Question

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 mark 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. $64.7$
• E. $65$

Answer 1

The correct option is D

$64.7$

Explanation for the correct option:

Step 1. Assume the number of students in X, Y, & Z:

Let the number of students in batches Y and Z be $6x$and $7x$respectively, and the number of students in batch X be $y$.
Since the average marks obtained Xand Y is $72,60$. and the average mark of batches X and Y taken together is $69$ . then

$$\begin{gathered} \Rightarrow 72y+60\times 6x=69(6x+y) \\ \Rightarrow 72y+360x=414x+69y \\ \Rightarrow 3y=54x \\ \Rightarrow y=18x \end{gathered}$$

Step2. Calculate the average score of all the three batches put together :

Therefore the required average

$$\begin{gathered} =\frac{72\times y+60\times 6x+50\times 7x}{18x+6x+7x} \\ =\frac{72\times 18x+60\times 6x+50\times 7x}{18x+6x+7x}[puty=18x] \\ =\frac{1296+360+350}{31} \\ =2006/31 \\ =64.7 \end{gathered}$$

Hence, the correct option is (D).