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The 9 horizontal and 9 vertical lines on an 8X8 chessboard form 'r' rectangles& 's' squares. The ratio of s/r in the lowest terms is?

The 9 horizontal and 9 vertical lines on an 8X8 chessboard form 'r' rectangles& 's' squares. The ratio of s/r in the lowest terms is?

Grade:12

2 Answers

Badiuddin askIITians.ismu Expert
148 Points
13 years ago

Dear jee king

number of rectangles = r = 9C2 *9C2  =1296

number of square=s = (12 + 22 + 32 + ................+ 82 ) =204

 

so s/r =204/1296

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Spandan Mallick
12 Points
4 years ago
First for squares,
We see in a 8×8 chessboard, there are 64, 1×1 squares. Similarly 32, 2×2 squares and 16, 3×3 squares. Hence it forms a pattern. Also if you divide the area... total area is 64 sq. units. 
For 1×1 squares, hence 64 are possible. For 2×2squares you can get it. Hence,
No of squares = (12+22+32+...+82) = 204.
For rectangles, consider the number of lines rather than squares. There are 9 lines. A rectangle can be formed by combination of any of these two four lines taking two from each pair of 9. Hence,
9C2 × 9C2 = 1296.
 
Spandan Mallick, IIT-JEE Aspirant

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