Answer :
To solve the problem, we need to use a system of equations based on the number of men (M), women (W), and children (C) in the group. We know from the problem:
The total number of people is 20:
[tex]M + W + C = 20[/tex]The total weight of the backpacks is 137 kg:
[tex]20M + 5W + 3C = 137[/tex]
Now we have a system of two equations with three unknowns. To solve this system, we can express C in terms of M and W from the first equation:
[tex]C = 20 - M - W[/tex]
Substituting this expression for C into the second equation gives us:
[tex]20M + 5W + 3(20 - M - W) = 137[/tex]
Now, let's simplify this equation:
Distribute the 3:
[tex]20M + 5W + 60 - 3M - 3W = 137[/tex]Combine like terms:
[tex](20M - 3M) + (5W - 3W) + 60 = 137[/tex]
[tex]17M + 2W + 60 = 137[/tex]Subtract 60 from both sides:
[tex]17M + 2W = 77[/tex]
Now, we have a second equation:
[tex]17M + 2W = 77[/tex]
We can express W in terms of M:
[tex]2W = 77 - 17M[/tex]
[tex]W = \frac{77 - 17M}{2}[/tex]
Now we will consider possible integer values for M (since M, W, and C must be whole numbers) and find corresponding values for W. We also need to ensure that W is a non-negative integer.
Calculating values for M:
If [tex]M = 0[/tex]:
[tex]W = \frac{77 - 17(0)}{2} = \frac{77}{2} = 38.5[/tex] (not an integer)If [tex]M = 1[/tex]:
[tex]W = \frac{77 - 17(1)}{2} = \frac{60}{2} = 30[/tex] (not possible since M + W + C = 20)If [tex]M = 2[/tex]:
[tex]W = \frac{77 - 17(2)}{2} = \frac{43}{2} = 21.5[/tex] (not an integer)If [tex]M = 3[/tex]:
[tex]W = \frac{77 - 17(3)}{2} = \frac{26}{2} = 13[/tex]
Then: [tex]C = 20 - 3 - 13 = 4[/tex]
Thus, one solution is M = 3, W = 13, C = 4.
If [tex]M = 4[/tex]:
[tex]W = \frac{77 - 17(4)}{2} = \frac{9}{2} = 4.5[/tex] (not an integer)If [tex]M = 5[/tex]:
[tex]W = \frac{77 - 17(5)}{2} = \frac{-8}{2} = -4[/tex] (not possible)
After checking all whole numbers, we find that:
- [tex]M = 3[/tex]
- [tex]W = 13[/tex]
- [tex]C = 4[/tex]
Finally, the number of men, women, and children in this group is:
- Men: 3
- Women: 13
- Children: 4
