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If a cube with the length of the side of 4 cm is cut into smaller cubes with the length of the side of 1 cm, then what is the percentage increase in the surface area of the resulting cubes?

Re: A 4 cm cube is cut into 1 cm cubes. [#permalink]

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25 Sep 2008, 14:44

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4 cm cube has 6 facets of 16 sq. cm each (96 sq. cm in all). After cutting the cube into 1 cm cubes we'll end up with 64 1 cm cubes. Each will have the surface area of 6 sq. cm. \(\frac{6*64}{6*16} = \frac{4}{1} = 400%\). Therefore the increase in surface area must have been 300%. Did that without looking into the OE . Let's check now...
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m01-Q11.....explanation not clear...help please [#permalink]

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02 Oct 2008, 12:53

A 4 cm cube is cut into 1 cm cubes. What is the percentage increase in the surface area after such cutting?

4% 166% 266% 300% 400% The easiest way to solve this problem is to calculate the original surface area and then the final. The original area is 4*4*6. The new area is 1*1*6*4*4*4. So, the difference is 1:4. Therefore, the increase is 300%. You can also solve it logically, but that's more risky.

The correct answer is D.

I couldnt quite understand the explanation here...

The original area is 4*4*6 i agree, but the new area should be 1*1*6*4, why has 4 been multiplied 3 times. Could someone please explain. The question says that the original cube has been cut into 1cm cubes so there are 4 cubes in all now, and every cube will have 1cm side so the SA of every cube will be 1*1*6 and since we have four such cubes the Area of all these will be 6*4.

Re: m01-Q11.....explanation not clear...help please [#permalink]

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03 Oct 2008, 10:10

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Hi Siddarth,

You have a 4 cm cube, so the surface area is 4*4(area of one side)*6 sides..= 96, which you agree with. Then figure out how many 1 cm cubes can fit into 4 cm cube. 4 cm cube's Volume is 4*4*4 = 64. A 1 cm cube's Volume is 1*1*1 = 1 so 64/1 = 64 little cubes.

Now for the surface area of each little cube: 1*1(area of one side of one cube) * 6 sides= 6 surface area of one cube. So, 64 cubes * 6 surface area/cube = 384 Total surface area.

So, 384 new/96 old is 4 times or 400% so 300% difference or 288 (difference in area)/96 (original) = 3 or 300% difference.

total 4*4*4 = 64 squares with side 1cm total surface area = 64 * 6 cm square surface area for 4cm cube is = 16 * 6 cm square

so surface area increase = (64-16)/16 * 100 = 300%

how did you know that there were going to be 64 additional squares?

it is 300% - I like how one of the participants just used l and did not input a variable in. one knows that there are 64 additional squares because discussed 3-dim shapes are cubes therefore all sides are equal. Consequently one can fit (4)/(1) lengths into one dimension = 4. So 4 little cubes in on dimension and then cube it.

Hmmm that was a bit complicated - hope that made sense.

surface area of 4 m cube = 6*4*4 surface area of 1 m cube = 6 no of cubes = 4*4*4 increase in surface area = (4*4*4*6 - 6*4*4)*100/6*4*4 = 300% hence D
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Surface area of 4 cm cube = (surface area of one side)*(number of sides) = (4*4) * (6) Surface area of all cubes: = (number of cubes)*(surface area of cube) = (4*4*4) * (6)

By simply looking at these two equations we see that there is 4 times more surface area with the little cubes, thus 300% increase.

The surface area of a cube with the length of the side of 4 cm is 6*4^2=6*16 cm^2.

Now, since the volume of the big cube is 4^3=64 cm^3 and the volume of the smaller cubes is 1^3=1 cm^3, then when the big cube is cut into the smaller cubes we'll get 64/1=64 little cubes. Each of those little cubes will have the surface area equal to 6*1^2=6 cm^2, so total surface are of those 64 little cubes will be 6*64 cm^2.

6*64 is 4 times more than 6*16 which corresponds to 300% increase.

Answer: D.

Or: general formula for percent increase or decrease, (percent change): \(Percent=\frac{Change}{Original}*100\)

So the percent increase will be: \(Percent=\frac{Change}{Original}*100=\frac{6*64-6*16}{6*16}*100=300%\).