Question

for each card, determine if \\(\frac{2}{3}\\) of the items are headphones.

Explanation:

Step1: Count total items

In the blue card, total items: \( 4 + 2 = 6 \) (4 headphones, 2 watches).

Step2: Calculate \( \frac{2}{3} \) of total

\( \frac{2}{3} \times 6 = 4 \).

Step3: Check headphones count

Headphones count is 4, which matches \( \frac{2}{3} \) of 6. Wait, no—wait, the first card (Yes section) has 3 items: 1 watch, 2 headphones. \( \frac{2}{3} \) of 3 is 2, which matches. The blue card: total items 6, \( \frac{2}{3} \times 6 = 4 \), and headphones are 4. Wait, but the question is about each card. Wait, the blue card: total items 6, headphones 4. \( \frac{4}{6} = \frac{2}{3} \), so yes? Wait, no, the first card (Yes box) has 3 items: 2 headphones (which is \( \frac{2}{3} \) of 3). The blue card: 6 items, 4 headphones. \( \frac{4}{6} = \frac{2}{3} \), so it should go to Yes? Wait, maybe I misread. Wait, the blue card has 4 headphones and 2 watches: total 6. \( \frac{2}{3} \times 6 = 4 \), so headphones are 4, which is \( \frac{2}{3} \). So for the blue card, it's Yes? Wait, no, the first card (Yes section) has 3 items: 2 headphones (correct, \( \frac{2}{3} \) of 3 is 2). The blue card: 6 items, 4 headphones ( \( \frac{4}{6} = \frac{2}{3} \) ), so it should be Yes. Wait, but maybe I messed up. Wait, let's re-express:

For the blue card:
Total items \( n = 6 \) (4 headphones, 2 watches).
\( \frac{2}{3}n = \frac{2}{3} \times 6 = 4 \).
Number of headphones is 4, so \( \frac{\text{headphones}}{n} = \frac{4}{6} = \frac{2}{3} \). So it satisfies. So the blue card should go to Yes? Wait, but the first card (Yes box) has 3 items: 2 headphones (correct). The blue card: 6 items, 4 headphones (correct). Wait, maybe the question is about the blue card. Let's confirm:

Total items: 4 headphones + 2 watches = 6.
\( \frac{2}{3} \) of 6 is 4. Headphones are 4, so yes. So the blue card should be placed in Yes.

Answer:

The blue card (with 4 headphones and 2 watches) should be placed in the "Yes" section.