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.