Application Problem 2-3 (Static) Answer Key

Problem 2 Solution: Detailed Answer Key

2 Classroom Seating Arrangement

Problem 2: Original Problem Statement

Scenario: Mrs. Johnson has 36 students in her class. She wants to arrange them in equal rows with no empty seats. What are all the possible arrangements she can make? Which arrangement would be most practical for a classroom setting?

Solution Requirements:

Complete Answer: All possible arrangements listed with factors

Practical Recommendation: Best arrangement based on classroom constraints

Verification: Confirmation that all arrangements work

Solution methodology:

Step 1: Find all factors of 36
Step 2: Form factor pairs (rows × columns)
Step 3: List all possible arrangements
Step 4: Evaluate practicality of each arrangement
Step 5: Recommend optimal solution

Step 1

Find factors

Step 2

Form pairs

Step 3

List arrangements

Step 1: Find all factors of 36

Systematic approach: 1, 2, 3, 4, 6, 9, 12, 18, 36

Verification: 1×36=36, 2×18=36, 3×12=36, 4×9=36, 6×6=36

Step 2: Form factor pairs

(1, 36), (2, 18), (3, 12), (4, 9), (6, 6)

These represent (rows, columns) arrangements

Step 3: List all possible arrangements

1 row × 36 columns: 1 row with 36 students

2 rows × 18 columns: 2 rows with 18 students each

3 rows × 12 columns: 3 rows with 12 students each

4 rows × 9 columns: 4 rows with 9 students each

6 rows × 6 columns: 6 rows with 6 students each

9 rows × 4 columns: 9 rows with 4 students each

12 rows × 3 columns: 12 rows with 3 students each

18 rows × 2 columns: 18 rows with 2 students each

36 rows × 1 column: 36 rows with 1 student each

Step 4: Evaluate practical arrangements

1×36: Too wide for classroom (impractical)

2×18: Very wide, difficult to manage (impractical)

3×12: Wide but manageable (somewhat practical)

4×9: Good balance of width and depth (practical)

6×6: Square arrangement, very practical (highly practical)

9×4: Tall but narrow, good for visibility (practical)

12×3: Very tall, difficult to see front (less practical)

18×2: Extremely tall, poor visibility (impractical)

36×1: Single file, poor for interaction (impractical)

Step 5: Identify optimal arrangement

Most practical options: 4×9 or 6×6

4×9: Good visibility for all students, manageable width

6×6: Square arrangement, maximizes interaction, balanced

All arrangements: 9 possibilities
Most practical: 4×9 or 6×6
Recommended: 4×9 (best visibility)

Complete Answer Key:

Part A - All possible arrangements:

1. 1 row × 36 columns

2. 2 rows × 18 columns

3. 3 rows × 12 columns

4. 4 rows × 9 columns

5. 6 rows × 6 columns

6. 9 rows × 4 columns

7. 12 rows × 3 columns

8. 18 rows × 2 columns

9. 36 rows × 1 column

Part B - Most practical arrangements: 4 rows × 9 columns or 6 rows × 6 columns

Part C - Recommendation: 4 rows × 9 columns (provides best visibility and classroom management)

Grading criteria:

• Full Credit: All 9 arrangements listed correctly

• Partial Credit: Some arrangements missing but method correct

• Practice Evaluation: Practical recommendations considered

3 Synchronized Event Planning

Problem 3: Original Problem Statement

Scenario: Two buses leave the station at the same time. Bus A returns every 12 minutes, and Bus B returns every 18 minutes. When will both buses return to the station at the same time again? How many trips will each bus have made by then?

Solution Requirements:

Complete Answer: Time until synchronization with LCM calculation

Individual Trip Counts: Number of trips per bus

Verification: Confirmation that synchronization occurs

Bus A Interval

12 minutes

Bus B Interval

18 minutes

Synchronization

36 minutes

Step 1: Identify the mathematical concept needed

When two events with different periods synchronize, use LCM

LCM(12, 18) = time until both return together

Step 2: Find prime factorization of each interval

12 = 4 × 3 = 2² × 3

18 = 9 × 2 = 3² × 2

Step 3: Calculate LCM(12, 18)

12 = 2² × 3¹

18 = 2¹ × 3²

LCM = 2² × 3² = 4 × 9 = 36 minutes

Step 4: Calculate number of trips for each bus

Bus A: 36 ÷ 12 = 3 trips

Bus B: 36 ÷ 18 = 2 trips

Step 5: Verify the solution

Bus A returns at: 12, 24, 36 minutes

Bus B returns at: 18, 36 minutes

Both return together at: 36 minutes ✓

Synchronization time: 36 minutes
Bus A trips: 3
Bus B trips: 2

Complete Answer Key:

Part A - Time until synchronization: 36 minutes

Part B - Number of trips:

• Bus A: 3 trips

• Bus B: 2 trips

Part C - Mathematical concept: Least Common Multiple (LCM)

Part D - Verification: Both buses return to the station at 36 minutes.

Grading criteria:

• Full Credit: Correct LCM calculation and trip counts

• Partial Credit: Correct method but computational error

• Method Credit: Proper use of LCM concept

Solution Analysis & Error Prevention

LCM(a,b) = (a×b)/GCD(a,b)

LCM Formula

Factor Pairs

1×36, 2×18, 3×12, 4×9, 6×6

36's factor pairs

LCM Calculation

LCM(12,18) = 36

Correct solution

Common Errors

Missing factors, wrong LCM

Typical mistakes

Key definitions:

Answer Key: Complete solution with verification and error analysis

Factor: A number that divides another number evenly with no remainder

Multiple: The product of a number and any integer

Least Common Multiple (LCM): The smallest positive integer divisible by two or more numbers

Greatest Common Divisor (GCD): The largest positive integer that divides two or more numbers without remainder

Solution verification methodology:

Re-read Problem: Ensure all parts addressed
Check Calculations: Verify all mathematical operations
Validate Logic: Confirm reasoning is sound
Test Solution: Substitute back into original problem
Consider Alternatives: Check for other possible answers

Tip 1: Always list factors systematically to avoid missing any.

Tip 2: Use prime factorization to calculate LCM efficiently.

Tip 3: Verify LCM by checking that both numbers divide it evenly.

Tip 4: Consider practical constraints when evaluating solutions.

Tip 5: Always verify that your solution makes sense in context.

Common errors: Missing factor pairs, miscalculating LCM, forgetting to consider all constraints, stopping too early in factorization.

Memory aid: "Factor pairs multiply to the original number" - always verify by multiplying.

Verification rules:

• Check that all factor pairs multiply to the original number

• Verify LCM is divisible by both original numbers

• Ensure all calculations are mathematically correct

• Confirm solution addresses all parts of the question

• Validate that answer makes sense in real-world context

Advanced Solution Techniques

4 Garden Planning Solution

Advanced Problem: Garden Planning Solution

Original Problem: A gardener has 60 flower bulbs to plant in equal rows. He wants to maximize the number of plants per row while keeping the number of rows reasonable (between 3 and 8). What is the optimal arrangement?

Solution Requirements:

Complete Answer: All possible arrangements within constraints

Optimization: Best arrangement based on criteria

Verification: Confirmation of optimality

Total Bulbs

Constraint

3-8 rows

Optimal

6×10

Step 1: Find all factors of 60

Systematic approach: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

Factor pairs: (1,60), (2,30), (3,20), (4,15), (5,12), (6,10)

Step 2: Apply row constraint (3-8 rows)

Valid arrangements: (3,20), (4,15), (5,12), (6,10)

3 rows × 20 bulbs per row

4 rows × 15 bulbs per row

5 rows × 12 bulbs per row

6 rows × 10 bulbs per row

Step 3: Evaluate optimization criteria

Maximize bulbs per row: 20 bulbs per row (3 rows)

Most balanced: 6 rows × 10 bulbs per row

Best compromise: 6 rows × 10 bulbs (practical and balanced)

Step 4: Provide complete answer

Optimal arrangement: 6 rows × 10 bulbs per row

Alternative maximum: 3 rows × 20 bulbs per row

Valid arrangements: 4 options
Mathematical maximum: 3×20
Practical optimum: 6×10

Complete Answer Key:

Part A - All valid arrangements:

• 3 rows × 20 bulbs per row

• 4 rows × 15 bulbs per row

• 5 rows × 12 bulbs per row

• 6 rows × 10 bulbs per row

Part B - Mathematical maximum: 3 rows × 20 bulbs per row

Part C - Practical recommendation: 6 rows × 10 bulbs per row

Part D - Reasoning: Balances practicality with the goal of maximizing bulbs per row.

Advanced techniques:

• Constraint Optimization: Balance competing objectives

• Multiple Criteria: Consider both mathematical and practical goals

• Trade-off Analysis: Evaluate different solutions

5 Pattern Recognition Solution

Pattern Problem: Tile Design Solution

Original Problem: A designer wants to create a rectangular tile pattern using 48 tiles. She wants to create a border using prime numbers of tiles on each side. Can she achieve this?

Solution Requirements:

Complete Analysis: All possibilities examined

Impossibility Proof: Mathematical demonstration of why no solution exists

Alternative: Suggestion for achievable solution

Total Tiles

Prime Numbers

2, 3, 5, 7, 11, 13...

Result

Impossible

Step 1: Find all factor pairs of 48

1 × 48 = 48

2 × 24 = 48

3 × 16 = 48

4 × 12 = 48

6 × 8 = 48

Step 2: Check if any pair uses two primes

(1,48): 1 is not prime

(2,24): 2 is prime, 24 is not

(3,16): 3 is prime, 16 is not

(4,12): Neither is prime

(6,8): Neither is prime

Step 3: Analyze perimeter constraint (alternative interpretation)

Perimeter = 2(length + width)

For 6×8: Perimeter = 28 (not prime)

For 4×12: Perimeter = 32 (not prime)

For 3×16: Perimeter = 38 (not prime)

For 2×24: Perimeter = 52 (not prime)

For 1×48: Perimeter = 98 (not prime)

Step 4: Provide complete answer

No arrangement of 48 tiles can have both dimensions as prime numbers

No arrangement of 48 tiles can have a prime perimeter

Therefore, the design is impossible with 48 tiles

Solution: Impossible
Reason: No prime factor pair exists
Alternative: Use 15 tiles (3×5)

Complete Answer Key:

Part A - Possibility: No, it's impossible to create such a design with 48 tiles

Part B - Mathematical proof: No factor pair of 48 consists of two prime numbers

Part C - Factor pairs: (1,48), (2,24), (3,16), (4,12), (6,8) - none have both primes

Part D - Alternative: To achieve the desired design, use a number that is the product of two primes, such as 15 tiles (3×5) or 21 tiles (3×7)

Advanced techniques:

• Impossibility Proof: Demonstrate why no solution exists

• Alternative Suggestions: Provide achievable modifications

• Multiple Interpretations: Consider different problem constraints

Complete Answer Key Reference Guide

Static Problem: Fixed Parameters

Definition

Factor Applications

Arrangements, distributions, equal groups

Common use cases

Multiple Applications

Synchronization, timing, cycles

Common use cases

Verification Methods

Check calculations, test solutions

Quality control

Complete definitions:

Answer Key: A comprehensive solution guide that includes the correct answer, solution method, and verification steps

Factor: A number that divides another number evenly with no remainder

Multiple: The product of a number and any integer

Least Common Multiple (LCM): The smallest positive integer divisible by two or more numbers

Greatest Common Divisor (GCD): The largest positive integer that divides two or more numbers without remainder

Complete answer key methodology:

Problem Restatement: Clearly state the original problem
Solution Steps: Provide detailed, step-by-step solution
Complete Answer: Give the final answer with all parts
Verification: Confirm the solution is correct
Error Analysis: Identify common mistakes
Grading Criteria: Specify how to award credit

Tip 1: Always address every part of the original question.

Tip 2: Show all steps to allow partial credit for correct methods.

Tip 3: Verify solutions by substituting back into the original problem.

Tip 4: Consider practical constraints when evaluating real-world problems.

Tip 5: If no solution exists, explain why mathematically.

Tip 6: Provide alternative approaches when possible.

Quality indicators: Complete solution, verification steps, clear explanation, proper formatting, addressing all parts of the question.

Best practices: Consistent notation, logical flow, verification of calculations, consideration of alternative interpretations.

Answer key standards:

• Address all parts of the original question

• Provide clear, step-by-step solutions

• Include verification methods

• Identify common errors

• Specify grading criteria

• Consider practical applications

Answer Key Components:

Essential Elements:

Original problem statement
Step-by-step solution process
Complete final answer
Solution verification
Error prevention tips
Grading criteria
Alternative approaches

Quality answer keys enhance learning and assessment!

Application Problem 2-3 (Static) Complete Answer Key

Answer Key Questions