Else if statement not working C++

Look at what you are doing. I'll shorten it a bit so identical bits are more obvious:

if (a || b || c)
   x;
else if ( d || b || c)
   y;

That refactors - just to make it more obvious what is happening - to this:

if (a || b || c)
   x;
else 
   if ( d || b || c)
      y;

And "||" means "OR" - if either of the conditions on the left or right hand side are true, the whole expression is true.
So if either b or c are true, then the whole of the first condition is automatically true. Which means that if either b or c is true, the else condition will always be false, and they never contribute to the condition at all: the code in the else block will only get executed is d is true.

In your case, the only time the else if expression can be executed is if both b and c are false: if startcall is between 1800 and 800 and that remains true for the second test, so the second test will always be just duration < 60

I'd suggest that you start by reading the question again, and applying the discount in separate tests:

if (right time of day)
   {
   apply 50% discount;
   if (call is long enough)
      {
      apply extra discount;
      }
   }

Thanks so much for your time. Program is working properly.
I had some missing curly brackets and some statements were not in a logical order either. for example I was calculating grosscost and deducting a discount value before calculating discount.

if(duration >60 && (startcall >=1800 || startcall < 800)) {
    discount = (duration * rate /2)*0.15;
    grosscost = ((((duration * rate/2)-discount)) * 0.040) + (((duration * rate/2)-discount));
    netcost = grosscost;

When you use the debugger you find out what your code is doing.

Advice: write some else code for the case that the else if isnt true.

else if(duration <60 || startcall >=1800 ||startcall <800){
        grosscost = ((duration *rate /2)*0.40) + (duration *rate/2);
        netcost = grosscost;
    } else {
      //some code 
    }