math word problem help

Question

Mrs. Stuart invested part of $8400 at 7% and the rest at 7.5%. If her total annual interest from these investments was $616, how much did she invest at 7%?

Answer 1

Simple interest formula is

    I = Prt

    Where I = Interest earned
    P = principal amount invested
    r = rate
    t = time (in years)

 

Let x = amount invested at 7%

Let y = amount invested at 7.5%  

 

FOR THE 7% INVESTMENT:

 

   I = Prt
     = X(.07)(1)

     = .07X

 

FOR THE 7.5% INVESTMENT:

    I = Prt
      = y(.075)(1)

      = .075y

Since the total interest earned was 616, we can set the individual amounts above to that value, and constuct a system of equations

 

     .07x + .075y = 616       (total interest earned) ... (Equation 1)

         x  +  y   = 8400         (total amount invested) ... (Equation 2)

 

Multiply equation 2 by -.07 ... then add the 2 equations to eliminate x 

 

     .07x + .075y =  616

    -.07x - 0.07y  = -588  ... (.07•8400 = 588)  

    ------------------------------

                .005y = 28

                       y = 28/.005

                       y = 5600

 

Since the total investment was 8400,

 

     8400 - 5600 = 2800

 

Thus

   $2800 was invested at 7% 

   $5600 was invested at 7.5%

 

CHECK:

 

   .07(2800)   = 196

   .075(5600) = 420

 

    196 + 420 = 616 ... total interested earned                     

 

     

Answer 2

So first lets write down in words what we are trying to solve:

 

Mrs. Stuart gets $616 in annual interest from xdollars times 7% +(8400-x)dollars times 7.5% (8400-x+x=8400)

 

now we can put this into a numeric equation:

 

616 = x(.07) + (8400-x)(.075)         since 7 and 7.5% convert to .07 and .075

616 = .07x +630 - .075x                 all i did here was multiply everthing out

616 = 630 -.005x                             combine .07x and -.075x to get -.05x

-14 = -.005x                                    subtract the 630 from both sides

2800 = x                                         solve for x and get the answer

 

Hope this helps!

Case