The total number of students in three sections A, B and C of a class in a school is 340. The number of students in sections A and B are in the ratio 3:5 and those in sections B and C are in the ratio 3:2. What is the mean proportional between the number of students in section A and the number of students in section C?

Given the ratios:
A:B=3:5
B:C=3:2
To find the combined ratio A:B:C, make the term for B common by multiplying the first ratio by 3 and the second ratio by 5:

A:B=3×3:5×3=9:15
B:C=3×5:2×5=15:10
Thus, A:B:C=9:15:10.

Let the number of students in sections A, B, and C be 9k, 15k, and 10k respectively.
The total number of students is 340:

9k+15k+10k=340
34k=340⟹k=10
Therefore, the number of students in each section is:

Number of students in section A = 9×10=90

Number of students in section C = 10×10=100

The mean proportional between A and C is given by
A×C

​
:

Mean Proportional=
90×100

​
=
9000

​
=30
10

​
=
9000

​

Let us simplify and check the options.

3010

​
 doesn’t match directly. Wait, let’s re-verify option D content.
If we write 30
10

​
as a single square root, it is
900×10

​
=
9000

​
.
Looking at the question text's typo options, "3010 \sqrt" inside the question block indicates 30
10

​
.

Mean Proportional=30
10

​

This corresponds exactly to Option D.