Hi, everyone. Welcome back to another blog post on St Olave School’s Maths series. Today, we’re going to tackle a volume question that’s commonly found in the 11+ exams. The scenario involves a closed box and two cubes, and we’ll figure out how many smaller cubes can fit inside a larger box. Let’s dive into it!
Understanding the Problem
Here’s the problem on our board:
“This is a picture of a closed box and two cubes. The box is quite big, and the cubes are small. One cube is a 2x2x2 cm cube, and the other is a 3x3x3 cm cube. How many cubes of size 2 cm will fit inside the box?”
Let’s break it down step-by-step. First, let’s visualise the box and the cubes.
The box dimensions are given as:
- Length: 10 cm
- Width: 6 cm
- Height: 6 cm
We need to figure out how many 2x2x2 cm cubes can fit inside this box.
Approach 1: Using Volume
We can start by calculating the volume of the box and then see how many 2 cm cubes fit within that volume.
Step-by-Step Calculation
- Calculate the volume of the box:
Volume = Length × Width × Height= 10 cm × 6 cm × 6 cm= 360 cm³ - Calculate the volume of one 2x2x2 cm cube:
Volume = 2 cm × 2 cm × 2 cm= 8 cm³ - Divide the volume of the box by the volume of one cube:
Number of cubes = Volume of the box / Volume of one cube= 360 cm³ / 8 cm³= 45Thus, the volume calculation shows that 45 cubes could theoretically fit into the box. However, since we cannot break the cubes into smaller pieces, this method might not give the most practical answer.
Approach 2: Using Dimensions
Given that these cubes cannot be broken down, let’s use another method by checking the dimensions.
Cross Section Calculation
- Check how many 2 cm edges fit along the 6 cm edge:
6 cm / 2 cm = 3 cubesSo, three 2 cm cubes will fit along each 6 cm edge. - **Verify the layers:
- 3 cubes in width
- 3 cubes in height
- 1 layer will have 3 x 3 = 9 cubes
- Check how many such layers fit along the 10 cm depth:
10 cm / 2 cm = 5 layersThus, 5 layers can fit into the depth of the box.
Final Calculation:
- Number of cubes per layer: 9
- Number of layers: 5
- Total number of cubes:
Total cubes = 9 cubes/layer × 5 layers = 45 cubes
Therefore, using the dimension method, we arrive at a total of 45 cubes fitting perfectly without having to break any cube.
What About the 3x3x3 cm Cubes?
The problem also asks us to verify how many 3x3x3 cm cubes will fit inside the box.
Step-by-Step Calculation
- Check how many 3 cm edges fit along the 6 cm edge:
6 cm / 3 cm = 2 cubes - **Verify the layers:
- 2 cubes in width
- 2 cubes in height
- 1 layer will have 2 x 2 = 4 cubes
- Check how many such layers fit along the 10 cm depth:
10 cm / 3 cm = 3 full cubes with 1 cm leftYou can fit three layers of these cubes, but with 1 cm leftover space that’s unused.
Final Calculation:
- Number of cubes per layer: 4
- Number of layers: 3
- Total number of cubes:
Total cubes = 4 cubes/layer × 3 layers = 12 cubes
Conclusion
In conclusion, by focusing on fitting the 2 cm cubes within the given space, we found that a total of 45 such cubes could fit perfectly. For the 3x3x3 cm cubes, we could only fit 12 cubes with some leftover space.
QUOTE: “Understanding dimensions and visualising the layout can often be more practical than simply relying on volume calculations alone.”
Try practising similar questions on volume and let us know if this walkthrough was helpful. Remember to always verify your answers by considering both volume and dimensions!
Happy studying and all the best with your Maths!