Three batches of students have average ages 17, 19, and 23 years respectively. If their numbers are in the ratio 3 : 4 : 6, what is the overall average age of all the students?

The correct option is 20.38.

To find the overall average age of all the students across the three batches, we use the formula for the weighted average age:

$\text{Weighted Average}=\frac{\sum (\text{Number of Students in a Batch}\times \text{Average Age of that Batch})}{\text{Total Number of Students}}$

Step 1: Define the variables from the given data.

We are given the average ages of three batches of students:

Average age of 1st batch ($A_1$) = 17 years
Average age of 2nd batch ($A_2$) = 19 years
Average age of 3rd batch ($A_3$) = 23 years

The ratio of the number of students in the three batches is given as 3 : 4 : 6.
Let the number of students in the three batches be $3x$, $4x$, and $6x$ respectively, where $x$ is a common multiplier.

Step 2: Calculate the total age of students in each batch.

Total age of 1st batch = $17\times 3x=51x$ years

Total age of 2nd batch = $19\times 4x=76x$ years

Total age of 3rd batch = $23\times 6x=138x$ years

Step 3: Calculate the total age of all students combined.

$\text{Total Age}=51x+76x+138x=265x$

Step 4: Calculate the total number of students.

$\text{Total Students}=3x+4x+6x=13x$

Step 5: Compute the overall average age.

$\text{Overall Average Age}=\frac{265x}{13x}=\frac{265}{13}$

Dividing 265 by 13:

$\frac{265}{13}\approx 20.3846$

Rounding to two decimal places, we get 20.38 years.