21 replies to this topic

[rookie1ja]

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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Spoiler for Solution

Belt - solution
The original length of belt was 96 cm.

Spoiler for old wording

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?

[ezkoalmc]

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?

[iknowall]

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]

[ezkoalmc]

Oh - I get it - Thanks!

[mdsl]

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.

[steve1011]

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

[Mythx]

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.

[Genius]

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.

[ibrahim_zulu18]

this s not a puzzle at all....

[bowfinger]

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