Lesson Two very different approaches to one word problem

Two very different approaches to one word problem

Problem 1

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).

Let x dollars be the amount Ben paid.
Then (x+10) dollars is the amount Cathy paid.

The amount Alisha paid was 3/5 of the total that Ben and Cathy paid:

    


So the total amount spent was

    


Daniel repaid $24 to Alisha, so Alisha's equal share was .


Since Alisha's share was , the total spent was

    .


The two expressions for the total amount spent must be equal:

    

    

    


Results so far....

Ben    spent x = $55;
Cathy  spent x+10 = $65;
Alisha spent (6/5)x+6 = $66+$6 = $72.


The total amount spent was $55+$65+$72 = $192.
So, the equal shares were $192/4 = $48.

Daniel repaid $24 to Alisha, making Alisha's share correct: $72-$24 = $48
Since the equal shares were $48 each, and since Daniel repaid $24 to Alisha, the total amount he repaid 
to Ben and Cathy together was $48-$24 = $24.


If we want to find the amounts repaid to Ben and Cathy separately, then we need to divide the $24 Daniel repaid 
to Ben and Cathy into two parts with Cathy's part being $10 more than Ben's.  That leads to

ANSWER.  Daniel repaid $17 to Cathy and $7 to Ben.