There is another question, however, which we feel is just as interesting, but has escaped media scrutiny.
The question runs thus:
Jack has 8 identical large cubes and some identical small cubes. He packs all the cubes tightly into a rectangular box such that cubes of the same size are stacked on top of each other. The box is filled to its brim exactly.
The figure below shows the first layer of cubes packed into the box.
(a) How many small cubes does Jack have? [1 mark]
(b) The volume of the box is 4032 cm³. The total volume of the 8 large cubes is 3/7 of the volume of the box. What is the length of one edge of the small cube? [3 marks]
We posed this question to several of our Primary 6 students this year and made several interesting observations of how they went about solving this question. Most struggled for several minutes before they discovered the ‘key’ that unlocks the question. Some even worked backwards, solving part (b) first and then using the answer from (b) to solve part (a)!
The first thing that struck students about the question is the lack of numbers given in the question stem. Indeed, the only number that is stated explicitly is 8. The other two numbers have to be derived from the diagram: 2 for the number large cubes, and 6 for the number small cubes.
The numbers 8, 2 and 6 all relate to the number of cubes; not one of these numbers refers to the dimensions of the cubes. Once students are cognisant of this fact, they’ll come to the conclusion that ratio or proportion must come into play.
Since only cubes of the same size can be stacked on top of each other, the maximum number of large cubes that can be stacked vertically in the box = 8 ÷ 2 = 4.
Here is where most students got stuck, until they reread the question more carefully and recalled the true meaning of ‘cube’ – a 3-dimensional solid formed by 6 identical square faces joined along their edges.
From the diagram we can see that 2 edges of the large cube is as long as 3 edges of the small cube. Now, the 2:3 ratio relationship that applies to the large and small cubes laid side by side horizontally also holds true when the cubes are stacked vertically because all edges of any cube are equal in length.
Since 4 large cubes can be stacked vertically in the box, the number of small cubes that can be stacked vertically in the box = 4 ÷ 2 x 3 = 6.
With the height, we can calculate the total number of small cubes.
6 x 2 x 3 = 36
Total volume of 36 small cubes = 4032 cm³ ÷ 7 x 4 = 2304 cm³
Volume of 1 small cube = 2304 cm³ ÷ 36 = 64 cm³
Length of one edge of the small cube = ∛64 cm³ = 4 cm
Solving for (b) seems straightforward as compared to (a), relative to their mark weightings.
Here’s how students who had problems solving for (a) first worked around their problem.
Starting with part (a)
Number of large cubes that can be stacked vertically in the container = 8 ÷ 2 = 4
Moving on to part (b)… Total volume of the 8 large cubes = 4032 cm³ ÷ 7 x 3 = 1728 cm³
Volume of 1 large cube = 1728 cm³ ÷ 8 = 216 cm³
Length of one edge of the large cube = ∛216 cm³ = 6 cm
(Some students stumble at this step because they do not know how to perform the cube root function on the calculator.)
Length of one edge of the small cube = 6 cm x 2 ÷ 3 = 4 cm
Coming back to part (a)…
Volume of 1 small cube = 4 cm x 4 cm x 4 cm = 64 cm³
Total volume of all the small cubes = 4032 cm³ – 1728 cm³ = 2304 cm³
Total number of small cubes = 2304 cm³ ÷ 64 cm³ = 36
Here’s what we learnt from this question:
- If you can’t solve part (a) at first, try solving part (b) first. While the solution tends to be easier if you solve the questions in sequence, taking a longer time to solve the question is better than throwing away an entire question because you’re stuck at an earlier part. For this particular question, doing so also makes sense in terms of the mark allocation because part (b) carries more marks than part (a).
- Make sure your child knows how to use the cube root function on his or her scientific calculator!
Watch how our resident Mathematics problem-solver, Ahsan, demolished the question.
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