Let's categorize each letter according to the definitions of rational and irrational numbers:
### Step-by-step Classification:
1. A. 2.125
- Explanation: 2.125 is a terminating decimal.
- Classification: Rational number
2. B. 0
- Explanation: 0 is an integer, and all integers are rational numbers.
- Classification: Rational number
3. C. π
- Explanation: π (pi) is a well-known constant that cannot be expressed as a fraction.
- Classification: Irrational number
4. D. [tex]\( -\frac{15}{5} \)[/tex]
- Explanation: [tex]\( -\frac{15}{5} \)[/tex] simplifies to -3, which is an integer.
- Classification: Rational number
5. E. [tex]\( 3 \frac{1}{4} \)[/tex]
- Explanation: [tex]\( 3 \frac{1}{4} \)[/tex] can be written as [tex]\( \frac{13}{4} \)[/tex], which is a fraction.
- Classification: Rational number
6. F. [tex]\( 0.\overline{4} \)[/tex]
- Explanation: [tex]\( 0.\overline{4} \)[/tex] is a repeating decimal, which can be represented as a fraction.
- Classification: Rational number
7. G. 23
- Explanation: 23 is an integer and thus can be expressed as a fraction (23/1).
- Classification: Rational number
8. H. -1
- Explanation: -1 is an integer and can be expressed as a fraction (-1/1).
- Classification: Rational number
9. I. [tex]\( \sqrt{\frac{16}{25}} \)[/tex]
- Explanation: [tex]\( \sqrt{\frac{16}{25}} = \frac{4}{5} = 0.8 \)[/tex], which is a terminating decimal.
- Classification: Rational number
10. J. [tex]\( -\sqrt{49} \)[/tex]
- Explanation: [tex]\( -\sqrt{49} = -7 \)[/tex] is an integer.
- Classification: Rational number
11. K. [tex]\( -\sqrt{35} \)[/tex]
- Explanation: [tex]\( \sqrt{35} \)[/tex] is not a perfect square and therefore cannot be expressed as a fraction.
- Classification: Irrational number
### Conclusion:
From this analysis, we can categorize the numbers into two groups:
- Irrational Numbers: C, K
- Rational Numbers: A, B, D, E, F, G, H, I, J
Thus, placing each letter in its location in the real number system would be as follows:
- Irrational Numbers: C, K
- Rational Numbers: A, B, D, E, F, G, H, I, J
