How many pencil boxes of dimensions 15cm x 5cm x 2cm can be packed in a carton, each of whose edges is 30cm?
The correct answer and explanation is:
To find how many pencil boxes can be packed into a carton, we need to compare the volume of the carton to the volume of one pencil box, and also consider how they can be arranged inside the carton.
Step 1: Volume of the pencil box
Each pencil box has dimensions:
- Length = 15 cm
- Width = 5 cm
- Height = 2 cm
So, the volume of one pencil box is: 15×5×2=150 cm315 \times 5 \times 2 = 150 \text{ cm}^3
Step 2: Volume of the carton
Each edge of the carton is 30 cm. This makes it a cube: 30×30×30=27,000 cm330 \times 30 \times 30 = 27,000 \text{ cm}^3
Step 3: Theoretical number of boxes (volume-wise)
27000150=180\frac{27000}{150} = 180
So theoretically, 180 boxes can fit based on volume alone.
Step 4: Check for actual fit by arrangement
Let’s check how many boxes can fit along each dimension of the carton:
- Along 30 cm edge:
- 30 ÷ 15 = 2 (boxes of length 15 cm)
- 30 ÷ 5 = 6 (boxes of width 5 cm)
- 30 ÷ 2 = 15 (boxes of height 2 cm)
So, by placing boxes 15 cm × 5 cm × 2 cm in a 30 cm × 30 cm × 30 cm carton:
- 2 boxes can fit along length (15 cm)
- 6 along width (5 cm)
- 15 along height (2 cm)
Now multiply: 2×6×15=1802 \times 6 \times 15 = 180
✅ Final Answer: 180 pencil boxes
Explanation (300 words)
To determine how many pencil boxes can be packed into a cube-shaped carton, we consider both volume and physical arrangement. Each pencil box measures 15 cm long, 5 cm wide, and 2 cm high. The carton has a cubic shape, each edge being 30 cm long. First, calculate the volume of one pencil box: 15×5×2=150 cm315 \times 5 \times 2 = 150 \text{ cm}^3
Next, calculate the volume of the carton: 30×30×30=27,000 cm330 \times 30 \times 30 = 27,000 \text{ cm}^3
Now, dividing the total volume of the carton by the volume of one pencil box: 27,000÷150=18027,000 \div 150 = 180
This gives a theoretical maximum of 180 boxes, assuming perfect packing. However, we must ensure the boxes can actually fit inside the carton in terms of orientation and dimensions. We evaluate how many boxes can fit along each axis of the carton:
- Along the 30 cm length: 30 ÷ 15 = 2 boxes
- Along the 30 cm width: 30 ÷ 5 = 6 boxes
- Along the 30 cm height: 30 ÷ 2 = 15 boxes
This configuration gives: 2×6×15=180 boxes2 \times 6 \times 15 = 180 \text{ boxes}
This confirms that not only do the boxes fit in terms of volume, but they also fit exactly with no wasted space, provided they are aligned correctly. Therefore, the carton can hold exactly 180 pencil boxes.