A solid cube of side 12 cm is cut into small solid cubes of side 4 cm each . What will be the relation between the total surface area of the original cube and the new total surface area of the small cubes so formed ?

The correct answer is: New total surface area will be thrice the original total surface area.

Step 1: Calculate the volume and total surface area of the original cube.
Let the side length of the original cube be S = 12 cm.
The volume of the original cube (V_original) is given by the formula:
$V_{\text{original}}=S^3$
$V_{\text{original}}={12}^3=1728{\text{cm}}^3$

The total surface area of the original cube (A_original) is given by:
$A_{\text{original}}=6\times S^2$
$A_{\text{original}}=6\times {12}^2=6\times 144=864{\text{cm}}^2$

Step 2: Find the number of small cubes formed.
Let the side length of each small cube be s = 4 cm.
The volume of one small cube (V_small) is:
$V_{\text{small}}=s^3$
$V_{\text{small}}=4^3=64{\text{cm}}^3$

The number of small cubes (N) formed by cutting the original cube is:
$N=\frac{V_{\text{original}}}{V_{\text{small}}}=\frac{1728}{64}=27$

Step 3: Calculate the total surface area of one small cube and the combined new total surface area.
The total surface area of one small cube (A_{single_small}) is:
$A_{\text{single\_small}}=6\times s^2$
$A_{\text{single\_small}}=6\times 4^2=6\times 16=96{\text{cm}}^2$

The combined new total surface area of all 27 small cubes (A_{new_total}) is:
$A_{\text{new\_total}}=N\times A_{\text{single\_small}}$
$A_{\text{new\_total}}=27\times 96=2592{\text{cm}}^2$

Step 4: Determine the relationship between the new total surface area and the original total surface area.
We compare A_{new_total} with A_original:
$\frac{A_{\text{new\_total}}}{A_{\text{original}}}=\frac{2592}{864}=3$

Thus, $A_{\text{new\_total}}=3\times A_{\text{original}}$, which means the new total surface area will be thrice the original total surface area.