
- C. 4.32891789376287184… and 0.2323 - The first number has an ellipsis suggesting it is non-repeating and therefore not rational, but 0.2323 is a terminating decimal, so it's rational.
- D. 8.3182706837521 and 0.232323 - The first number could potentially be non-repeating and non-terminating, and thus not rational, but 0.232323 is a repeating decimal (which can be written as the fraction 23/99) and is rational.
However, the only option where we are certain both numbers are rational is A. 0753 and 0.232323… as both are either terminating or repeating decimals. Terminating decimals like 0.753 can be expressed as the fraction 753/1000. As for repeating decimals like 0.232323..., they can be represented by a fraction where the repeating part dictates the numerator and denominator (in this case, 23/99).