Max was on vacation twice as long is jared and half as long as wesley. the boys were on vacation a total of three weeks. how many days with each boy on vacation?

To begin with, the question is asking for the answer in days, so let's change 3 weeks to days. There are 7 days in 1 week; 3 weeks times 7 days = 21 days.
21 total days of vacation.

Max + Jared + Wesley = 21 days
(Let's use the first letter of their name to represent their vacation time)

M + J + W = 21 [This is the equation we'll be coming back to]

Now, we can use the clues given in the question to have an equation for each variable/boy. 

Max was on vacation twice as long as Jared. We can interpret this as  M= 2J Max was on vacation only half as long as Wesley. We can interpret this as M = (1/2)W.

So far we have these three equations: M + J + W = 21M = 2J M = (1/2)W

To have an equation for each individual boy, we must rearrange the last two equations in the list.

First, M = 2J.
Divide both sides by 2
M/2 = 2J/2
(1/2)M = J

Second, M = (1/2)W
Multiply both sides by 2
2M = W

New equations:
M + J + W = 21 [From the old list]
J = (1/2)W
W = 2M

Now we can substitute the last two equations into the first one. 
M + J + W = 21
M + (1/2)W + 2M = 21
[Combine Like Terms]
(7/2)M = 21 

Then, solve for M (Max's vacation days):
Multiply both sides by 2/7

M = 6

Now we know Max was on vacation for 6 days.

If Max was on vacation twice as long as Jared, that means Jared was on vacation HALF as long as Max.

So...
J = (1/2)M 
J = (1/2) * 6
J = 3
Jared was on vacation for 3 days 

Wesley was on vacation twice as long as Max so...  W = 2M 
W= 2*6
W = 12
Wesley was on vacation for 12 days. 

Let's double check our
M + J + W = 21 days6 + 3 + 12 = 21   
The numbers work out so the math is correct. Hope this helps and makes sense!

Subtracting is the opposite of adding