Question

Coloured cubes (red and yellow) of equal size are arranged and stacked to form the cuboidal structure as shown in the image. If the ratio of the number of yellow cubes to that of the red cubes is \(1:3\), how many yellow cubes are NOT visible in the image?

Hint:

When solving problems involving 3D structures, always account for overlapping cubes when calculating the visible ones. Be sure to systematically analyze hidden interiors or cubes shared between faces to avoid overcounting.

Answer 1

To determine the number of yellow cubes that are not visible, we need to first find the total number of cubes and then apply the given ratio of yellow to red cubes. Then, we count the visible yellow cubes in the image and subtract this number from the total number of yellow cubes.
Step 1
Count the total number of cubes in the cuboidal structure. The structure is a 3x3x3 cube, so the total number of cubes is $3 \times 3 \times 3 = 27$ cubes.
Step 2
Using the given ratio of yellow cubes to red cubes, which is $1:3$, we can determine the number of yellow and red cubes. Let the number of yellow cubes be $y$ and the number of red cubes be $r$. According to the ratio, $y:r = 1:3$. Therefore, y = \(\frac{1}{4} \times 27 = 6.75\) and \( r = \frac{3}{4} \times 27 = 20.25.\) Since the number of cubes must be an integer, we round $y$ to 7 and $r$ to 20.
Step 3
Count the number of visible yellow cubes in the image. From the image, we can see that there are 5 yellow cubes visible.
Step 4 Subtract the number of visible yellow cubes from the total number of yellow cubes to find the number of yellow cubes that are not visible. $7 - 5 = 2$.
Step 5 Therefore, the number of yellow cubes that are not visible in the image is 2.