Math Word Problem

Question

Terry invests her inheritance of $350,000, part at 7% per annum and the remainder at 10% per annum. After one year, the total interest was $32,600. How much was invested at each rate?

 

For the question, intro the variables, set up a system of equations and show full solution 

Answer 1

Hi Amahl,

 

Let's begin by representing our two unknowns with two letters:

    x = amount invested at 7%

    y = amount invested at 10%

 

Next, we'll write our two equations which we can use to solve for our two unknowns:

    Equation #1 (based upon amounts invested):

        x + y = 350,000

    Equation #2 (based upon interest accrued):

        (0.07)(x) + (0.10)(y) = 32,600

    

Next, we'll rewrite Equation #1 to express y in terms of x and then use that "x" expression for y in

Equation #2 to solve for x:

    x + y = 350,000 (Equation # 1)

          y = 350,000 - x 

    (0.07)(x) + (0.10)(y) = 32,600 (Equation #2)

    (0.07)(x) + (0.10)(350,000 - x) = 32,600

    0.07x + 35,000 - 0.10x = 32,600

    0.07x - 0.10x = 32,600 - 35,000

    -0.03x = -2400

    (-1)(-0.03x) = (-1)(-2400)

    0.03x = 2400

          x = 2400/0.03

          x = $80,000 (amount invested at 7%) 

 

Next, we'll plug our value for x into Equation #2 and solve for y:

    (0.07)(x) + (0.10)(y) = 32,600

    (0.07)(80,000) + (0.10)(y) = 32,600

    5600 + 0.10y = 32,600

    0.10y = 32,600 - 5600

    0.10y = 27,000

          y = 27,000/0.10

          y = $270,000 (amount invested at 10%)

 

Finally, we can verify our answers by plugging them into Equation #1 and check to see that both sides are equal:

    x + y = 350,000 (Equation #1)

    80,000 + 270,000 = 350,000

                 350,000 = 350,000 (both sides are equal)

 

Since both sides are equal, we are confident that our answers are correct.

 

The key to this problem was realizing that we could express one unknown in terms of the other, which allowed us to solve for one of the unknowns and then plug in its value to solve for the other unknown.

 

Thanks for submitting this problem and glad to help.

 

God bless, Jordan.