The 9 horizontal and 9 vertical lines on an 8 × 8 chessboard form ‘r’

Question

The 9 horizontal and 9 vertical lines on an 8 × 8 chessboard form ‘r’ rectangles and ‘s’ squares, The ratio s/r in its lowest terms is

(A) 17/108
(B) 1/6
(C) 4/27
(D) 5/27
(E) 6/27

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Saasingh · Answer

A.

Number of squares in a grid = summation(n^2)

Which is n(n+1)(2n+1)/6. 
Putting n = 8 we get squares = 17*12 = 204.

For number of rectangles, we select 2 horizontal and 2 verticals.

9c2 * 9c2 = 9*8*9*8/4 = 1296.

So 204/1296 = 17/108.

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ShaikhMoice · Answer

Experts kindly help with the solution
Thanks in advance

GMATWhizTeam · Answer

Solution:

Choosing any 2 verticles and 2 horizontal lines would give us a rectangle.

Note: few, in this case, will also be squares. We can count them as rectangles only.

For number of Rectangles  r :

Choosing 2 vertical lines and 2 horizontal line  = 9C2 	imes 9C2 = 36 	imes 36 = 1296

For number of squares  s :

If we see there will be  8	imes 8 = 64  1*1 squares. 
There will be  7 	imes 7 = 49  2*2 squares. 
and so on..

So number of squares (s)  = 8^2+7^2+6^2....1^2  (sum of n squares)  = \frac{n(n+1)(2n+1)}{6} = \frac{8	imes 9 	imes 17}{6} = 204

So Ratio  \frac{s}{r} = \frac{204}{1296} = \frac{17}{108}  (in lowest terms)

Hence the right answer is  Option A .

Note: if in  r  only and only rectabgles are considered then the answer will be  = \frac{204}{1296-204} = \frac{204}{1092}

Kushalp5 · Answer

Bunuel  please share detailed answer. Thanks in advance

Fdambro294 · Answer

Every square is a rectangle. So essentially, the question is similar to asking the following: out of every possible rectangle that can be made on the 8 by 8 board, how many are squares?

(1st) total number of rectangles (including squares)

If you think about a rectangle drawn on the coordinate plane, you can identify the rectangle by:

2 distinct X coordinates

And

2 distinct Y coordinates.

For instance, a square/rectangle drawn from the origin that has length of 4 and is located in quadrant 1 will be:

(0.0) —-(4.0) ——(0.4) —-(4.4)

Letting the 2 X coordinates be A and B
and
Letting the 2 Y coordinates be P and Q

(A , P) —- (B , P) —— (A, Q) —- (B , Q)

Thus, if you can picture the 8 by 8 chessboard on the coordinate plane, we would be able to identify any unique rectangle by:

Picking 2 unique horizontal lines out of the 9 drawn

And

Picking 2 unique vertical lines out of the 9 drawn

Total rectangles = “9 choose 2” * “9 choose 2” = 36 * 36

(2nd) how many of these total rectangles are squares

1st we can easily find the 1 by 1 squares - they will be the number or squares on the 8 by 8 chess board ——-> (8)^2

2nd, to figure out how many 2 by 2 square exist - you can count from the first row up - there will be 7 unique squares in the 1st column, and there will be 7 columns more

———> (7)^2

This pattern will continue all the way until you get to the amount of 8 by 8 squares = (1)^2

So we are looking for the sum of the first 8 consecutive perfect squares which is given by:

= (1/6) (2n + 1) n (n + 1) =

= (1/6) (2*8 + 1) * 8 * (8 + 1)

=. (1/6) (17) (8) (9)

= 17 * 4 * 3

17 * 12

Final answer:

(17 * 12) / (36 * 36)

Cancel the factor of 12 in the NUM and the DEN

(17) / (3 * 36) =

17/108

Answer is B not A?

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bumpbot · Answer

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The 9 horizontal and 9 vertical lines on an 8 × 8 chessboard form ‘r’

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