Question
Megan added two numbers and the resulting sum was an irrational number. If the first number was 2/3 , which of the following could be the second number?
A. 6/5
B. 3π
C. 5
D. √9
A. 6/5
B. 3π
C. 5
D. √9
Answers
GPT-4o mini
To determine which of the given options could be the second number such that the sum with \(\frac{2}{3}\) results in an irrational number, we need to consider the nature of numbers:
1. A rational number added to another rational number always results in a rational number.
2. A rational number added to an irrational number results in an irrational number.
Since \(\frac{2}{3}\) is a rational number, we need to find an option that is irrational to ensure the sum is irrational.
Now we'll analyze each option:
A. \( \frac{6}{5} \) - This is a rational number.
B. \( 3\pi \) - This is an irrational number because \(\pi\) is irrational and multiplying by a non-zero rational number keeps it irrational.
C. \( 5 \) - This is a rational number.
D. \( \sqrt{9} \) - This simplifies to \( 3 \), which is a rational number.
From these analyses, the only option that is irrational is:
**B. \(3\pi\)**
Thus, the correct answer is **B. \(3\pi\)**.
1. A rational number added to another rational number always results in a rational number.
2. A rational number added to an irrational number results in an irrational number.
Since \(\frac{2}{3}\) is a rational number, we need to find an option that is irrational to ensure the sum is irrational.
Now we'll analyze each option:
A. \( \frac{6}{5} \) - This is a rational number.
B. \( 3\pi \) - This is an irrational number because \(\pi\) is irrational and multiplying by a non-zero rational number keeps it irrational.
C. \( 5 \) - This is a rational number.
D. \( \sqrt{9} \) - This simplifies to \( 3 \), which is a rational number.
From these analyses, the only option that is irrational is:
**B. \(3\pi\)**
Thus, the correct answer is **B. \(3\pi\)**.