To determine which number produces a rational number when added to 0.25, let's evaluate each option step-by-step:
1. Option A: 0.45
When we add 0.25 and 0.45:
[tex]\[
0.25 + 0.45 = 0.70
\][/tex]
0.70 is a rational number because it can be expressed as the fraction [tex]\( \frac{70}{100} \)[/tex].
2. Option B: [tex]\( -\sqrt{15} \)[/tex]
When we add 0.25 and [tex]\( -\sqrt{15} \)[/tex]:
[tex]\[
0.25 + (-\sqrt{15})
\][/tex]
Since [tex]\( \sqrt{15} \)[/tex] is an irrational number, adding it to a rational number (0.25) will still result in an irrational number. Therefore, this sum is irrational.
3. Option C: [tex]\( \pi \)[/tex]
When we add 0.25 and [tex]\( \pi \)[/tex]:
[tex]\[
0.25 + \pi
\][/tex]
Since [tex]\( \pi \)[/tex] is an irrational number, adding it to a rational number (0.25) will also result in an irrational number. Hence, this sum is irrational.
4. Option D: 0.54732871...
When we add 0.25 and 0.54732871:
[tex]\[
0.25 + 0.54732871 = 0.79732871
\][/tex]
0.79732871 is a rational number because it can be expressed as a fraction (assuming the decimal representation terminates or repeats).
Thus, the numbers that produce a rational sum when added to 0.25 are:
- Option A: 0.45
- Option D: 0.54732871...
So the correct numbers are 0.45 and 0.54732871.
