Key Concepts

• Packaging in Limited Spaces: When packing items into limited spaces (e.g., a box or suitcase), how items are packed determines how much can fit.

Examples include packing school items like books and lunch into a backpack.

• Key Factors to Consider When Packing:
• Cost of Extra Containers: Sending additional containers or boxes increases costs.
• Weight Restrictions: There are weight limits for transporting packages, which can influence cost (e.g., shipping).
• Fragility of Items: Fragile items need protective material to avoid damage.
• Ease of Carrying: Ensure packed items are not too heavy to carry easily.

Worked Example 1

A scenario involves packing biscuit boxes into larger shipping boxes for transport.

Questions:

1. How many boxes?

• Vuyo has 600 small boxes of ginger biscuits, and each large box holds 15 small boxes.
• Calculate the number of large boxes needed for ginger biscuits.

2. Packing Chocolate Biscuits:

• If 500 chocolate biscuits fit into 20 small boxes, and 25 small boxes fit into one large box:
• Determine the total number of large boxes required for chocolate biscuits.

3. Cost Comparison:

• Large shipping boxes cost R5.50 each. Which biscuits will cost more to pack: ginger or chocolate biscuits?

4. Weight Limits:

• Large boxes can hold a maximum weight of 3.5 kg.
• (i) If each small box weighs 200 g, calculate how many can fit into one large box.
• (ii) If each small box weighs 300 g, determine the maximum that can fit without exceeding the weight limit.

5. New Shipping Boxes:

• New boxes have a surface area of 720 cm². Calculate the number of smaller boxes (surface area = 25 cm²) that can fit into the larger boxes.
• Demonstrate the packing layout using scaled diagrams.

Solutions

• Ginger Biscuits:
• 600 small boxes ÷ 15 = 40 large boxes.
• Chocolate Biscuits:
• 500 cans ÷ 20 small boxes = 25 small boxes.
• 25 small boxes ÷ 1 large box = 25 large boxes.
• Chocolate biscuits are cheaper to pack because they require fewer large boxes.
• Cost Comparison:
• Ginger biscuits: 40 × R5.50 = R220.00.
• Chocolate biscuits: 25 × R5.50 = R137.50.
• Chocolate biscuits cost less to pack.
• Weight Limit Calculations:
• (i) For ginger biscuits (200 g per small box): 3,500 g ÷ 200 g = 17 small boxes per large box.
• (ii) For chocolate biscuits (300 g per small box): 3,500 g ÷ 300 g = 11 small boxes per large box.
• Surface Area Packing:
• Large box area = 720 cm², small box area = 25 cm².
• 720 ÷ 25 = 28 small boxes fit into one large box (on one layer).