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The image displays a problem where a sandwich shop offers menus for sandwiches, soups, and salads. The number of salads is twice the number of sandwiches and one more than the number of soups. There are two statements given about customer choices:

1. The total number of choices when selecting one item from two menus is 63.
2. The total number of choices when selecting one item from all three menus is 90.

We need to determine how many soups are listed on the soup menu and check the sufficiency of the given statements.

### Let:
- \( s \) = number of sandwiches,
- \( p \) = number of soups,
- \( a \) = number of salads.

From the problem:
- \( a = 2s \),
- \( a = p + 1 \), i.e., \( p = a - 1 \).

Using the statements:
- From Statement (1), choosing from two menus gives \( s \times a = 63 \),
- From Statement (2), choosing from all three menus gives \( s \times p \times a = 90 \).

We will solve these equations now.It seems that the system of equations did not yield a solution. Let's reevaluate the approach:

The conditions were:
- \( a = 2s \) (the number of salads is twice the number of sandwiches),
- \( p = a - 1 \) (the number of soups is one less than the number of salads),
- From Statement (1): \( s \times a = 63 \),
- From Statement (2): \( s \times p \times a = 90 \).

I will now attempt solving this again manually, step by step, to determine the correct number of soups.Let's reconsider the system of equations step-by-step.

We have:

1. \( a = 2s \) (number of salads is twice the number of sandwiches),
2. \( p = a - 1 \) (number of soups is one less than the number of salads),
3. From Statement (1), \( s \times a = 63 \),
4. From Statement (2), \( s \times p \times a = 90 \).

### Step 1: Use \( a = 2s \) in \( s \times a = 63 \)
This gives:
\[
s \times (2s) = 63 \quad \Rightarrow \quad 2s^2 = 63 \quad \Rightarrow \quad s^2 = 31.5
\]
Since \( s \) must be an integer, there seems to be an inconsistency here.

It looks like neither of the statements alone is sufficient to determine the number of soups because the system leads to non-integer results.

Therefore, **Statements (1) and (2) together are NOT sufficient** to find a solution, matching the correct answer marked on the question.

Would you like me to clarify any part of this reasoning or expand further on related problems?

Here are 5 related questions to explore:
1. How would changing the relationships between salad, soup, and sandwich quantities affect the solution?
2. What happens if the total choices from one menu change?
3. How can we use trial and error for finding integer solutions in similar cases?
4. How can other forms of equations help solve systems where quadratic results arise?
5. Could alternative forms of data sufficiency questions lead to a more straightforward solution?

**Tip:** For data sufficiency problems, ensure you translate each piece of information into equations or inequalities before combining statements.

A sandwich shop offers a sandwich menu, a soup menu, and a salad menu. The number of salads listed on the salad menu is twice the number of sandwiches listed on the sandwich menu and 1 more than the number of soups listed on the soup menu. How many soups are listed on the soup menu?

Explore how to solve a system of algebraic equations to determine the number of soups listed on a menu, given relations between sandwiches, salads, and soups and customer choices from these menus. This problem is suited for high school students studying algebra.

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