IPhone 3GS runs out of battery life at 70%~?

I recently bought a 3GS, and there seems to be some kind of issue concerning my battery. No matter how much I charge the iPhone it gets to between 60-80% and then shuts down. When I try to turn it back on, it says there's no battery left and is unusable until charged.
Curiously, when put on charge and upon restarting after roughly 20 minutes, the battery shows itself to be, again, around 60-70%. The iPhone does not charge fast enough to reach 60-70% in less than half an hour.
The charge lasts little over an hour and a half of use, and as far as I'm aware this is a normal amount of time for the charge to decrease to around 70%. Therefore it is inexplicable why it 'runs out' of battery while displaying 70% remaining, it's not even like there's a clear loss of battery life displayed by reaching 0%.
This has been a persistent problem for several days, despite the 9 hours or so it has been left on charge during the night in hope of giving it a 'full charge'.

Realistically, I think my only solution is to choke up £35 to get the battery replaced (which I'm not even entirely sure would fix the problem), or to spend £55 and get a reconditioned model from Apple themselves, but I was just curious on the off-chance that anyone else recognises this issue and has a (cheaper) fix for it.

I wouldn't have the battery replaced if I were you. Husband had the exact
same issue and wasted his money that way. Made no difference. The fault
was in the 3GS. He bought the phone off our son so we know it was well
used. The phone needs a repair as a component has failed.