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.

The ratio of total staff is in college D to lab assistant in A is 3:4. Total lab assistant in D is 20 less than that in C. Find the professor in D.

Correct Answer: 24

Let us break down the given information step-by-step to find the number of staff members in each college.

Let $L_A$, $L_B$, $L_C$ represent the number of lab assistants in colleges A, B, and C respectively, and $P_A$, $P_B$, $P_C$ represent the number of professors in colleges A, B, and C respectively.

Step 1: Analyze College A and College C
We are given that the total staff in college A is 82:
$L_A+P_A=82$

We are also given that the total staff in C is 24 more than that in A:
$L_C+P_C=82+24=106$

For college C, the total number of lab assistants is 6 less than the number of professors:
$L_C=P_C-6$

Substitute this value into the total staff equation for C:
$(P_C-6)+P_C=106$
$2P_C=112$
$P_C=56$

Now, calculate the lab assistants in C:
$L_C=56-6=50$

Step 2: Find staff details for College B and College A
The number of lab assistants in B is 4 more than in C:
$L_B=L_C+4=50+4=54$

The number of professors in A is 44 less than the lab assistants in B:
$P_A=L_B-44=54-44=10$

Using the total staff of A, we can find the number of lab assistants in A:
$L_A=82-P_A=82-10=72$

Step 3: Solve for College D
The ratio of total staff in college D to lab assistants in A is 3:4.
Let the total staff in D be $T_D$:
$\frac{T_D}{L_A}=\frac{3}{4}$
$T_D=\frac{3}{4}\times 72=54$

The number of lab assistants in D is 20 less than that in C:
$L_D=L_C-20=50-20=30$

Finally, find the number of professors in D ($P_D$):
$P_D=T_D-L_D$
$P_D=54-30=24$

Thus, the number of professors in D is 24.