To find which number produces a rational number when added to 0.25, we need to understand the properties of rational and irrational numbers.
A rational number is any number that can be expressed as the quotient or fraction [tex]\(\frac{p}{q}\)[/tex] of two integers, where [tex]\(p\)[/tex] and [tex]\(q\)[/tex] are integers and [tex]\(q \neq 0\)[/tex]. An irrational number is a number that cannot be expressed as a simple fraction; its decimal representation goes on forever without repeating.
Now, let's analyze each option to see whether adding it to 0.25 results in a rational number:
1. Option A: 0.45
[tex]\(0.45\)[/tex] is a rational number because it can be expressed as the fraction [tex]\(\frac{45}{100}\)[/tex] or [tex]\(\frac{9}{20}\)[/tex]. Adding two rational numbers results in another rational number. Therefore, [tex]\(0.25 + 0.45 = 0.7\)[/tex], which is [tex]\( \frac{7}{10} \)[/tex] and hence a rational number.
2. Option B: [tex]\(-\sqrt{15}\)[/tex]
[tex]\(\sqrt{15}\)[/tex] is an irrational number because it cannot be expressed as a fraction of two integers. Adding an irrational number to a rational number (0.25 in this case) results in an irrational number. So, [tex]\(0.25 + (-\sqrt{15})\)[/tex] is irrational and therefore not the answer.
3. Option C: 0.54732871...
The given number appears to be a non-repeating, non-terminating decimal, which indicates it is likely an irrational number. Adding this to 0.25, a rational number, would result in an irrational number. So, [tex]\(0.25 + 0.54732871...\)[/tex] is irrational and therefore not the answer.
4. Option D: [tex]\(\pi\)[/tex]
[tex]\(\pi\)[/tex] is a well-known irrational number. Similar to the above, adding [tex]\(\pi\)[/tex] to 0.25 (a rational number) results in an irrational number. Thus, [tex]\(0.25 + \pi\)[/tex] is also irrational and not the answer.
After evaluating all the options, the number that produces a rational number when added to 0.25 is:
A. 0.45
