Answer :
To solve this question, we need to determine which one of the given numbers produces a rational number when multiplied by 0.25. Let's analyze each option step by step.
### Definitions:
- Rational Number: 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].
### Analysis of each option:
1. Option A: [tex]\(0.54713218\)[/tex]
- Multiplying by 0.25:
- [tex]\(0.25 \times 0.54713218 = 0.136783045\)[/tex]
- 0.136783045 is a decimal that does not obviously simplify to a fraction of two integers, meaning we cannot confirm it’s rational just from this multiplication.
2. Option B: [tex]\( -\sqrt{15} \)[/tex]
- Multiplying by 0.25:
- [tex]\(0.25 \times -\sqrt{15} = -0.25 \sqrt{15}\)[/tex]
- Since [tex]\(\sqrt{15}\)[/tex] is an irrational number, multiplying by -0.25 (a rational number) will still result in an irrational number.
3. Option C: [tex]\(0.45\)[/tex]
- Multiplying by 0.25:
- [tex]\(0.25 \times 0.45 = 0.1125\)[/tex]
- 0.1125 is a decimal, and on further investigation (0.1125 = [tex]\(\frac{9}{80}\)[/tex]) it could potentially be rational but we need clear-cut identification.
4. Option D: [tex]\(\pi\)[/tex]
- Multiplying by 0.25:
- [tex]\(0.25 \times \pi = 0.25 \pi\)[/tex]
- [tex]\(\pi\)[/tex] is an irrational number, so any real multiple of [tex]\(\pi\)[/tex] will still be an irrational number.
### Conclusion:
None of the given options multiplied by 0.25 result in a clear-cut rational number without further conversion or simplification. Therefore, after detailed consideration, it is evident that there isn't a definitive rational result from any of the options provided when multiplied by 0.25. Thus, the correct conclusion is:
None of the options (A, B, C, D) produces a rational number when multiplied by 0.25.
### Definitions:
- Rational Number: 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].
### Analysis of each option:
1. Option A: [tex]\(0.54713218\)[/tex]
- Multiplying by 0.25:
- [tex]\(0.25 \times 0.54713218 = 0.136783045\)[/tex]
- 0.136783045 is a decimal that does not obviously simplify to a fraction of two integers, meaning we cannot confirm it’s rational just from this multiplication.
2. Option B: [tex]\( -\sqrt{15} \)[/tex]
- Multiplying by 0.25:
- [tex]\(0.25 \times -\sqrt{15} = -0.25 \sqrt{15}\)[/tex]
- Since [tex]\(\sqrt{15}\)[/tex] is an irrational number, multiplying by -0.25 (a rational number) will still result in an irrational number.
3. Option C: [tex]\(0.45\)[/tex]
- Multiplying by 0.25:
- [tex]\(0.25 \times 0.45 = 0.1125\)[/tex]
- 0.1125 is a decimal, and on further investigation (0.1125 = [tex]\(\frac{9}{80}\)[/tex]) it could potentially be rational but we need clear-cut identification.
4. Option D: [tex]\(\pi\)[/tex]
- Multiplying by 0.25:
- [tex]\(0.25 \times \pi = 0.25 \pi\)[/tex]
- [tex]\(\pi\)[/tex] is an irrational number, so any real multiple of [tex]\(\pi\)[/tex] will still be an irrational number.
### Conclusion:
None of the given options multiplied by 0.25 result in a clear-cut rational number without further conversion or simplification. Therefore, after detailed consideration, it is evident that there isn't a definitive rational result from any of the options provided when multiplied by 0.25. Thus, the correct conclusion is:
None of the options (A, B, C, D) produces a rational number when multiplied by 0.25.
