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Mathematics

If the length of each side of a cube is reduced by 50%, then the ratio of the total surface area of the original and the new cube is :

  1. 2 : 1

  2. 4 : 1

  3. 8 : 1

  4. 8 : 3

Mensuration

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Answer

Let the original side be a units and new side be a' units.

Reduction of 50% :

a' = a - 50% of a = a - 12a=a2\dfrac{1}{2}a = \dfrac{a}{2} units.

We know that,

Total surface area of cube (TSA) = 6(side)2.

Calculating the original total surface area of cube,

original TSA = 6a2

Calculating the new total surface area of a cube,

new TSA = 6(a')2

= 6 (a2)2\Big(\dfrac{a}{2}\Big)^2

= 6 × a24\dfrac{a^2}{4}

= 3a22\dfrac{3a^2}{2}.

Ratio of TSA of original cube to the new cube:

original TSA : new TSA

6a2 : 3a22\dfrac{3a^2}{2}

6a23a22\dfrac{6a^2}{\dfrac{3a^2}{2}}

41\dfrac{4}{1}

4 : 1.

Hence, option 2 is the correct option.

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