On combining two groups of students having 30 and 40 average marks respectively in an exam, the resultant group has an average score of 34. Find the ratio of the number of students in the first group to the number of students in the second group.

On combining two groups of students having 30 and 40 average marks respectively in an exam, the resultant group has an average score of 34. Find the ratio of the number of students in the first group to the number of students in the second group.
1). 2 : 1
2). 3 : 2
3). 3 : 1
4). 4 : 3

Let the number of students in the first group be $x$, and the number of students in the second group be $y$.

The total marks for the first group is $30x$, since the average marks of the first group is 30.

The total marks for the second group is $40y$, since the average marks of the second group is 40.

When both groups are combined, the total number of students is $x + y$, and the total marks is $30x + 40y$.

The average score of the combined group is 34, so:

$ \frac{30x + 40y}{x + y} = 34 $

Now, multiply both sides of the equation by $x + y$:

$ 30x + 40y = 34(x + y) $

Expand both sides:

$ 30x + 40y = 34x + 34y $

Rearrange the equation:

$ 30x - 34x = 34y - 40y $

Simplify:

$ -4x = -6y $

Divide by $-2$:

$ 2x = 3y $

Thus, the ratio of the number of students in the first group to the second group is:

$ \frac{x}{y} = \frac{3}{2} $

So, the correct answer is 3 : 2.

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