Models (Grade 12 NSC Matric Mathematical Literacy): Revision Notes
Models
Understanding packing models
Packing models are mathematical representations used to solve problems involving the arrangement of items within limited spaces. These models help us determine the most efficient ways to pack objects into containers like boxes, bags, or shipping containers.
When working with packing models, you need to consider several factors:
- Available space - the dimensions and capacity of containers
- Item sizes - the dimensions and quantities of objects being packed
- Weight limitations - maximum weight restrictions for containers
- Cost considerations - expenses related to containers and shipping
- Practical constraints - real-world limitations that affect packing arrangements
Key principles of packaging problems
Effective packing requires understanding that theoretical calculations often need adjustment for practical realities. The shape and size of items, along with container dimensions, create constraints that pure mathematical calculations might not capture.
Important considerations include:
- Protective packaging - fragile items need extra materials and space
- Weight distribution - items must be packed to avoid damage during transport
- Cost efficiency - balancing container costs with item protection
- Handling requirements - ensuring packed items remain manageable
Mathematical approaches to packing calculations
Basic capacity calculations
To find how many items fit in a container:
Always round up when determining containers needed, but round down when calculating items per container (since you cannot pack partial items).
Weight limit calculations
When weight restrictions apply:
Remember to round down to the nearest whole number since you cannot pack fractions of items.
Area-based packing
For rectangular containers and items:
- Calculate the area of the container base: Length × Width
- Calculate the area each item occupies
- Divide container area by item area to find theoretical capacity
- Consider practical spacing and arrangement constraints
Step-by-step problem-solving approach
Step 1: Identify the constraints
- Container dimensions and capacity
- Item sizes and quantities
- Weight limitations
- Cost factors
Step 2: Choose the appropriate calculation method
- Division for basic capacity problems
- Area calculations for spatial arrangements
- Weight calculations for loading limits
Step 3: Perform calculations
- Use the relevant formula
- Apply proper rounding rules
- Check your answer makes practical sense
Step 4: Consider real-world factors
- Account for packing inefficiencies
- Consider item shapes and arrangement patterns
- Factor in protective materials and spacing
Worked example: Multi-constraint packing problem
Worked Example: Multi-constraint packing problem
Problem setup: A business needs to pack small boxes of items into larger shipping containers. They must consider:
- Small box capacity in large containers
- Weight restrictions per container
- Cost comparison between different packing methods
- Spatial arrangement constraints
Solution approach:
Part A: Basic capacity calculation
- 600 small boxes ÷ 15 boxes per container = 40 large containers needed
Part B: Alternative packing method
- 500 items ÷ 20 items per container = 25 large containers needed
Part C: Cost comparison
- Method A: 40 containers × R5.50 = R220.00
- Method B: 25 containers × R5.50 = R137.50
- Method B is more cost-effective
Part D: Weight limit considerations
- Container weight limit: 3.5 kg = 3500 g
- Small box weight: 200 g
- Maximum boxes per container: 3500 g ÷ 200 g = 17.5 → 17 boxes (rounded down)
Part E: Spatial arrangement calculations
- Large container: 45 cm × 16 cm
- Small box base: 5 cm × 5 cm
- Length-wise: 45 cm ÷ 5 cm = 9 boxes
- Width-wise: 16 cm ÷ 5 cm = 3.2 → 3 boxes (rounded down)
- Total per layer: 9 × 3 = 27 boxes
Part F: Area calculation method
- Large container area: 45 cm × 16 cm = 720 cm²
- Small box area: 5 cm × 5 cm = 25 cm²
- Theoretical capacity: 720 ÷ 25 = 28.8 → 28 boxes
The area calculation (28 boxes) differs from the spatial arrangement calculation (27 boxes) because area calculations don't account for the actual shapes and how items fit together in practice.
Common exam tips and problem-solving strategies
Typical exam traps to avoid:
- Rounding errors - always round down for items/containers you can actually use
- Forgetting units - ensure all measurements use consistent units
- Ignoring practical constraints - theoretical calculations may not work in reality
- Mixing calculation methods - be clear about which approach you're using
Exam-style problem-solving tips:
- Read questions carefully to identify all constraints
- Show all working clearly with units
- State assumptions when necessary
- Check if your answer makes practical sense
- Use diagrams when helpful to visualise packing arrangements
Key formulas to remember:
- Basic capacity: Total items ÷ Capacity per container
- Weight limit: Weight limit ÷ Weight per item
- Area packing: Container area ÷ Item area
- Cost comparison: Number of containers × Cost per container
Key Points to Remember:
- Packing models help solve problems about fitting items into limited spaces efficiently
- Always consider multiple constraints - space, weight, cost, and practical limitations
- Round down when calculating how many whole items fit, but round up when determining containers needed
- Area calculations give theoretical maximums, but spatial arrangements show practical reality
- Check your answers against real-world common sense - if it seems impossible, recalculate!