### A coaching institute has students in 3 batches – X, Y and Z. In a certain examination, the average marks obtained by these batches are 72, 60 and 50 respectively. The average marks of batches X and Y taken together is 69. If the ratio of the number of students in batches Y and Z is 6 : 7. What is the average score of all the three batches put together?

A. 59

B. 61.6

C. 63.5

D. 64.7

Let the number of students in batches Y and Z be 6x and 7x respectively, and the number of students in batch X be y. Then, $\begin{array}{l}\Rightarrow72y+60\times6x=69(6x+y)\\ \Rightarrow72y+360x=414x+69y\\ \Rightarrow3y=54x\\ \Rightarrow y=18x\\ \text{Therefore,}\\ \text{Required average}\\ =\Large\frac{72 \times 18 x+60 \times 6 x+50 \times 7 x}{18 x+6 x+7 x} \\ =\Large\frac{1296+360+350}{31} \\ =\Large\frac{2006}{31} \\ =64.7 \end{array}$