In a group of 180 persons composed of men, women, and children, there are twice as many men as women, and three times as many women as children. How many are there of each?

m+w+c = 180

m = 2w
w = 3c
A little substitution gets the job done now.

To solve this problem, we need to set up a system of equations based on the given information. Let's use variables to represent the number of men, women, and children.

Let's say the number of men is "m," the number of women is "w," and the number of children is "c."

We are given the following information:

1) "there are twice as many men as women:" This can be written as m = 2w.

2) "three times as many women as children:" This can be written as w = 3c.

We also know that the total number of people in the group is 180, so we can write the equation:

m + w + c = 180.

Now, let's solve the system of equations:

Substitute w from the second equation into the first equation:

m = 2(3c) = 6c.

Plug these values into the third equation:

6c + 3c + c = 180.

Combine like terms:

10c = 180.

Divide both sides by 10:

c = 18.

Now, substitute c = 18 back into the second equation:

w = 3c = 3(18) = 54.

Finally, substitute c = 18 and w = 54 into the first equation:

m = 2w = 2(54) = 108.

Therefore, there are 108 men, 54 women, and 18 children in the group.