SOA FMFinancial Mathematics

A fund begins the year with 100. A deposit of 50 is made at $t=0.5$, immediately after which the fund is valued at 170. The fund value just before the deposit (at $t=0.5$) was 120, and the year-end value is 180. Calculate the time-weighted annual return.

Using the same fund as above (begin 100, deposit 50 at $t=0.5$, end value 180), calculate the (exact) money-weighted annual return $i$ solving $100(1+i)+50(1+i)^{0.5}=180$.