[NumberSix]

Couple thoughts.

If "got asked from a reputable university" means the client is a university, in USA at least universities and non-profit institutions tend to pay less than for-profit.

Generally, I avoid firm fixed price contracts and quote an hourly rate and a non-binding estimate of hours. The reason is it is usually very hard to accurately predict how long a software project will take. Sometimes if it is something very cookie-cutter that you have done several times before, you can make a fairly good estimate of how long it will take.