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

Class	1 - 3	3 - 5	5 - 7	7 - 9	9 - 11	11 - 13
Frequency	12	45	80	k	38	16

Formula used:

Mode = L + [ (f1 - f0) / (2f1 - f0 - f2) ] × h

Where:

L = lower limit of the modal class

h = size of the class interval

f1 = frequency of the modal class

f0 = frequency of the class preceding the modal class

f2 = frequency of the class succeeding the modal class

Calculation:

Given mode = 6 and 1/6 = 37/6

Since the mode (37/6 = 6.166...) lies in the class 5 - 7, the modal class is 5 - 7.

From the modal class 5 - 7:

L = 5 (lower limit of modal class)

h = 7 - 5 = 2 (class size)

f1 = 80 (frequency of modal class)

f0 = 45 (frequency of class preceding modal class, i.e., 3 - 5)

f2 = k (frequency of class succeeding modal class, i.e., 7 - 9)

Substitute these values into the mode formula:

37/6 = 5 + [ (80 - 45) / (2 × 80 - 45 - k) ] × 2

⇒ 37/6 - 5 = [ 35 / (160 - 45 - k) ] × 2

⇒ (37 - 30) / 6 = [ 35 / (115 - k) ] × 2

⇒ 7/6 = 70 / (115 - k)

⇒ 7 × (115 - k) = 6 × 70

⇒ 7 × (115 - k) = 420

⇒ 115 - k = 420 / 7

⇒ 115 - k = 60

⇒ k = 115 - 60

⇒ k = 55

∴ The correct answer is option 3.