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Belt

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Magic Belt - Back to the Cool Math Games

A magic wish-granting rectangular belt always shrinks to 1/2 its length and 1/3 its width whenever its owner makes a wish. After three wishes, the surface area of the belt’s front side was 4 cm2.

What was the original length, if the original width was 9 cm?

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Belt - solution

The original length of belt was 96 cm.

A magic rectangular belt always shrinks its length to 1/2 and width to 1/3 whenever its owner wishes something. After three such wishes, its surface (Edit: surface area of the front side) was 4 cm2. What was the original length, if the original width was 9 cm?

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I am new here and I may be all wrong, but when I worked this out I got 192 or double 96.

Here's my work:

Original width = 9

1st wish: 9 * 1/3 = 3

2nd wish: 3 * 1/3 = 1

3rd wish: 1 * 1/3 = 1/3

Current width is 1/3 cm. Total area must be 1/3 * L = 16 cm

L = 16 / 1/3 = 48

Current Length must be 48 cm.

Working backwards: 3rd wish = 48

2nd wish = 48 * 2 = 96

1st wish = 96 * 2 = 192 original length.

What did I miss?

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I am new here and I may be all wrong, but when I worked this out I got 192 or double 96.

Here's my work:

Original width = 9

1st wish: 9 * 1/3 = 3

2nd wish: 3 * 1/3 = 1

3rd wish: 1 * 1/3 = 1/3

Current width is 1/3 cm. Total area must be 1/3 * L = 16 cm

L = 16 / 1/3 = 48

Current Length must be 48 cm.

Working backwards: 3rd wish = 48

2nd wish = 48 * 2 = 96

1st wish = 96 * 2 = 192 original length.

What did I miss?

u got the area wrong

u said

Total area must be 1/3 * L = 16 cm

Its not 4 squared which does equal 16, it is 4 cm squared

ex: a square with a width and length of 2 = 4 cm2 not 16

so actually the total area of belt after 3 wishes is 1/3*L = 4 cm2

1/3= 0.33

4 * 0.33 = 12

12*2= 24

24*2= 48

48*2= 96

[if u really want to get technical 1/3 = 0.333333333333333333333333333333 and so on so its not exactly 96]

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Posted · Report post

Oh - I get it - Thanks!

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Posted · Report post

so actually the total area of belt after 3 wishes is 1/3*L = 4 cm2

1/3= 0.33

4 * 0.33 = 12

12*2= 24

24*2= 48

48*2= 96

[if u really want to get technical 1/3 = 0.333333333333333333333333333333 and so on so its not exactly 96]

1/3 does equal .33 repeated but your math is wrong.

4 * 0.33 does not = 12, but it doesn't matter because you were supposed to divide which does give you 12.

0.33 * L = 4 => L = 4 / 0.33

4 / 0.33 = 4 * 3 = 12, no decimals.

So the answer is exactly 96.

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Posted · Report post

If you want to do it with the fraction instead of breaking it down to decimals it is even simpler.

4/(1/3)=4*3=12

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Width after three wishes:

9*(1/3)*(1/3)*(1/3) = 9*(1/3)^3 = 9*(1/27) = 1/3

Length after three wishes:

surface = length * width

length = surface / width

length = 4/(1/3) = 4*3 = 12

Original length:

12/(1/2)/(1/2)/(1/2) = 12/(1/2)^3 = 12/(1/8) = 96

4 / (9 * (1/3)^3) / (1/2)^3 = 96 cm

4 / (9/27) * 8 = 96 cm

And the correct denotation of 1/3 would be 0.33... (with two or three dots) in which case the dots mean that the number is being repeated.

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Posted · Report post

One other method of solving it would be,

If x,y are the original length and width resp.

xy=area

After each wish, length and width become 1/2 and 1/3 of the original resp.

Thus, the new area, after a wish = 1/6*old area.

As three wishes were made, and the final area being 4, the initial area is thus = 4*6*6*6 = 864

Width = 9 (given)

Length = 864/9=92.

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Posted · Report post

this s not a puzzle at all....

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let's assume original length is a and original width is b :

after 1st wish the area = a/2* b/3

after 2nd wish the area = a/4*b/9

after 3rd wish the area = a/8*b/27

the area after 3rd wish is given as 4 and the width is 9

therefore a/8*9/27=4 , 9a = 4*8*27

thus a = 96 cms

I hope it's the right way

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Posted · Report post

96 if I got my halvsies and thirdsies right.

by the by

heres the question..............

If a person with a 96cm waist was wearing the belt, would he/she be suffocated on the first, second, or third wish?

I would have wished the belt didn't shrink

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Posted · Report post

Width is (((9/3)/3)/3) = 1/3 after 3 wishes

Area is 4cm (squared) so final length is 4 / 1/3 = 12

So original length is (((12*2)*2)*2) = 96

I know everyone has said it before, just confirming my working!

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Posted · Report post

i tought he wished to make it a diffrent size....but if you want to go though all of that complication

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Belt - Back to the Logic Puzzles

A magic rectangular belt always shrinks its length to 1/2 and width to 1/3 whenever its owner wishes something. After three such wishes, its surface was 4 cm2. What was the original length, if the original width was 9 cm?

I hate to be anal, but a belt is three dimensional. Even if the thickness of the belt were small enough to be neglegable, the belt would still have a front and a back. At most, the surface area of one side is 2 cm^2, resulting in an original length of 48 cm. You could just change the puzzle to state that "After three such wishes, the surface area of the front side was 4 cm^2."

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Posted · Report post

Belt - Back to the Logic Puzzles

A magic rectangular belt always shrinks its length to 1/2 and width to 1/3 whenever its owner wishes something. After three such wishes, its surface was 4 cm2. What was the original length, if the original width was 9 cm?

I hate to be anal, but a belt is three dimensional. Even if the thickness of the belt were small enough to be neglegable, the belt would still have a front and a back. At most, the surface area of one side is 2 cm^2, resulting in an original length of 48 cm. You could just change the puzzle to state that "After three such wishes, the surface area of the front side was 4 cm^2."

puzzle edited

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Belt - Back to the Logic Puzzles

A magic rectangular belt always shrinks its length to 1/2 and width to 1/3 whenever its owner wishes something. After three such wishes, its surface was 4 cm2. What was the original length, if the original width was 9 cm?

I hate to be anal, but a belt is three dimensional. Even if the thickness of the belt were small enough to be neglegable, the belt would still have a front and a back. At most, the surface area of one side is 2 cm^2, resulting in an original length of 48 cm. You could just change the puzzle to state that "After three such wishes, the surface area of the front side was 4 cm^2."

puzzle edited

I believe this is one of those puzzles that you take literally and when the owner makes a wish the belts lengt goes to 1/2. Not that it is half each wish. It never says that it is each time.

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Hello!

i am new to this.

Let me put the solution in this way.

Let the original lenght = x cm.

and the given width = 9 cm.

1st wish:

lenght =x/2 cm; width =9/3=3 cm;

2nd wish:

lenght =x/4 cm; width =1 cm;

3rd wish:

lenght =x/8 cm; width =1/3 cm;

After three wishes:

The given Surface area A (say) = 4 cm2;

Surface area A = length * width;

4 cm2 =(x/8 )*(1/3) cm2;

x = 4 *24= 96 cm;

The original length x= 96 cm;

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I did it kinda like everyone else...

1. I divided 9 ( the original width) by 3, 3(new width) by 3, then 1( new, new , width) by 3, for the 3 wishes. This gave me 1/3 for the resulting width.

2. Next, i divided 4cm squared by 1/3, and got 12.

3. I doubled thise 3 times ( for the 3 wishes) to get 96, the original width.

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simple i just used an equation to solve:

L= Length

W= Width

1/8L (1/27W) = 4

then just put 9 in for the width and solved for the length

(FYI i got 1/8 from placing 1/2 to the third power since it was wished to shrink 3 times, and did the same for the length- 1/3 ^ 3)

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Posted (edited) · Report post

96 cm

Edited by scuttill
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Posted · Report post

l/8*w/27=4

l=4*8*27/9=96

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I am new here and I may be all wrong, but when I worked this out I got 192 or double 96.

Here's my work:

Original width = 9

1st wish: 9 * 1/3 = 3

2nd wish: 3 * 1/3 = 1

3rd wish: 1 * 1/3 = 1/3

Current width is 1/3 cm. Total area must be 1/3 * L = 16 cm

L = 16 / 1/3 = 48

Current Length must be 48 cm.

Working backwards: 3rd wish = 48

2nd wish = 48 * 2 = 96

1st wish = 96 * 2 = 192 original length.

What did I miss?

you got the first bit right, the width after three wishes is 1/3cm, but then you said the area was 16cm. its meant to be 4cm^2, which is not the same as 16cm

so instead of 1/3 * L = 16cm, it should be

1/3 * L = 4 to find length after 3 wishes

i havent checked my answer, but looking at your post, that jumped out at me.

hope it helps

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