Answer :
The individual masses of the four men are 90 kg, 98 kg, and two men with 101 kg each, calculated using the given mean, modal, and range values.
The question involves finding the mass of each of the four men given the mean mass, modal mass, and range of their masses. The mean mass of the four men is given as 97.5 kg, which is the average of their individual masses.
The modal mass is the most frequently occurring mass among the four men and is given as 101 kg. The range is the difference between the highest and lowest mass, which is 8 kg.
To find the individual masses, we can first multiply the mean mass by the number of men to get the total mass:
97.5 kg x 4 = 390 kg.
Since the modal mass is 101 kg and it occurs more than once, let's assume two men have this mass.
Now, the total mass of these two men would be 101 kg x 2 = 202 kg.
Subtracting this from the total mass, we get 390 kg - 202 kg = 188 kg remaining for the other two men.
Now, with a range of 8 kg, if one man has the least mass, the other has the greatest. Let's denote the smallest mass as x.
Therefore, x + (x + 8 kg) = 188 kg.
Solving this equation: 2x + 8 = 188
= 2x = 180,
x = 90 kg.
The masses of the four men are: 90 kg, 98 kg (90 kg + 8 kg), and 101 kg for the two men with the modal mass.
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.
