Lesson Using systems of equations to solve problems on investment

Using systems of equations to solve problems on investment

Problem 1

Let x be the amount of money Madison invested at 3.5%, and
let y be the amount of money Madison invested at 4.5%

At the 3.5% account Madison earned 0.035x dollars per year   (0.035 = 3.5% !)
At the 4.5% account Madison earned 0.045x dollars per year   (0.045 = 4.5% !)

You are given that

0.035x + 0.045y = 1320     dollars.

You also know that

x + y = 30000              dollars.


So, you have this system of two equations

     x +      y = 30000,   (1)
0.035x + 0.045y =  1320.   (2)


To solve it, express x = 30000 - y from (1) and sibstitute it into equation (2). You will get
      
0.035*(30000-y) + 0.045y = 1320,   or

1050 - 0.035y + 0.045y = 1320,

0.01y = 1320 - 1050  ====>  0.01y = 270  ====>  y =  = 27000.

Thus you found that Madison invested $27000 at 4.5%.

Then the investment at 3.5% is 30000 - 27000 = 3000 dollars.


Check.  0.035*3000 + 0.045*27000 = 1320.  ! Correct !.

Answer. $3000 was invested at 3.5% and $27000 was invested at 4.5%.

Problem 2

From the condition, you have these two equations

    A +      B = 80000,    (1)
0.07A + 0.084B =  6300.    (2)


To solve this system, multiply the second equation by 1000 (both sides). You will get the two equations in this form

  A +   B =   80000,       (3)
70A + 84B = 6300000.       (4)


Next, multiply equation (3) by 70 (both sides). You will get the two equations in this form

70A + 70B = 5600000,       (5)
70A + 84B = 6300000.       (6)


Subtract from (5) from equation (6) to eliminate A:

70A + 84B - 70A - 70B = 6300000-5600000,

14B = 700000 ====>  B =  = 50000.

Thus the amount of money for certificate of deposit is $50000.

Then the amount of money for saving account is 80000 - 50000 = 30000 dollars.


Check.  0.07*30000 + 0.084*50000 = 6300 dollars.   ! Correct !

Answer. $30000 for the saving account at 7%  and  $50000 for the certificate of deposit at 8.4%.

Problem 3

From the condition, you have THIS system of 2 equations in 2 unknowns

    x +     y = 15074            (1)    (counting investments)
0.14x - 0.05y =  1694.07         (2)    (the difference between earnings)


To make your calculations easier, multiply eq(2) by 100 (both sides). The modified system is

    x +     y = 15074            (3)   
  14x -    5y = 169407           (4)  


Now multiply eq(3) by 5. The modified system takes the form

   5x +    5y = 5*15074          (5)
  14x -    5y = 169407           (4)  


Next add the equations (5) and (6) (both sides).  The terms with 5y will cancel each other, and you will get a single equation for x:


  19x = 5*15074 + 169407 = 244777.

    x =  = 12883.


Answer.  It was invested $12883 at %14.  The rest 15074 - 12883 = 2191 was invested at  5%.


Check.  0.14*12883 - 0.05*2191 = 1694.07.   ! Correct !

Problem 4

I will solve the problem by reducing to two equations in two 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 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 an 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 !

Problem 5

Let x be the amount which was invested and gained 6.7%.
Let y be the amount which was invested and lost   7.8%.


Then you can write two equations.
First equation is OBVIOUS: it is

    x + y = 10000  dollars.              (1)


It says that the total of the two accounts is $10,000, exactly as the problem states.


Second equation is about the net loss.
It is also obvious:

    0.078y - 0.067x = 562.50  dollars.   (2)


Indeed, 0.078y is the lost amount, while 0.067x is the gained amount.

We take 0.078y with the sign "+", since in equation (2) we consider the lost amount as positive.

You may consider the loss as negative value, but then you should be consistent and write all the terms of equation (2) with opposite signs.


So, you either write the "loss" equation in the form (2),
or you change all the terms in this equation to opposite.

It is your choice to write (2) as is or with opposite signs - the meaning of the equation (2) will be the same.


So, for what follows, let's write these two equations as

      x +      y = 10000      (1)
-0.067x + 0.078y = 562.50     (2)


So, we consider the loss in (2) with the sign "+". As soon as you wrote these equations, the setup is completed.


To solve the system, let's multiply equation (1) by 0.067  (all the terms).
Keep equation (2) as is.  You will get then

 0.067x + 0.067y =  670.00    (1')
-0.067x + 0.078y =  562.50    (2')


Now add equations (1') and (2').  The terms 0.067x and -0.067x will kill each other,
and you will get a single equation with unknown y

          0.067y + 0.078y = 670 + 562.50,

    or        0.145y      =   1232.5,  which gives  y = 1232.5/0.145 = 8500.


Then from equation (1),  x = 10000 - 8500 = 1500.


So, the invested amount to the account which gained 6.7% was 1500 dollars.

    the invested amount to the account which lost   7.8% was 8500 dollars.


CHECK.  -0.067*1500 + 0.078*8500 = (I use my calculator) = 562.5  dollars, the net loss.   ! correct !