High School

A group of 20 people went on a hike, consisting of men, women, and children, all carrying backpacks. Men's backpacks weigh 20 kg, women's backpacks weigh 5 kg, and children's backpacks weigh 3 kg. The total weight of the backpacks is 137 kg. How many men, women, and children are in the group?

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:

  1. The total number of people is 20:
    [tex]M + W + C = 20[/tex]

  2. 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:

  1. Distribute the 3:
    [tex]20M + 5W + 60 - 3M - 3W = 137[/tex]

  2. Combine like terms:
    [tex](20M - 3M) + (5W - 3W) + 60 = 137[/tex]
    [tex]17M + 2W + 60 = 137[/tex]

  3. 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