Evaluate the two sums and determine which statement is true enter the correct number associated with your response

Column A Column B
0.5 + 3/4 sqrt 3 + 2/5

statement # 1: Only column A has a rational sum
statement # 2: Only Column B has a rational sum
statement # 3: Column A and B both have rational sums

The simplified values of both columns indicate that statement # _____ is true

User Icon for Bot GPT 4-o mini Bot GPT 4-o mini answered
2 months ago

Let's evaluate the sums in both columns.

**Column A:**
\[ 0.5 + \frac{3}{4} \]

First, convert \(0.5\) to a fraction:
\[ 0.5 = \frac{1}{2} \]

Now, find a common denominator to add \( \frac{1}{2} \) and \( \frac{3}{4} \).
The common denominator of \(2\) and \(4\) is \(4\).

Convert \( \frac{1}{2} \) to have a denominator of \(4\):
\[ \frac{1}{2} = \frac{2}{4} \]

Now add:
\[ \frac{2}{4} + \frac{3}{4} = \frac{2 + 3}{4} = \frac{5}{4} \]

So, the sum in Column A is:
\[ \frac{5}{4} \]
This is a rational number.

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**Column B:**
\[ \sqrt{3} + \frac{2}{5} \]

The term \( \sqrt{3} \) is an irrational number, while \( \frac{2}{5} \) is a rational number.

Since the sum of a rational number and an irrational number is always irrational, we conclude:
\[ \sqrt{3} + \frac{2}{5} \]
is irrational.

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Now we evaluate the statements:

- **Statement # 1**: Only column A has a rational sum. (True, since Column A = \( \frac{5}{4} \) and Column B is irrational)
- **Statement # 2**: Only Column B has a rational sum. (False, since Column B is irrational)
- **Statement # 3**: Column A and B both have rational sums. (False)

Therefore, the correct statement is statement # 1.

So, the answer is:
**1**