Grades 3-4 Video Solutions 2022-2022_3-4

Video Transcription

Problem number 16. The cube in the picture is built using the three kinds of wooden blocks shown. How many white wooden blocks are used? 8, 11, 13, 16, or 19?  First, let's compare the sizes of the three types of blocks. This white one is the smallest, and it's just a cube, so we'll call this side a size 1.  So this one has 1 on all three sides. The grey one is made up of the equivalent of 3 of these small cubes.  So it has 3, 1, and 1.  And the grey one, which we can see over here, is made up of 3 cubes in length and width.  Something like this.  So 3, 3, and 1. So you can think of the white cube as taking up 1 unit of volume. This dark grey one taking up 3, and this light grey one taking up 1, 2, 3, 4, 5. 5 units of volume.  Now let's look at our cube. It has a height equal to 3 little cubes, a width equal to 3 little cubes, and a depth also equal to 3 little cubes.  So we can think of it as being made up of 3 x 3 x 3, 27 unit cubes, or the equivalent of 27 white cubes.  Now we see there's a light grey piece here, and then another light grey piece kind of in the back. It has to go like this.  And that together is the equivalent of 10 cubes, because each one is the equivalent of 5. So 10 for light grey.  Then we have the dark grey. There's one here.  And then there's the one here going backwards.  We actually cannot fit another dark grey one anywhere. We know that the light grey that we only see the ends of here and here kind of takes up the back,  so there's no way to lay a dark grey one back there. And it cannot be here, because this top one is white. So these are the only two dark grey blocks in this cube.  Each one takes up the equivalent of 3 white cubes, so that takes up 6.  Since we figured out that the only grey blocks are the ones that we can see, at least partially here,  that means that everything that's not covered by the 2 light and the 2 dark are white cubes.  The grey cubes take up 16 units of volume, and we already figured out that there are 27 units of volume altogether.  So 27 minus 16, or the total minus the dark and light grey blocks, gives us 11. And since each white cube is 1 unit of volume, there are 11 of them. The answer is B.

Video Summary

The problem involves determining the number of white wooden blocks used to construct a cube made of three kinds of blocks: white (1 unit), dark grey (3 units), and light grey (5 units). The entire cube occupies a volume of \(3 \times 3 \times 3 = 27\) units. The light grey and dark grey blocks together take up 16 units of volume (10 units from two light grey blocks and 6 units from two dark grey blocks). Subtracting the grey blocks' volume from the total gives \(27 - 16 = 11\), hence there are 11 white blocks used. The answer is B.

Keywords

white wooden blocks

cube construction

block volume

grey blocks

unit calculation