Read the information and answer the following question given below.

The information is about the staffs (lab assistant and professor) in three colleges.
For A – total lab assistant and total professor is 82.
For B – total lab assistant is 4 more than total lab assistant in C.
For C – total lab assistant is 6 less than total professor
Note – total professor in A is 44 less than total lab assistant in B.
Total staffs in C is 24 more than that in A and total professor in al the college is 102.

Find the ratio of total professors in B to total lab assistant in A.

The correct option is 1:2.

Let us define the variables for the staff members in colleges A, B, and C:
Let $L_A$, $L_B$, and $L_C$ represent the number of lab assistants in colleges A, B, and C respectively.
Let $P_A$, $P_B$, and $P_C$ represent the number of professors in colleges A, B, and C respectively.

Step 1: Find the total number of staff in College A and College C
We are given that the total number of staff (lab assistants + professors) in college A is 82:
$L_A+P_A=82$
The total staff in C is 24 more than that in A:
$\text{Total staff in C}=L_C+P_C=82+24=106$

Step 2: Find the staff details for College C
For College C, the number of lab assistants is 6 less than the number of professors:
$L_C=P_C-6$
Substitute this into the total staff equation for College C:
$(P_C-6)+P_C=106$
$2P_C-6=106$
$2P_C=112$
$P_C=56$
Now, find the number of lab assistants in College C:
$L_C=56-6=50$

Step 3: Find the number of lab assistants in College B
The number of lab assistants in College B is 4 more than in C:
$L_B=L_C+4=50+4=54$

Step 4: Find the staff details for College A
The number of professors in College A is 44 less than the number of lab assistants in B:
$P_A=L_B-44=54-44=10$
Using the total staff of College A to find its lab assistants:
$L_A+10=82$
$L_A=72$

Step 5: Find the number of professors in College B
The total number of professors in all three colleges is 102:
$P_A+P_B+P_C=102$
Substitute the known values:
$10+P_B+56=102$
$66+P_B=102$
$P_B=36$

Step 6: Calculate the required ratio
We need to find the ratio of total professors in B to total lab assistants in A:
$\text{Ratio}=\frac{P_B}{L_A}=\frac{36}{72}=\frac{1}{2}$
Thus, the required ratio is 1:2.