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Geometry Smart Math

[Smart Math] Geometry Problem 11

Here’s and example of a SMART MATH problem for GEOMETRY.

Geometry

Problem

A large cube is formed from the material obtained by melting three smaller cubes of side 3, 4 and 5 cms. What is the ratio of the total surface areas of the smaller cubes and the large cube?

  1. 4 : 3
  2. 3 : 2
  3. 25 : 27
  4. 27 : 20
  5. 32 : 15

The Usual Method

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Surface area of cube with side 3 cms = 6\times 3^{2}=6\times 9=54 cms2

Surface area of cube with side 4 cms = 6\times 4^{2}=6\times 16=96 cms2

Surface area of cube with side 5 cms = 6\times 5^{2}=6\times 25=150 cms2

Hence, total surface area of the smaller cubes = 54 + 96 + 150 = 200 cms2

Volume of cube with side 3 cms = 3^{3}=27 cms3

Volume of cube with side 4 cms = 4^{3}=64 cms3

Volume of cube with side 5 cms = 5^{3}=125 cms3

Hence, total volume of the smaller cubes = 27 + 64 + 125 = 216 cms3

Hence, side of the larger cube = \sqrt[3]{216}=6 cms

Hence, surface area of the larger cube = 6\times 6^{2}=6\times 36=216 cms2

Hence, ratio of surface areas = 200 : 216

i.e. 25 : 27

(Ans: 3)

Estimated Time to arrive at the answer = 60 seconds.

Using Technique

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Add squares of 3, 4 and 5 to get 9 + 16 + 25 = 50.

50 is a factor of the total surface areas of the three smaller cubes and hence 50 or its factor should be one of the values in the proportion. Only option ‘3’ has the factor of 50; 25, so option ‘3’ has to be the answer.

(Ans: 3)

Estimated Time to arrive at the answer = 10 seconds.
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