Question

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

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 Column B both have rational sums.

(1 point)
The simplified values of both columns indicate that Statement # is true.

Answer 1

Let's evaluate each column.

**Column A:**
\(0.5 + \frac{3}{4}\)

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

Now we find a common denominator for \(\frac{1}{2}\) and \(\frac{3}{4}\). The common denominator is \(4\):
\[
\frac{1}{2} = \frac{2}{4}
\]

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

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

**Column B:**
\(3 - \sqrt{3} + \frac{2}{5}\)

First, let's consider \(3 - \sqrt{3}\). The number \(\sqrt{3}\) is irrational, so \(3 - \sqrt{3}\) is also irrational.

Next, we have:
\[
3 - \sqrt{3} + \frac{2}{5}
\]

To combine these, we can express \(3\) as a fraction:
\[
3 = \frac{15}{5}
\]

Now we combine:
\[
\frac{15}{5} - \sqrt{3} + \frac{2}{5} = \frac{17}{5} - \sqrt{3}
\]

Since \(\sqrt{3}\) is irrational, \(\frac{17}{5} - \sqrt{3}\) remains irrational as well.

**Conclusion:**
- Column A has a rational sum: \(\frac{5}{4}\).
- Column B has an irrational sum: \(\frac{17}{5} - \sqrt{3}\).

Based on the evaluations:
- Statement #1: Only Column A has a rational sum. **(True)**
- Statement #2: Only Column B has a rational sum. **(False)**
- Statement #3: Column A and Column B both have rational sums. **(False)**

Therefore, the correct answer is Statement #1 is true.

Final answer:
**1**