Paul has $90,000 to invest. his intent is to earn 12% interest on his investment. he can invest part of his money at 7% interest and part at 13% interest. how much does paul need to invest in each option to make a total 12% return on his $90,000?

If Paul wants to gain 12% interest, he would expect that the money he will have after a certain period of time is:

Future worth = 90000(1+0.12) = $100,800

Let's distribute the future worth to the different interest rates. The solution is as follows:

x + y = 90,000 --> eqn 1
100,800 = x(1+0.07) + y(1+0.13) --> eqn 2

Solving the equations simultaneously,

100,800 = x(1+0.07) + (90,000 - x)(1+0.13)
Solving for x:
x = 15,000
y = 90 000 - 15 000 = 75,000

Therefore, Paul should place $15,000 to the 7% interest and $75,000 to the 13% interest.