If the mode of the following distribution is $$6\frac{1}{6}$$​, then what is the value of k?

Correct option is B

​$$\textbf{Given:} \\\text{Mode} = 6\frac{1}{6} = \frac{37}{6} \\\text{Class intervals: } 1\!-\!3,\;3\!-\!5,\;5\!-\!7,\;7\!-\!9,\;9\!-\!11,\;11\!-\!13 \\\text{Frequencies: } 12,\;45,\;80,\;k,\;38,\;16 \\\textbf{Concept Used:} \\\text{Mode of grouped frequency distribution} \\\textbf{Formula Used:} \\\text{Mode} = l + \frac{f_1 - f_0}{2f_1 - f_0 - f_2}\,h \\\textbf{Solution:} \\\text{Modal class is } 5\!-\!7 \\ \ \\l = 5,\; h = 2 \\ \ \\f_1 = 80,\; f_0 = 45,\; f_2 = k \\ \ \\\frac{37}{6} = 5 + \frac{80 - 45}{2(80) - 45 - k}\times 2 \\ \ \\\frac{37}{6} - 5 = \frac{35}{160 - 45 - k}\times 2 \\ \ \\\frac{7}{6} = \frac{70}{115 - k} \\ \ \\7(115 - k) = 6 \times 70 \\ \ \\805 - 7k = 420 \\ \ \\7k = 385 \\ \ \\k = 55$$​​