The mean age of a group of 40 students is 17.45 years. Find the missing frequencies.

Age (in years)	Number of students
15	3
16	?
17	9
18	11
19	?
20	3

Let missing frequencies be:

Frequency at age 16 = a

Frequency at age 19 = b

By formula,

Mean = $\dfrac{\sum f$ixi}{\sum f_i}

where

x_i = Age (in years)

f_i = Number of students

∑f_i = 3 + a + 9 + 11 + b + 3 = 40

⇒ a + b + 26 = 40

⇒ a + b = 14

⇒ b = 14 - a ……..(1)

Mean = $\dfrac{\sum f$ixi}{\sum f_i}

⇒ 17.45 = $\dfrac{45 + 16a + 153 + 198 + 19b + 60}{40}$

⇒ 17.45 × 40 = 456 + 16a + 19b

⇒ 698 = 456 + 16a + 19b

⇒ 242 = 16a + 19b ……….(2)

Substitute equation 1 in equation 2

⇒ 16a + 19(14 − a) = 242

⇒ 16a + 266 − 19a = 242

⇒ −3a + 266 = 242

⇒ −3a = −24

⇒ a = 8.

⇒ Put a = 8 in equation 1

⇒ b = 14 - 8

⇒ b = 6.

Hence, the missing frequencies are 8 & 6.