Answer: Let's denote the masses of the four men as a, b, c, and d. Then we can use the given information to set up a system of equations:
Mean mass of 4 men is 97.5kg:
(a + b + c + d) / 4 = 97.5
a + b + c + d = 390
Modal mass is 101kg:
We know that one of the men has a mass of 101kg. Let's say that man is a. Then we have two cases:
Case 1: b, c, and d each have a mass less than 101kg.
In this case, the second largest mass must be the mode. Let's say that mass is b. Then we have:
b = a - x (where x is some positive number less than 4)
c = a - y (where y is some positive number less than 4 and not equal to x)
d = a - z (where z is some positive number less than 4 and not equal to x or y)
Note that we subtract x, y, and z from a because b, c, and d have masses less than a.
Then we have:
a + (a - x) + (a - y) + (a - z) = 390
4a - (x + y + z) = 390
Case 2: b, c, and d each have a mass greater than 101kg.
In this case, the second smallest mass must be the mode. Let's say that mass is b. Then we have:
b = a + x (where x is some positive number less than 4)
c = a + y (where y is some positive number less than 4 and not equal to x)
d = a + z (where z is some positive number less than 4 and not equal to x or y)
Note that we add x, y, and z to a because b, c, and d have masses greater than a.
Then we have:
a + (a + x) + (a + y) + (a + z) = 390
4a + (x + y + z) = 390
Range is 8kg:
The range is the difference between the largest and smallest masses. Let's say that the smallest mass is e and the largest mass is f. Then we have:
f - e = 8
Now we have three equations (either from Case 1 or Case 2) and three unknowns (a, x, and y or a, x, and z) that we can solve for to find the masses of the four men. However, the system of equations is quite complicated and solving it by hand can be tedious. One way to solve it is to use a numerical method, such as Newton's method or the bisection method. Alternatively, we can use a computer algebra system to solve it.