The organizers of a conference offered a certain number of simultaneous seminars with the intention that each seminar would be...

1. Translate the problem requirements: Understand that we have some seminars limited to 15 attendees each, plus 4 remaining seminars that together must accommodate at least 93 attendees. Originally, all seminars were planned for 18 attendees each.
2. Set up the constraint equation: Use the fact that the total number of attendees remains the same under both scenarios - the original plan and the actual arrangement with space limitations.
3. Express the attendee distribution: Calculate total attendees as \(\mathrm{number\ of\ limited\ seminars} \times 15) + (\mathrm{at\ least\ 93\ attendees\ in\ 4\ seminars})\), and set this equal to \((\mathrm{total\ seminars} \times 18)\).
4. Solve for the total number of seminars: Work backwards from the constraint that exactly balances the attendee count between original and modified plans.

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what's happening in this conference scenario in plain English:

• Originally, the organizers planned for ALL seminars to have exactly 18 attendees each
• Due to space limitations, SOME seminars can only accommodate 15 attendees (instead of 18)
• There are exactly 4 seminars that are NOT space-limited
• These 4 unrestricted seminars must together handle at least 93 attendees
• The total number of conference attendees stays the same

The key insight is that when some seminars lose capacity (from 18 to 15), those "displaced" attendees must go somewhere - they end up in the 4 unrestricted seminars.

Process Skill: TRANSLATE - Converting the problem's language into a clear mathematical scenario

2. Set up the constraint equation

Since the total number of attendees doesn't change, we can set up an equation:

Original plan total attendees = Actual arrangement total attendees

Let's say there are S total seminars.
• Let's call the number of space-limited seminars: \((\mathrm{S} - 4)\)
• The remaining 4 seminars are unrestricted

Original plan: \(\mathrm{S\ seminars} \times 18\ \mathrm{attendees\ each} = 18\mathrm{S}\ \mathrm{total\ attendees}\)

Actual arrangement:
• \((\mathrm{S} - 4)\ \mathrm{limited\ seminars} \times 15\ \mathrm{attendees\ each} = 15(\mathrm{S} - 4)\ \mathrm{attendees}\)
• 4 unrestricted seminars with at least 93 attendees total

So our equation becomes: \(18\mathrm{S} = 15(\mathrm{S} - 4) + (\mathrm{attendees\ in\ 4\ unrestricted\ seminars})\)

3. Express the attendee distribution

Since we want to find the exact number of seminars, let's think about what "at least 93 attendees" means in context.

If the 4 unrestricted seminars have exactly 93 attendees (the minimum), then:
\(18\mathrm{S} = 15(\mathrm{S} - 4) + 93\)

Let's solve this:
\(18\mathrm{S} = 15\mathrm{S} - 60 + 93\)
\(18\mathrm{S} = 15\mathrm{S} + 33\)
\(18\mathrm{S} - 15\mathrm{S} = 33\)
\(3\mathrm{S} = 33\)
\(\mathrm{S} = 11\)

Let's verify this makes sense: If there are 11 total seminars, then 7 are space-limited and 4 are unrestricted.

Process Skill: INFER - Recognizing that "at least 93" likely means "exactly 93" for the equation to balance

4. Solve for the total number of seminars

Let's double-check our answer of \(\mathrm{S} = 11\):

Original plan: \(11\ \mathrm{seminars} \times 18\ \mathrm{attendees} = 198\ \mathrm{total\ attendees}\)

Actual arrangement:
• \(7\ \mathrm{limited\ seminars} \times 15\ \mathrm{attendees} = 105\ \mathrm{attendees}\)
• 4 unrestricted seminars with 93 attendees
• Total: \(105 + 93 = 198\ \mathrm{attendees}\) ✓

The numbers match perfectly! Also, let's verify that having 93 attendees across 4 seminars is reasonable:
\(93 \div 4 = 23.25\ \mathrm{attendees\ per\ seminar\ on\ average}\), which is more than the original 18, making sense since these seminars absorbed the displaced attendees.

Final Answer

The total number of seminars is 11.

This corresponds to answer choice (B) 11.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the constraint "at least 93 attendees"

Students often get confused about what "at least 93 attendees" means in the context of this problem. They may think this creates a range of possible answers rather than understanding that for the equation to balance with the given constraints, there must be exactly 93 attendees in the 4 unrestricted seminars. This leads them to set up inequalities instead of equations, making the problem much more complex than necessary.

2. Incorrectly identifying which seminars are space-limited

Students frequently misread the problem and assume that ALL seminars except 4 have the space limitation, when the problem actually states that "only up to 15 conference attendees" are allowed in "a number of" seminars. They may incorrectly assume that only 4 seminars are space-limited instead of understanding that (total seminars - 4) seminars are space-limited.

3. Failing to recognize the conservation principle

Many students don't realize that the total number of attendees remains constant between the original plan and the actual arrangement. They may try to solve the problem by focusing only on the individual seminar capacities without setting up the crucial equation: Original total attendees = Actual total attendees.

Errors while executing the approach

1. Algebraic manipulation errors

When solving the equation \(18\mathrm{S} = 15(\mathrm{S} - 4) + 93\), students commonly make arithmetic mistakes such as incorrectly expanding \(15(\mathrm{S} - 4)\) as \(15\mathrm{S} - 4\) instead of \(15\mathrm{S} - 60\), or making sign errors when moving terms across the equation.

2. Setting up the wrong variable relationships

Students may incorrectly define their variables, such as letting the number of space-limited seminars be S instead of \((\mathrm{S} - 4)\), which leads to completely wrong equations and thus incorrect answers.

Errors while selecting the answer

1. Failing to verify the solution

Students may arrive at \(\mathrm{S} = 11\) but fail to check their work by substituting back into the original constraints. Without verification, they might miss calculation errors or select a wrong answer choice. The verification step (198 total attendees in both scenarios) is crucial to confirm the answer is correct.

Alternate Solutions

Smart Numbers Approach

Step 1: Make a strategic assumption
Since the 4 remaining seminars must accommodate "at least 93 attendees," let's use the minimum case and assume they accommodate exactly 93 attendees. This gives us concrete numbers to work with.

Step 2: Set up the scenario with concrete values
Let's say there are n seminars total.
• Number of space-limited seminars: \((\mathrm{n} - 4)\)
• Attendees in space-limited seminars: \(15 \times (\mathrm{n} - 4)\)
• Attendees in remaining 4 seminars: 93
• Total attendees under new arrangement: \(15(\mathrm{n} - 4) + 93\)

Step 3: Apply the constraint
The total number of attendees must equal the original plan of 18 attendees per seminar:
\(15(\mathrm{n} - 4) + 93 = 18\mathrm{n}\)

Step 4: Solve with concrete arithmetic
\(15\mathrm{n} - 60 + 93 = 18\mathrm{n}\)
\(15\mathrm{n} + 33 = 18\mathrm{n}\)
\(33 = 3\mathrm{n}\)
\(\mathrm{n} = 11\)

Step 5: Verify our smart number choice
With 11 total seminars:
• 7 seminars with 15 attendees each = 105 attendees
• 4 seminars with 93 total attendees
• Total: 105 + 93 = 198 attendees
• Original plan: \(11 \times 18 = 198\ \mathrm{attendees}\) ✓

The smart numbers approach works because assuming the minimum value (exactly 93) gives us the boundary condition that determines the unique solution.