Let d = Danesh's starting number, m = Meg's starting number, and c = Chris' starting number. We use letters to represent their starting numbers since we do not know their values yet. These are called unknowns/variables in an equation.
For example, x + 4 = 5. x = 1. We have 'solved' the equation and found the value of the unknown/variable.
If Danesh started with d, then took away 4 from it, and finished with 12, we can write this as the expression:
d - 4 = 12 (he started with d, took away 4, which made it equal to 12).
This is a linear equation and can be solved by adding 4 to both sides:
d - 4 + 4 = 12 + 4
d = 16
Danesh started with 16, subtracted 4, and finished with 12.
Meg started with m and added 8 to it. She ended up with 12. This translates to:
m + 8 = 12
We want to get rid of the 8 and leave m by itself so we know it's value. To do this, we have to minus 8 from both sides of the equation. Since one side 'equals' the other side in any equation, what we do to one side has to be done to the other side so that both sides remain equal.
m + 8 = 12
m + 8 - 8 = 12 - 8
m = 4
Meg started with 4, added 8, and ended up with 12.
Finally, to find Chris' starting number, we use the same technique's as above. You should have noticed the rough outline of what we should do by now. First, we construct an equation relating the variables and the outcome. In this case, the outcome of the mathematical operations is 12. Since Chris 'halved' his starting number to get 12, we simply right c/2 = 12. Dividing by two is the same as multiplying by 1/2, because c, is technically c/1. Any integer (whole number) is in fact a fraction, but we just right it without the fraction to make life easy for our selves. For example, 5, is actually 5/1. (5/1=5) but we just write 5. In this case, c is actually c/1, and halving is multiplying by 1/2, so c/1 x 1/2 = c / 2 = 12 (we are told they all finish with 12). To solve this, we multiply both side by 2. Doing this gets rid of the 2 on the left side, because c/2 x 2/1 = 2c/2 = c. And the right side ends up as 12 x 2 = 24. Remember, an equation needs both sides to remain equal, what we do to one side has to be done to the other, otherwise the equations will not remain equal most of the time. (Try and find something that leaves both sides the same when it is applied only to one side! Experiment :D)
We conclude from the working out above, that the numbers the three people started with are:
d = 16
m = 4
c = 24
A good idea when you have finished solving an equation is to substitute your value back into the equation, for example for Chris' equation, we got 24, so to check if our answer is right, we put it in place of c (since c = 24):
c/2 = 12
substituting 24, we get:
24/2 = 12
12 = 12
Out answer is correct.
When faced with a word problem, try to first of all convert the words into mathematics, and then solve the problem that way.