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

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,
$$\eqalign{ & \Rightarrow 72y + 60 \times 6x = 69\left( {6x + y} \right) \cr & \Rightarrow 72y + 360x = 414x + 69y \cr & \Rightarrow 3y = 54x \cr & \Rightarrow y = 18x \cr} $$
? Required average

$$\eqalign{ & = \frac{{72 \times 18x + 60 \times 6x + 50 \times 7x}}{{18x + 6x + 7x}} \cr & = \frac{{1296 + 360 + 350}}{{31}} \cr & = \frac{{2006}}{{31}} \cr & = 64.7 \cr} $$