Question

how many blocks with dimensions of 1/4 x 1 x 1 are needed to fill the gap in the prism? 3 3 3/4 2 6 12 18 24

Explanation:

Step1: Calculate the missing length in the \(x -\) direction

The length of the prism is \(3\frac{3}{4}=\frac{15}{4}\). If we assume the blocks are placed along the \(x -\) direction (where the dimension of the block is \(\frac{1}{4}\)), and let the number of blocks along \(x\) be \(n_x\). But since we can also think in terms of volume. First, assume the height of the gap is \(h = 3\) (same as the height of the prism) and the width of the gap is \(w=2\) (same as the width of the prism). The length of the gap: the full - length is \(l = 3\frac{3}{4}=\frac{15}{4}\), assume the non - gap part (from the figure, if we consider the unit blocks) has a length that is a multiple of \(1\). Let's use the volume formula. The volume of each block \(V_{block}=\frac{1}{4}\times1\times1=\frac{1}{4}\).

Step2: Calculate the volume of the gap

The volume of a rectangular prism is \(V=l\times w\times h\). The volume of the gap: assume the length of the gap \(l_{gap}\), width \(w = 2\), height \(h = 3\). If we consider the number of \(1\times1\times1\) blocks in the non - gap part (from the figure's pattern), and then find the length of the gap. Another way: the number of blocks \(n=\frac{V_{gap}}{V_{block}}\). The volume of the gap: if we assume the gap has a length of \(\frac{3}{4}\) (by observing the fraction \(3\frac{3}{4}\), and if we consider the unit of \(\frac{1}{4}\)), width \(2\) and height \(3\). Then \(V_{gap}=\frac{3}{4}\times2\times3=\frac{18}{4}\).

Step3: Calculate the number of blocks

Since \(V_{block}=\frac{1}{4}\), then \(n=\frac{V_{gap}}{V_{block}}=\frac{\frac{18}{4}}{\frac{1}{4}} = 18\)

Answer:

\(18\)