Question

The $9$ horizontal and $9$ vertical lines on an $8\times 8$ chess-board form $r$ rectangles and $s$ squares. Then, the ratio $s:r$ in its lowest terms is

• A. $\frac{1}{6}$
• B. $\frac{17}{108}$
• C. $\frac{4}{27}$
• D. None of the above

Answer 1

The correct option is B $\frac{17}{108}$

Let horizontal lines (m) $=9$, vertical lines (n) $=9$

Number of rectangles $={}^m{C}_2\times {}^n{C}_2$

$={}^9{C}_2\times {}^9{C}_2$

$=1296$.

Numbers of squares $=(1^2+2^2+.....+7^2+8^2)$ $=204$

So, $\frac{\text{Number of squares}}{\text{Number of rectangles}}$$=\frac{204}{1296}$$=\frac{17}{108}$