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We have to find the Ratio of number of rectangles (not squares) and number of squares in a chess board.

For this type of question do we have to manually count the squares and rectangles or is there any other method?

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The number of squares is $1^2+2^2+\dots+ 8^2=\frac{8\times9\times 17}{6}=204$ and the number of rectangles is $\binom{9}{2}^2=36^2=1296$. So the answer is $\frac{204}{1296-204}=\frac{17}{91}\approx 0.1868$

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    Can you please explain , how do you get the sum of square terms and $^9C_2$ – 2017-01-10
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    for the sum of squares you use the formula $\sum\limits_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6}$ and for the binomial squared notice that selecting a rectangle is equivalent to selecting two vertical lines and selecting two horizontal lines. There are $9$ options for each orientation. – 2017-01-10
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    For sum of squares, I want to know why you have used it . – 2017-01-10
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    oh, because there are $8^2$ squares of size $1$, $7^2$ squares of size $2$ etc. – 2017-01-10
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For a typical $n\times n $ chess board,

$(1) $ The number of squares are given by $$\sum_{i=1}^{n} i^2 =\frac{n (n+1)(2n+1)}{6} $$

$(2) $ A rectangle is formed by two vertical and two horizontal lines. On a $n\times n $ chessboard, we can choose both two horizontal and two vertical lines in $\binom {n+1}{2} $ ways. Thus the number of rectangles are given by $$ \binom {n+1}{2}^2 = \frac {n^2 (n+1)^2}{4} $$

Hope you can take it from here.