With mutual funds, you buy shares. The annualized percentage change in the share price between any 2 dates is what they quote as the fund's return. It would be the same return you would calculate if you didn't have money going in and out.
In your example, let's say the share price is $10 on day one. Your $100 buys you 10 shares. Next month, the share price is $20 so you get 5 shares. At that point, you have 15 shares worth $20 each. At the end of 12 months, you've got x shares at a final price of y. In your example, x times y equals $1500. The mutual fund company would quote their own performance as 100 times (y-10)/10. Your own personal return is calculated by solving for i such that 100*[(1+i)+(1+i)^(11/12)+(1+i)^(10/12)+ ... +(1+i)^(1/12)]=1500 which I think works out to 47.6%.
