Question 1207197: Hi
Four friends Aisha Ben Cathy and Daniel shared lunch equally. Aisha paid 3/5 of the total amount Ben and Cathy paid. Cathy paid $10 more than Ben. Daniel repaid $24.00 to Aisha and some money to Ben and Cathy.
How much did Daniel repay to Ben and Cathy
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52776)   (Show Source):
You can put this solution on YOUR website! .
Four friends, Aisha, Ben, Cathy, and Daniel shared lunch equally.
Aisha paid 3/5 of the total amount Ben and Cathy paid.
Cathy paid $10 more than Ben.
Daniel repaid $24.00 to Aisha and some money to Ben and Cathy.
How much did Daniel repay to Ben and Cathy
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It is clear that the major task in solution to this problem is to make setup.
I will make setup in several steps.
At step 1, I introduce variables in general form.
We have a network (like a circuit). You can imagine it as a graph with vertices (nodes) A, B, C and D and edges,
connecting the nodes. The nodes represent the people; the edges represent cash flows between people.
Aisha (node A) pays A dollars to the restaurant; Ben (node B) pays B dollars to the restaurant;
Cathy (node C) pays C dollars to the restaurant; Daniel (node D) pays D dollars to the restaurant.
Edge Ab represents cash flow from A to B; edge Ac represents cash flow from A to C;
. . . and so on . . . edge Dc represents cash flow from D to C.
There is no cash flow from A to A; no cash flow from B to B; . . . no cash flow from D to D.
So, there are 3*4 = 12 edges Ab, Ac, . . . , Dc.
At step 2, I will give the values to some variables, based on given data.
A, Ab=0, Ac=0, Ad=0
B, Ba=0, Bc=0, Bd=0
C, Ca=0, Cb=0, Cd=0
D=0, Da=24, Db, Dc
As you see from this table, many variables are 0 (zero). Non-zero values are A, B, C, Da=24, Db and Dc.
My next task is to write equations for non-zero unknowns.
At step 3, I will write equations for non-zero variables
A = 0.6B + 0.6C (1) A - 24 = x (4) part for A
B - Db = x (5) part for B
C = B + 10 (2) C - Dc = x (6) part for C
Da = 24 (3) 24 + Db + Dc = x (7) part for D
Equations follow to given information LITERALLY, so I will not explain their origin and meaning.
Here x is the amount representing the PART of everyone.
According to the problem, this part is the same for all 4 participants.
As you see, there are 7 unknown and 7 independent equations - so, there is a hope that the problem is solvable to the end.
These equations, written in one column, are presented below one more time.
Equation (3) is excluded, since it is just used in (7).
A = 0.6B + 0.6C (8)
A - 24 = x (9)
B - Db = x (10)
C = B + 10 (11)
C - Dc = x (12)
24 + Db + Dc = x (13)
So, there are 6 equations for 6 unknowns.
The unknown A can be easily excluded together with equation (8), by replacing A = 0.6B + 0.6C in equation (9).
Thus we arrive to 5 equations in 5 unknowns
0.6B + 0.6C - 24 = x (14)
B - Db = x (15)
C = B + 10 (16)
C - Dc = x (17)
24 + Db + Dc = x (18)
Now I rewrite these equations in canonical form of a system of linear equations
0.6B + 0.6C - x = 24 (19)
B - dB - x = 0 (20)
B - C = -10 (21)
C - dC - x = 0 (22)
dB + dC - x = -24 (23)
At this point, my setup is complete.
Probably, this system can be solved manually, step by step.
But I decided do not spend my time for technical exercises and used an online solver
http://www.math.odu.edu/~bogacki/cgi-bin/lat.cgi
Below are the numbers from this online solver
B = 55
C = 65
Db = 7
Dc = 17
x = 48
The ANSWER to the problem's question is: Daniel repaid $7 to Ben (Db=7) and $17 to Cathy (Dc=17).
Solved.
You may check, that the obtained numbers satisfy the problem's conditions and equations (19) - (23).
This check is easy straightforward task, so I leave it to you.
This problem and this my solution seems very educative to me,
since they represent new type of thinking, which I never saw before in school Math problems.
Answer by greenestamps(13198)  (Show Source):
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