Help Solving for compound interest using a system of equations with 2 variables

Question

Today, a couple invested $92,000 separately They figure that in 10 years, they will have a combined total interest of $168,132.75 . Person X has an interest rate of 10% and person Y has and interest rate of 12% compounded annually. How much did each person invest?

Answer 1

Interest=fv=p(1+int/c)^(nt)

future value

premium

int%

compound

years

times/yr

Answer 2

Let x = the amount person X invested at 10% per year, using an assumed annual compounding.

Let y = the amount person Y invested at 12% per year, using annual compounding.

When A dollars earns interest at the rate of i per compounding interval, the investment grows in n compounding units of time to a value of F given by:

F = A*(1+i)ⁿ

and the interest earned I is computed as

I = F - A = A * [ 1 - (1+i)ⁿ]

Let I_x and I_y be the interest earned by X and Y. Thus

I_x = x * [(1+i)ⁿ - 1] (where i is the growth rate of x's investment)

I_y = y * [(1+j)ⁿ - 1] (where j is the growth rate of y's investment)

We are given the following info:

92000 = x + y

168,132.75 = I_x + I_y

Now can proceed as follows:

I_x = x * [ (1 + 0.10)¹⁰ - 1 ] = 1.5937425 x

I_y = y * [ (1 + 0.12)¹⁰ - 1 ] = 2.1058482 y

168,132.75 = I_x + I_y

y = 9200 - x

I_y = 2.1058482 * (92000 - x) = 193738.034 - 2.1058482 x

168,132.75 = I_x + I_y = 1.5937425 x + ( 193738.034 - 2.1058482 x )

-25605.284 = -0.5121057 x

x = 49,996,60

y = 92000 - x = 42,003.40

if the compounding assumption made above is incorrect, then at least this gives an idea of how to proceed.