To determine which number produces a rational number when added to 0.25, we need to identify whether each given option results in a rational number. A rational number is a number that can be expressed as the quotient of two integers, i.e., \( \frac{a}{b} \) where \(a\) and \(b\) are integers and \( b \neq 0 \).
Let's evaluate each option:
Option A: \( -\sqrt{15} \)
- The square root of 15 (\( \sqrt{15} \)) is an irrational number because it cannot be expressed as a fraction of two integers.
- Adding an irrational number (\( -\sqrt{15} \)) to a rational number (0.25) will result in an irrational number.
- Thus, \( 0.25 + (-\sqrt{15}) \) is irrational.
Option B: 0.54732871...
- The number 0.54732871... is a non-terminating, non-repeating decimal which indicates it is an irrational number.
- Adding an irrational number (0.54732871...) to a rational number (0.25) will also result in an irrational number.
- Hence, \( 0.25 + 0.54732871... \) is irrational.
Option C: 0.45
- The number 0.45 is a terminating decimal, and it can be represented as the fraction \( \frac{45}{100} \), which simplifies to \( \frac{9}{20} \). This means that 0.45 is a rational number.
- Adding two rational numbers (0.25 and 0.45) results in another rational number.
- \( 0.25 + 0.45 = 0.70 \) which is also \( \frac{70}{100} \) or \( \frac{7}{10} \).
Option D: \( \pi \)
- \( \pi \) (pi) is a well-known irrational number because it cannot be represented as a fraction of two integers and its decimal representation is non-terminating and non-repeating.
- Adding an irrational number (\( \pi \)) to a rational number (0.25) will result in an irrational number.
- Thus, \( 0.25 + \pi \) is irrational.
Based on the evaluations above, the correct option is:
Option C: 0.45
This number, when added to 0.25, produces a rational number.
