Answer:
A = 181 marbles
B = 249 marbles
C = 86 marbles
D = 566 marbles
Explanation:
Let A, B and C represent the number of marbles Ali,(A), Billy(B) and Charles(C) have, respectively.
We are told that "Ali has 68 marbles fewer than Billy." This can be written as A = B - 68
We also learn that "Dan has 50 more marbles more than the total number that Ali, Billy and Charles have." We can write this as D = A + B + C + 50. [Dan has not lost his marbles].
Finally, we are told that A + B + C + D = 1082
We have 3 equations and 4 unknowns. We should be able to find answers using substitution. The idea is to rearrange some of the equations to isolate just one of the 4 variables (A, B, D, or D) and then use these new definitions of A, B, C, and D in the other equations. We can slowly eliminate some of the variables until we solve for just one. That can be used in further substitutions until four answers can be found.
Where to start? Let write the four equations from above:
(1) A = B - 68
(2) D = A + B + C + 50
(3) A + B + C + D = 1082
Equation (1) has already isolated A. It is B - 68. Let's use that definition of A in equation (2):
D = A + B + C + 50
D = (B-68) + B + C + 50
D = 2B + C - 18
Now lets use these definitions of A and C in the third equation:
A + B + C + D = 1082
(B - 68) + B + C + (2B - C - 18) = 1082
4B + (-68 -18) = 1082
4B + (-68 -18) = 1082 [Note that the C is elimated]
4B - 86 = 1082
4B = 996
B = 249 marbles
Now use B - 249 in equation 1:
A = B - 68
A = 249 - 68
A = 181 marbles
Since we know A and B, lets next find a way to eliminate either C or D. We can see that equations 2 and 3 can be rewritten with the actual values of A and B:
(2) D = A + B + C + 50
D = 181 + 249 + C + 50
D = 181 + 249 + C + 50
D = C + 480
(3) A + B + C + D = 1082
181 + 249 + C + D = 1082
C + D = 652
Now lets use the definition of D (from above) in the second equation:
C + D = 652
C + (C + 480) = 652
2C = 172
C = 86 marbles
We now know A, B, and D. Use equation (3) to find D:
(3) A + B + C + D = 108
(181) +(249) + (86) + D = 1082
D = 566 marbles
We now have:
A = 181 marbles
B = 249 marbles
C = 86 marbles
D = 566 marbles
CHECK:
1. Does Ali have 68 marbles fewer than Billy?
(1) A = B - 68?
181 = 249 - 68?
181 = 181 YES
2. Does Dan have 50 marbles more than the total number that Ali, Billy and Charles have?
D = A + B + C + 50
566 = 181 + 249 + 86 + 50?
566 = 566 YES
3. Do the children have a total of 1082 marbles?
A + B + C + D = 1082
181 +249 + 86 + 566 = 1082?
1082 = 1082 YES
The solutions above are valid:
A = 181 marbles
B = 249 marbles
C = 86 marbles
D = 566 marbles
[Note that there is no one approach to how to go about eliminating variables through substitution. Simply imagine the easiest way to isolate and substitute to get started.]