SOLUTION: Jack inherited 250,000 pesos and invested money in SM, Meralco, and Manila Water. After a year, he got a small return of 16,200 pesos from the three investments. SM returned 6%, Me

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Question 1099909: Jack inherited 250,000 pesos and invested money in SM, Meralco, and Manila Water. After a year, he got a small return of 16,200 pesos from the three investments. SM returned 6%, Meralco returned 7%, and Manila Water returned 8%. There was 60,000 more invested in Meralco than in Manila Water. How much did he invest in SM, Meralco, and Manila Water?
Found 2 solutions by richwmiller, ikleyn:
Answer by richwmiller(17219)   (Show Source): You can put this solution on YOUR website!
.06x+.07y+.08z=16200
x+y+z=250000
0x+y-z=60000
.06,.07,.08,16200
1,1,1,250000
0,1,-1,60000
SM=150000, Meralco=80000, Manila Water=20000
Solved by pluggable solver: Using Cramer's Rule to Solve Systems with 3 variables







First let . This is the matrix formed by the coefficients of the given system of equations.


Take note that the right hand values of the system are , , and and they are highlighted here:




These values are important as they will be used to replace the columns of the matrix A.




Now let's calculate the the determinant of the matrix A to get . To save space, I'm not showing the calculations for the determinant. However, if you need help with calculating the determinant of the matrix A, check out this solver.



Notation note: denotes the determinant of the matrix A.



---------------------------------------------------------



Now replace the first column of A (that corresponds to the variable 'x') with the values that form the right hand side of the system of equations. We will denote this new matrix (since we're replacing the 'x' column so to speak).






Now compute the determinant of to get . Again, as a space saver, I didn't include the calculations of the determinant. Check out this solver to see how to find this determinant.



To find the first solution, simply divide the determinant of by the determinant of to get:



So the first solution is




---------------------------------------------------------


We'll follow the same basic idea to find the other two solutions. Let's reset by letting again (this is the coefficient matrix).




Now replace the second column of A (that corresponds to the variable 'y') with the values that form the right hand side of the system of equations. We will denote this new matrix (since we're replacing the 'y' column in a way).






Now compute the determinant of to get .



To find the second solution, divide the determinant of by the determinant of to get:



So the second solution is




---------------------------------------------------------





Let's reset again by letting which is the coefficient matrix.



Replace the third column of A (that corresponds to the variable 'z') with the values that form the right hand side of the system of equations. We will denote this new matrix






Now compute the determinant of to get .



To find the third solution, divide the determinant of by the determinant of to get:



So the third solution is




====================================================================================

Final Answer:




So the three solutions are , , and giving the ordered triple (150000, 80000, 20000)




Note: there is a lot of work that is hidden in finding the determinants. Take a look at this 3x3 Determinant Solver to see how to get each determinant.




gauss jordan
same as elimination without the variables
.06x+.07y+.08z=16200
x+y+z=250000
0x+y-z=60000
original 3*4 matrix
0.06,0.07,0.08,16200
1,1,1,250000
0,1,-1,60000
divide row 1 by 0.06
1,1.16666667,1.33333333,270000
1,1,1,250000
0,1,-1,60000
add -1*row 1 to row 2
1,1.16666667,1.33333333,270000
0,-0.16666667,-0.33333333,-20000
0,1,-1,60000
add 0*row 1 to row 3
1,1.16666667,1.33333333,270000
0,-0.16666667,-0.33333333,-20000
0,1,-1,60000
divide row 2 by -0.16666667
1,1.16666667,1.33333333,270000
0,1,2.0,120000.0
0,1,-1,60000
add -1*row 2 to row 3
1,1.16666667,1.33333333,270000
0,1,2.0,120000.0
0,0,-3.0,-60000.0
divide row 3 by -3.0
1,1.16666667,1.33333333,270000
0,1,2.0,120000.0
0,0,1,20000.0
add -2.0*row 3 to row 2
1,1.16666667,1.33333333,270000
0,1,0,80000.0
0,0,1,20000.0
add -1.33333333*row 3 to row 1
1,1.16666667,0,243333.333
0,1,0,80000.0
0,0,1,20000.0
add -1.16666667*row 2 to row 1
1,0,0,150000.0
0,1,0,80000.0
0,0,1,20000.0

1 150000.0
2 80000.0
3 20000.0
done
check
.06*150000+.07*80000+.08*20000=16200
9000+5600+1600=16200
16200=16200
Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
.
Jack inherited 250,000 pesos and invested money in SM, Meralco, and Manila Water. After a year, he got a small return of 16,200 pesos
from the three investments. SM returned 6%, Meralco returned 7%, and Manila Water returned 8%. There was 60,000 more invested
in Meralco than in Manila Water. How much did he invest in SM, Meralco, and Manila Water?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

I will solve the problem by reducing to 2 equations in 2 unknown.


Let S be the amount invested in SM, and
let W be the amount invested in Manila Water.

Then investment in Meralco is (W + 60000), according to the condition.


First equation is for the total investment

S + (W + 60000) + W = 250000,    or, equivalently,

S + 2W = 250000 - 60000 = 190000.     (1)


Second equation is for earning

0.06*S + 0.07*(W + 60000) + 0.08*W = 16200.

Multiplying by 100 and simplifying, you can get this form

5*S + 15W = 1620000 - 420000,    or, equivalently,

5S + 15W = 1200000.                     (2)


Now, to make my and your life and calculations easier, I will take off 3 zeroes in the right side of equations (1) and (2). 
Later I simply will multiply the solution by 1000.


Thus I get these two equations

 S +  2W =  190       (3)                 ( instead of (1) )    and
6S + 15W = 1200       (4)                 ( instead of (2) )


Now the solution is EASY EXERCISE.   Multiply eq(3) by 6 (both sides) and then subtract from eq(4). You will get

15W - 12W = 1200 - 6*190 = 60,   or

3W = 60  ====>  W =  = 20.

Then from (3)  S = 190 - 2W = 190 - 2*20 = 150.


Thus it was invested   20000  in  Manila Water;  
                      150000  in  SM;  and
                       20000+60000 = 80000 in Meralco.

Check.  20000 + 150000 + 80000 = 250000                (total).     ! Correct !

        0.06*150000 + 0.07*80000 + 0.08*20000 = 16200  (earning)    ! Correct !

-------------------
*** SOLVED ***
-------------------

Now compare it with the solution by the other tutor.

Should I explain more about how the problem SHOULD be solved ?


My teachers and all my life taught me to solve problems AS SIMPLE AS POSSIBLE.

        (if there are no special reasons to do it by another way).

I think that it is really the  ONLY  way   for a homo sapiens.



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