Application Problems 2-3 (Static) | Factor and Multiple Applications

Static Application Problems 2-3

2 Static Factor Application

Problem 2: Classroom Seating Arrangement

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?

Definition:

Equal Arrangement: Using factors to determine possible rectangular arrangements

Static Problem: A problem with fixed parameters that doesn't change over time

Factor-based arrangement method:

Find all factors of the total number
Form factor pairs (rows × columns)
Identify practical arrangements
Evaluate for real-world constraints

Total Students

Factors

{1,2,3,4,6,9,12,18,36}

Arrangements

9 possible

Step 1: Find all factors of 36

Start with 1: 1 × 36 = 36 → Factors: 1, 36

Try 2: 2 × 18 = 36 → Factors: 2, 18

Try 3: 3 × 12 = 36 → Factors: 3, 12

Try 4: 4 × 9 = 36 → Factors: 4, 9

Try 6: 6 × 6 = 36 → Factors: 6

Try 7: 7 doesn't divide 36 evenly

Stop when factors repeat

All factors: {1, 2, 3, 4, 6, 9, 12, 18, 36}

Step 2: Form factor pairs for arrangements

1 row × 36 columns

2 rows × 18 columns

3 rows × 12 columns

4 rows × 9 columns

6 rows × 6 columns

9 rows × 4 columns

12 rows × 3 columns

18 rows × 2 columns

36 rows × 1 column

Step 3: Evaluate practical arrangements

1 row × 36 columns: Too wide for classroom

2 rows × 18 columns: Very wide, impractical

3 rows × 12 columns: Quite wide

4 rows × 9 columns: Good balance

6 rows × 6 columns: Square arrangement, good

9 rows × 4 columns: Tall but narrow, good

12 rows × 3 columns: Very tall

18 rows × 2 columns: Extremely tall

36 rows × 1 column: Impractical

Step 4: Determine most practical arrangement

Best options: 4 rows × 9 columns or 6 rows × 6 columns

4 rows × 9 columns: Good visibility for all students

6 rows × 6 columns: Square, compact arrangement

All arrangements: 9 possibilities
Most practical: 4×9 or 6×6
Factors: {1,2,3,4,6,9,12,18,36}

Final Answer:

Mrs. Johnson has 9 possible arrangements:

• 1 row × 36 columns

• 2 rows × 18 columns

• 3 rows × 12 columns

• 4 rows × 9 columns (recommended)

• 6 rows × 6 columns (recommended)

• 9 rows × 4 columns

• 12 rows × 3 columns

• 18 rows × 2 columns

• 36 rows × 1 column

The most practical arrangements are 4 rows × 9 columns or 6 rows × 6 columns.

Applied rules:

• Factor Pairs: Each pair represents a possible arrangement

• Real-world Application: Factors relate to arranging objects

• Practical Constraints: Consider classroom limitations

3 Static Multiple Application

Problem 3: Synchronized Event Planning

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?

Definition:

Least Common Multiple (LCM): The smallest number that is a multiple of two or more numbers

Synchronized Events: Using LCM to find when periodic events coincide

Bus Intervals

12, 18 minutes

LCM

36 minutes

Trips

3, 2 trips

Step 1: Find prime factorization of each interval

12 = 4 × 3 = 2² × 3

18 = 9 × 2 = 3² × 2

Step 2: Find LCM(12, 18)

12 = 2² × 3¹

18 = 2¹ × 3²

LCM = 2² × 3² = 4 × 9 = 36

Step 3: Calculate number of trips

Bus A: 36 ÷ 12 = 3 trips

Bus B: 36 ÷ 18 = 2 trips

Step 4: 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: 36 minutes
Bus A: 3 trips
Bus B: 2 trips

Final Answer:

Both buses will return to the station at the same time after 36 minutes.

By that time:

• Bus A will have made 3 complete trips

• Bus B will have made 2 complete trips

This is a classic application of LCM in scheduling and timing.

Applied rules:

• LCM Rule: For synchronized events, use LCM of intervals

• Prime Factorization: Efficient method for finding LCM

• Division Rule: Number of cycles = Total time ÷ Cycle time

Static Problem Solving Concepts

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

Example solution

Static Problems

Fixed parameters, no time dependency

Problem characteristic

Key definitions:

Static Problem: A mathematical problem with fixed parameters that don't change over time or context

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 that is divisible by two or more numbers

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

Static problem-solving approach:

Identify Fixed Parameters: Recognize what doesn't change
Find Relevant Concepts: Determine if factors, multiples, or patterns apply
Apply Systematic Methods: Use factorization, LCM, or GCD as appropriate
Verify Solutions: Check that answers satisfy all conditions

Tip 1: Static problems often involve finding arrangements or combinations with fixed totals.

Tip 2: Look for words like "arrange," "distribute," or "share" to identify factor problems.

Tip 3: For synchronized events, always use LCM to find when they coincide.

Tip 4: Consider real-world constraints when evaluating solutions.

Tip 5: Always verify that your answer makes sense in the problem context.

Common characteristics: Fixed totals, unchanging relationships, predetermined constraints, time-independent parameters.

Memory aid: "Static" means "unchanging" - look for fixed totals and arrangements.

Essential rules for static problems:

• Identify all fixed parameters first

• Determine if the problem involves factors, multiples, or patterns

• Use systematic approaches to find all possibilities

• Consider practical constraints when applicable

• Verify solutions against original problem conditions

Advanced Static Applications

4 Static Optimization

Advanced Problem: Garden Planning

Scenario: 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? How many bulbs will be planted per row?

Definition:

Optimization: Finding the best solution among all possible arrangements

Total Bulbs

Factor Pairs

1×60, 2×30, 3×20, 4×15, 5×12, 6×10

Optimal

6 rows × 10 bulbs

Step 1: Find all factors of 60

1 × 60 = 60

2 × 30 = 60

3 × 20 = 60

4 × 15 = 60

5 × 12 = 60

6 × 10 = 60

Stop at √60 ≈ 7.7 (after 6×10)

Step 2: Identify arrangements with 3-8 rows

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: Maximize bulbs per row within constraints

3 rows: 20 bulbs per row

4 rows: 15 bulbs per row

5 rows: 12 bulbs per row

6 rows: 10 bulbs per row

Highest: 20 bulbs per row with 3 rows

Step 4: Evaluate reasonableness

3 rows × 20 bulbs: Long, narrow garden

4 rows × 15 bulbs: Good balance

5 rows × 12 bulbs: Good balance

6 rows × 10 bulbs: Compact, square-like

Optimal: 6 rows × 10 bulbs (most compact arrangement)

Optimal arrangement: 6 rows × 10 bulbs
Maximum bulbs per row: 20 (with 3 rows)
Recommended: 6 rows × 10 bulbs

Final Answer:

The optimal arrangement is 6 rows with 10 bulbs per row.

While maximizing bulbs per row would mean 3 rows with 20 bulbs each, the most balanced and practical arrangement is 6 rows with 10 bulbs per row.

This creates a more square-like garden that's easier to maintain.

Applied rules:

• Optimization: Balance mathematical maximum with practical considerations

• Constraint Satisfaction: Meet all specified requirements

• Practical Evaluation: Consider real-world implications

5 Static Pattern Recognition

Pattern Problem: Tile Design

Scenario: 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? If so, what are the possible dimensions?

Definition:

Prime Number: A number greater than 1 with exactly two factors: 1 and itself

Border Pattern: Using perimeter constraints in geometric arrangements

Total Tiles

Prime Numbers

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

Possibility

No valid solution

Step 1: Clarify the problem requirements

48 tiles total in a rectangle

Border of rectangle uses prime numbers of tiles on each side

Need to determine if this is possible

Step 2: Find all factor pairs of 48

1 × 48 = 48

2 × 24 = 48

3 × 16 = 48

4 × 12 = 48

6 × 8 = 48

Step 3: Check if any dimensions use prime numbers

Rectangle 1×48: Dimensions are 1 and 48 (1 is not prime)

Rectangle 2×24: Dimensions are 2 and 24 (2 is prime, 24 is not)

Rectangle 3×16: Dimensions are 3 and 16 (3 is prime, 16 is not)

Rectangle 4×12: Dimensions are 4 and 12 (neither is prime)

Rectangle 6×8: Dimensions are 6 and 8 (neither is prime)

Step 4: Consider if "border" means perimeter

If perimeter must be prime: Perimeter = 2(length + width)

For 6×8: Perimeter = 2(6+8) = 28 (not prime)

For 4×12: Perimeter = 2(4+12) = 32 (not prime)

For 3×16: Perimeter = 2(3+16) = 38 (not prime)

For 2×24: Perimeter = 2(2+24) = 52 (not prime)

For 1×48: Perimeter = 2(1+48) = 98 (not prime)

Step 5: Analyze all possibilities

No rectangle with area 48 has both dimensions as prime numbers

No rectangle with area 48 has a prime perimeter

It's impossible to create such a design with 48 tiles

Solution: Not possible
Reason: No factor pair uses two primes
Alternative: Use 15 tiles (3×5)

Final Answer:

No, it's not possible to create a rectangular tile pattern using 48 tiles with prime numbers of tiles on each side.

The factor pairs of 48 are: (1,48), (2,24), (3,16), (4,12), (6,8).

In no case do both numbers in the pair represent prime numbers.

For such a design to work, you'd need a number that is the product of two primes.

Applied rules:

• Prime Testing: Verify if numbers are prime

• Systematic Analysis: Check all possibilities

• Impossibility Proof: Demonstrate why no solution exists

Complete Static Problem Guide

Static: Fixed Parameters

Definition

Factor Applications

Arrangements, distributions, equal groups

Common use cases

Multiple Applications

Synchronization, timing, cycles

Common use cases

Static Characteristics

Fixed, unchanging, predetermined

Problem features

Complete definitions:

Static Problem: A mathematical problem with fixed parameters that remain constant throughout the problem

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 static problem-solving methodologies:

Parameter Identification: Recognize all fixed values
Concept Matching: Determine if factors, multiples, or patterns apply
Systematic Enumeration: Find all possible solutions
Constraint Application: Apply any additional requirements
Solution Verification: Check all conditions are met

Tip 1: Static problems often involve equal distribution or arrangement.

Tip 2: Look for keywords like "total," "equal," "arrange," or "distribute."

Tip 3: Consider practical constraints when evaluating solutions.

Tip 4: Use factor trees for complex prime factorization.

Tip 5: Always verify that your solution satisfies all problem conditions.

Tip 6: If no solution exists, explain why mathematically.

Interesting Fact: Static problems form the foundation for dynamic problem solving!

Real-world Use: Static problems appear in construction, design, and resource allocation.

Advanced Concept: Static problems can become optimization problems with additional constraints.

Important rules for static problems:

• Identify all fixed parameters before solving

• Use systematic approaches to enumerate all possibilities

• Apply mathematical concepts appropriately (factors, multiples, patterns)

• Consider practical constraints and real-world applicability

• Verify solutions against original problem requirements

• Explain impossibility when no solution exists

Static Problem Mastery:

Key Characteristics:

Fixed, unchanging parameters
No time-dependent variables
Complete information provided upfront
Solutions don't change over time
Focus on mathematical relationships

Static problems build foundational problem-solving skills!

Application Problems 2-3: Static Factor and Multiple Problems

Static Problem Solving Questions