The mean marks (out of 100) of boys and girls in an examination are 70 and 73 respectively. If the mean marks of all the students in that examination is 71, find the ratio of the number of boys to the number of girls.

Given:

Mean marks of boys in the examination = 70

Mean marks of girls in the examination = 73

Let the number of boys and girls be a and b, respectively.

Mean = $\dfrac{\text{Sum of all observations }}{\text{Number of all observations }}$

Mean marks of boys = $\dfrac{\text{Sum of all marks of boys}}{\text{Number of all boys}}$

⇒ 70 = $\dfrac{\text{Sum of all marks of boys}}{a}$

⇒ Sum of all marks of boys = 70a

Mean marks of girls = $\dfrac{\text{Sum of all marks of girls}}{\text{Number of all girls}}$

⇒ 73 = $\dfrac{\text{Sum of all marks of girls}}{b}$

⇒ Sum of all marks of girls = 73b

Mean of all students = $\dfrac{\text{Sum of marks of all students}}{\text{Number of all students}}$

⇒ 71 = $\dfrac{70a + 73b}{a + b}$

⇒ 71(a + b) = 70a + 73b

⇒ 71a + 71b = 70a + 73b

⇒ 71a - 70a = 73b - 71b

⇒ a = 2b

⇒ $\dfrac{a}{b} = \dfrac{2}{1}$

Hence, the ratio of the number of boys to the number of girls is 2 : 1.