$V_1, V_2, V_3$ and $V_4$ are the volumes of four cubes of side lengths x cm, 2x cm, 3x cm and 4x cm respectively. The following statements regarding these volumes are given below.

(A) $V_1 + V_2 + 2V_3 < V_4$

(B) $V_1 + 3V_2 > V_3 + V_4$

(C) $2(V_1 + V_3) + V_2 = V_4$

(D) $V_1 + 4V_2 + V_3 < V_4$

Which of these statements is/are correct?

Choose the correct answer from the options given below:

The four cubes have side lengths $x$ cm, $2x$ cm, $3x$ cm, and $4x$ cm.

Volume of a cube $= \text{(side)}^3$

$V_1 = x^3$

$V_2 = (2x)^3 = 8x^3$

$V_3 = (3x)^3 = 27x^3$

$V_4 = (4x)^3 = 64x^3$

Quick tip: When you cube a number with $x$, cube the number separately. For example, $(2x)^3 = 2^3 \times x^3 = 8x^3$

Super shortcut: Since all volumes have $x^3$ in common, we can just compare the coefficients: $1, 8, 27, 64$

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Statement (A): $V_1 + V_2 + 2V_3 < V_4$

Left side:

$V_1 + V_2 + 2V_3 = x^3 + 8x^3 + 2(27x^3) = x^3 + 8x^3 + 54x^3 = 63x^3$

Right side: $V_4 = 64x^3$

Since $63x^3 < 64x^3$, Statement (A) is TRUE.

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Statement (B): $V_1 + 3V_2 > V_3 + V_4$

Left side:

$V_1 + 3V_2 = x^3 + 3(8x^3) = x^3 + 24x^3 = 25x^3$

Right side:

$V_3 + V_4 = 27x^3 + 64x^3 = 91x^3$

Since $25x^3 \not> 91x^3$, Statement (B) is FALSE.

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Statement (C): $2(V_1 + V_3) + V_2 = V_4$

Left side:

$2(V_1 + V_3) + V_2 = 2(x^3 + 27x^3) + 8x^3 = 2(28x^3) + 8x^3 = 56x^3 + 8x^3 = 64x^3$

Right side: $V_4 = 64x^3$

Since $64x^3 = 64x^3$, Statement (C) is TRUE.

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Statement (D): $V_1 + 4V_2 + V_3 < V_4$

Left side:

$V_1 + 4V_2 + V_3 = x^3 + 4(8x^3) + 27x^3 = x^3 + 32x^3 + 27x^3 = 60x^3$

Right side: $V_4 = 64x^3$

Since $60x^3 < 64x^3$, Statement (D) is TRUE.

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Statements (A), (C), and (D) are correct.