The correct option is b.
The pair where both decimals represent rational numbers is B. 0.115 and 0.12333..., as both can be expressed as fractions of two integers.
To determine which pair of decimals represents rational numbers, we need to check if each decimal can be expressed as a fraction of two integers (i.e., a rational number). Rational numbers can be written in the form p/q, where p and q are integers, and q is not equal to zero.
Let's analyze each pair of decimals step by step:
A. 9.0032812373025549382... and 0.12333...
- 9.0032812373025549382...: This is a non-repeating decimal and appears to be irrational.
- 0.12333...: This is a repeating decimal, and repeating decimals can be expressed as rational numbers. It can be written as 12333/100000.
B. 0.115 and 0.12333...
- 0.115: This is a finite decimal, and finite decimals can be expressed as rational numbers. It can be written as 115/1000.
- 0.12333...: As mentioned earlier, this is a repeating decimal and can be written as 12333/100000.
C. 3.10382947305718274... and 0.123
- 3.10382947305718274...: This is a non-repeating decimal and appears to be irrational.
- 0.123: This is a finite decimal and can be expressed as a rational number. It can be written as 123/1000.
D. 9.0032812373025549382... and 0.115
- 9.0032812373025549382...: This is a non-repeating decimal and appears to be irrational.
- 0.115: This is a finite decimal and can be expressed as a rational number. It can be written as 115/1000.
Complete question is here:
In which pair do both decimals represent rational numbers?
A. 9.0032812373025549382... and 0.12333..
B. 0.115 and 0.12333...
C. 3.10382947305718274... and 0.123
D. 9.0032812373025549382... and 0.115