
Forces of interest
Tawny makes a deposit into a bank account that credits interest at a nominal interest rate of 10% per year, compounded semiannually. At the same time, Fabio deposits 1000 into a different bank account, which is credited with simple interest. At the end of 5 years, the forces of interest on the two accounts are equal, and Fabio’s account has accumulated to Z. Determine Z.
Here is what I have ...
For Fabio, the accumulation function would be 1000(1 + 5i) and the force of interest at time t would be i/(1 + 5i)
The force of interest for Tawny would be ln(1 + i) = ln(1.1) for all t
So I think you set i/(1 + 5i) = ln(1.1) , which would give i = .1821
Then Fabio's account would equal 1000(1 + 5*.1821) = 1910.5 at 5 years
But our teacher said the answer is 1953 so I'm a little off somewhere and can't figure out where. Any help would be great thanks!

You forgot to factor in that the nominal interest is compounded semiannually. If you make this conversion, you get the answer your teacher provided;
so the force of interst for twany should have been $\displaystyle ln(1.1025)=0.09758$ once you take into account the semiannual compounding.
This gives $\displaystyle i=0.19055$; $\displaystyle 1000\cdot (1+5\cdot 0.19055)=1952.749$