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Ramon wants to cut a rectangular board into identical square

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sandy
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Ramon wants to cut a rectangular board into identical square [#permalink] New post   12 Aug 2018, 15:33
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Ramon wants to cut a rectangular board into identical square pieces. If the board is 18 inches by 30 inches, what is the least number of square pieces he can cut without wasting any of the board?

(A) 4
(B) 6
(C) 9
(D) 12
(E) 15
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E
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Re: Ramon wants to cut a rectangular board into identical square [#permalink] New post   15 Aug 2018, 05:20
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Explanation


Cutting a rectangular board into square pieces means that Ramon needs to cut pieces that are equal in length and width. “Without wasting any of the board” means that he needs to choose a side length that divides evenly into both 18 and 30.

“The least number of square pieces” means that he needs to choose the largest possible squares. With these three stipulations, choose the largest integer that divides evenly into 18 and 30, or the greatest common factor, which is 6. This would give Ramon 3 pieces going one way and 5 pieces going the other.

He would cut 3 × 5 = 15 squares of dimension 6" × 6". Note that this solution ignored squares with non-integer side length for the sake of convenience, a potentially dangerous thing to do. (After all, identical squares of 1.5" by 1.5" could be cut without wasting any of the board.) However, to cut squares any larger than 6" × 6", Ramon could only cut 2 squares of 9" or 1 square of 18" from the 18" dimension of the rectangle, neither of which would evenly divide the 30" dimension of the rectangle. The computed answer is correct.
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Re: Ramon wants to cut a rectangular board into identical square [#permalink] New post   03 Sep 2018, 00:38
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Another approach:
The area of the rectangle is 18*30 = 540
Area of any square =x^2
Now we are required to find the least number of squares that fit exactly inside the rectangle.
Form a equation nx^2 = 540
Replace ans choices for n, when n = 15 the equation reduces to x^2 = 36 therefore each square will be of side 6.
For all other values of n x will not be an integer hence there will be some wastage of board.
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Re: Ramon wants to cut a rectangular board into identical square [#permalink] New post   31 Aug 2018, 09:28
not understood, can it be a bit simpliar.
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Re: Ramon wants to cut a rectangular board into identical square [#permalink] New post   31 Aug 2018, 15:07
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IshanGre wrote:
not understood, can it be a bit simpliar.


Sorry about that I messed up the explanations. This is the correct explanation.
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Re: Ramon wants to cut a rectangular board into identical square [#permalink] New post   31 Aug 2018, 20:39
what is the correct approach?
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Re: Ramon wants to cut a rectangular board into identical square [#permalink] New post   01 Sep 2018, 16:21
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The answer above is a bit cumbersome.

30 is the length of the rectangular region and 18 is the width. Now, squares that fit into the region without leftover AND a single square must have 4 equal sides.

So a bit of logic, divide the measures by two. actually, you do have 3 and 5 which is the smallest area inside the rectangular area with which you can have a square considering the length and width of the rectangle.

3*5 = 15.

E is the answer. 30-second approach.

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Re: Ramon wants to cut a rectangular board into identical square [#permalink] New post   01 Sep 2018, 23:30
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I would not classify this question as easy!
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Re: Ramon wants to cut a rectangular board into identical square [#permalink] New post   02 Sep 2018, 00:46
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Hi,

it is also not the end of the world in terms of difficulty. If this is hard, then 170 on quant is far from being the reach.

See here for reference https://gre.myprepclub.com/forum/all-you-n ... -8898.html

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Re: Ramon wants to cut a rectangular board into identical square [#permalink] New post   02 Sep 2018, 00:51
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Carcass wrote:
Hi,

it is also not the end of the world in terms of difficulty. If this is hard, then 170 on quant is far from being the reach.

See here for reference https://gre.myprepclub.com/forum/all-you-n ... -8898.html

Regards


Well it shouldn't be easy, nor hard, should be in the range of medium relative to other questions classified here.
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Re: Ramon wants to cut a rectangular board into identical square [#permalink] New post   02 Sep 2018, 01:21
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In fact, it is in the easy medium range.

:)
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Re: Ramon wants to cut a rectangular board into identical square [#permalink] New post   02 Sep 2018, 03:10
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The following approach may look lengthy, but if you have practiced GCF word problems enough, you can attempt this questions in under 20 seconds.

You want to have least number of squares without any wastage. Meaning you need to maximise the dimension of the squares.
You have 30 by 18 inch of cardboard.
Ask this -> what is the greatest length will divide both 18 and 30 (without remainder = no wastage)? In mathematical terms you are looking at the GCF of 18 and 30. Which is 6.

Now you will be able to cut 3 squares along 18 inch side and 5 squares along 30 inch side. 3 by 5 squares = 15 squares
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Re: Ramon wants to cut a rectangular board into identical square [#permalink] New post   03 Sep 2018, 01:15
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amorphous wrote:
Another approach:
The area of the rectangle is 18*30 = 540
Area of any square =x^2
Now we are required to find the least number of squares that fit exactly inside the rectangle.
Form a equation nx^2 = 540
Replace ans choices for n, when n = 15 the equation reduces to x^2 = 36 therefore each square will be of side 6.
For all other values of n x will not be an integer hence there will be some wastage of board.


Add to that, nx^2 = 540

prime factorization of 540 = 54x10 = 2^2 x 3^3 x 5 = (2x3)^2 x (3x5) => n = 3x5 = 15
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Re: Ramon wants to cut a rectangular board into identical square [#permalink] New post   04 Sep 2018, 15:21
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The keyword here is identical squares. Otherwise you can end up with 4 squares.
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Re: Ramon wants to cut a rectangular board into identical square [#permalink] New post   28 Oct 2019, 10:42
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sandy wrote:
Ramon wants to cut a rectangular board into identical square pieces. If the board is 18 inches by 30 inches, what is the least number of square pieces he can cut without wasting any of the board?

(A) 4
(B) 6
(C) 9
(D) 12
(E) 15

If we aren't wasting any wood, the length and width must be divisible by one side of the square
So, this question is a clever way of asking us what the greatest common divisor (GCD) of 18 and 30
The GCD of 18 and 30 is 6, so if we cut squares that are 6 x 6, then we won't waste any wood.

We get something like this:
Image

So, we can cut 15 squares.
Answer: E

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Brent
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Re: Ramon wants to cut a rectangular board into identical square [#permalink] New post   08 Apr 2021, 16:19
sandy wrote:
IshanGre wrote:
not understood, can it be a bit simpliar.


Sorry about that I messed up the explanations. This is the correct explanation.



try to equate 540=n*L^2
and substitute value for n from option and the option satisfy above condition , is your ans...
and question is saying that we have rectangle=18*30 and we have to cut into square and we dont have waste material too
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Ramon wants to cut a rectangular board into identical square [#permalink] New post   10 Apr 2021, 08:25
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sandy wrote:
Ramon wants to cut a rectangular board into identical square pieces. If the board is 18 inches by 30 inches, what is the least number of square pieces he can cut without wasting any of the board?

(A) 4
(B) 6
(C) 9
(D) 12
(E) 15


\(18 = (2)(3)(3)\)
\(30 = (2)(3)(5)\)

Since we need to cut squares, the size would be the HCF of these two numbers i.e. \(6\)

Therefore, we can have least \((3)(5) = 15\) squares

Hence, option E
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Re: Ramon wants to cut a rectangular board into identical square [#permalink] New post   04 May 2022, 05:03
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chethanbp1989 wrote:
The keyword here is identical squares. Otherwise you can end up with 4 squares.

Thank you - I kept getting tripped up on why not 4
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Re: Ramon wants to cut a rectangular board into identical square [#permalink] New post   21 Jul 2022, 03:48
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Aren't we assuming here that the sides of the squares must be integers?
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Re: Ramon wants to cut a rectangular board into identical square [#permalink] New post   23 Nov 2022, 09:23
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This one I got in 45 seconds. I decided to find the prime factorization of both sides of the rectangle, finding that they both have 6 in common at the least. That is the smallest integer count that could be multiplied to fit into the larger rectangle!
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Re: Ramon wants to cut a rectangular board into identical square [#permalink]
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